Stable models of modular curves and applications.

Sebastiaan Johan Edixhoven 05/06/1989

Contents

1 Introduction. 3

2 The stable reduction at p of X_{0}(p^{2}N). 9 2.1 Construction of the stable model. . . . . . . . . . . . . . . . . 9 2.2 The inertia action. . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 A description of the special fibre. . . . . . . . . . . . . . . . . 16

3 The action of the Hecke algebra. 21 3.1 Reduction to the special fibre. . . . . . . . . . . . . . . . . . . 22 3.2 The inertia decomposition. . . . . . . . . . . . . . . . . . . . 24 3.3 Computation of T_{m} on the horizontal part. . . . . . . . . . . . 29 3.4 An algorithm for T_{2} on J_{0}(p^{2}). . . . . . . . . . . . . . . . . . 32 3.5 Some examples. . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Weil curves. 41 4.1 Stable models of elliptic curves. . . . . . . . . . . . . . . . . . 41 4.2 Reduction of Weil parametrizations. . . . . . . . . . . . . . . 45 4.3 Horizontal Weil curves. . . . . . . . . . . . . . . . . . . . . . . 50 4.4 Examples: conductor p^{2}. . . . . . . . . . . . . . . . . . . . . . 54 4.5 Vertical Weil curves. . . . . . . . . . . . . . . . . . . . . . . . 58 4.6 The constant c of a Weil curve. . . . . . . . . . . . . . . . . . 59

Samenvatting. 71

Dankwoord. 73

Curriculum Vitae. 75 2 CONTENTS

Chapter 1

Introduction.

The aim of this thesis is to get information on the action of the Hecke algebra on modular forms, using the geometry of modular curves over suitable finite fields.

To be more precise, let M and k be positive integers and let Γ_{0}(M) be the congruence subgroup of SL_{2}(Z) consisting of matrices

( \begin{matrix}a & b \\ c & d \end{matrix} )

with c ≡ 0(M). Recall that SL_{2}(Z) acts on the complex upper half plane H by fractional linear transformations. We define S_{k}(M) to be the (finite dimensional) complex vector space of cuspforms of weight k and level M. By definition, elements of S_{k}(M) are complex analytic functions on H, satisfying certain vanishing conditions at the rational numbers and at “i∞” (the “cusps” of Γ_{0}(M)) and transforming in a certain way with respect to the action of Γ_{0}(M). These spaces S_{k}(M) arise in the theory of automorphic representations of GL_{2,Q}.

The spaces S_{k}(M) are naturally equipped with a set of operators, called the Hecke operators. For every prime number p there is an operator T_{p} (if p does not divide M) or U_{p} (if p divides M). All the T_{p} and U_{p} together generate a commutative algebra called the Hecke algebra. The Hecke algebra respects the decomposition of S_{k}(M) into its old part (coming from lower levels) and its new part (the orthogonal complement of the old part with respect to the Petersson inner product). The new part splits as an orthogonal sum of (one dimensional) eigenspaces. Each eigenspace then gives a system (a_{p}) of eigenvalues.

These systems (a_{p}) play an important role in the theory of two dimensional representations (complex, ℓ-adic and modular) of the Galois group of \overline{Q} over Q. Given an eigenform, one knows (Kuga-Shimura, Deligne, Deligne-Serre) how to construct such a representation with the property that for almost all p, the Frobenius elements at p in the Galois group have trace a_{p}. In the opposite direction, there are conjectures of Shimura, Taniyama, Weil, Langlands and Serre.

A famous consequence of this construction of Deligne (and of Deligne’s proof of the Weil conjectures) is the proof of the Ramanujan conjecture:

|τ(p)| < 2p^{\frac{11}{2}} , where q ∏_{n>0}(1 − q^{n})^{24} = ∑_{n>0} τ(n)q^{n}.

Another surprising result is Ribet’s proof (using ideas of Frey, Serre and Mazur) that the Taniyama-Weil conjecture implies Fermat’s Lost Theorem.

One would like to have an algorithm (implemented on a computer) for decomposing the spaces S_{k}(M) and computing the numbers a_{p} for small p. Theoretically, this can be done using the Selberg trace formula, but in practice it seems (Mestre) that even in the easiest case (k = 2 and M is prime) there is no hope to treat levels larger than 5,000. There are other methods, using Brandt matrices (Eichler, Pizer) or modular symbols (Manin, Tingley), but the computations in them are quite involved.

In 1985, Mestre and Oesterlé developed an extremely efficient algorithm (for k = 2 and M prime), known as the “graph method’. As Serre writes, the examples verified by Mestre (using the graph method) convinced him to take his conjecture (concerning mod p Galois representations) serious. The graph method has had also an application to the class number problem for imaginary quadratic fields.

One obtains the graph method by reduction mod p, as we will now explain. The quotient H/Γ_{0}(M) of the upper half plane by the action of Γ_{0}(M) is a punctured Riemann surface. It can be compactified by adding a finite set of points (the cusps). The resulting compact Riemann surface is denoted by X_{0}(M)(C), the corresponding complete, smooth, complex algebraic curve is the modular curve X_{0}(M)_{C}. The genus of this curve is (very) roughly M/12. From now on we will only consider weight 2 forms, the reason for this is that S_{2}(M) can be interpreted as the space of differential forms on X_{0}(M)_{C}.

The program to be carried out is as follows: show that X_{0}(M)_{C} and the Hecke algebra are defined over Q, construct a model over Z and find a faithful action of the Hecke algebra on some object in characteristic p. At first sight, it seems very improbable that something like this can be done: one should define X_{0}(M)_{C} by polynomial equations with coefficients in Z and then be able to say something about its reduction mod p. Nevertheless, working in this way, Kronecker derived his congruence formula:

(x − y^{p})(x^{p} − y) = 0,

describing the reduction mod p of X_{0}(p)_{C}. It is clear, however, that this is not the way to proceed. The idea is to interpret the curves X_{0}(M)_{C} as moduli spaces for elliptic curves.

One can set up a bijection between H/Γ_{0}(M) and the set of isomorphism classes of complex elliptic curves with a cyclic subgroup of order M:

H/Γ_{0}(M) ˜ ⟶\{(E, G)\}/ \begin{matrix}∼ \\ = \end{matrix}

z ↦ (C/(Z · 1 + Z · z), ⟨1/M⟩).

The right hand side of this bijection makes sense not only over C but over arbitrary algebraically closed fields, and even over arbitrary schemes. It seems reasonable now that the model over Z of X_{0}(M)_{C} could be provided by a compactified moduli space of elliptic curves with a certain level structure. In the late 50’s, Igusa constructed such a moduli space over Z[1/M]. Ten years later, Deligne extended this moduli space to one over Z[1/m], where m is the largest square dividing M. Another ten years later, using “Drinfeldian” level structures, Katz and Mazur obtained a moduli space X_{0}(M) over Z. The importance of all this work is that the moduli interpretation of X_{0}(M) allows one to calculate rather explicitly the reduction mod p of X_{0}(M). Igusa showed that for p not dividing M, the characteristic p fibre X_{0}(M)_{p} is a smooth curve. For p not dividing N, Deligne gave a description of the reducible curve X_{0}(pN)_{p}. Katz and Mazur described in detail the reducible, non-reduced curves X_{0}(p^{n}N)_{p}.

The action of the Hecke operator T_{l} on S_{2}(M) is induced by a correspondence:

X_{0}(Ml)_{C}
↙  ↘
X_{0}(M)_{C}  X_{0}(M)_{C}

A differential form on X_{0}(M)_{C} is pulled back to X_{0}(Ml)_{C} along one of the two maps, and pushed down along the other. Doing the same with divisors gives an endomorphism (denoted T_{l} )of the jacobian variety J_{0}(M)_{C} of X_{0}(M)_{C}. The correspondence inducing T_{l} extends to the models of Katz and Mazur over Z. Restricting this correspondence to the characteristic p fibres (p a prime) gives an endomorphism T_{l} of the jacobian variety J_{0}(M)_{p}^{0} of X_{0}(M)_{p}.

The action of the Hecke algebra on J_{0}(M)_{p}^{0} is faithful if p^{2} does not divide M, since then J_{0}(M)_{p}^{0} is a semi-stable (commutative) algebraic group (an extension of an abelian variety by a torus). Eichler and Shimura proved their congruence relation “T_{l} = F + V ” (for l not dividing M) by taking p = l. Mestre and Oesterlé obtained their graph method by studying the action of the Hecke algebra on the toric part of J_{0}(pN)_{p}^{0} (p not a divisor of N).

In this thesis we study the next case: the action of the Hecke algebra on J_{0}(p^{2}N), where p does not divide N. The first problem that arises is the “instability” of J_{0}(p^{2}N)_{p}^{0} : the interesting part of it is the unipotent part. This implies that the action of the Hecke algebra on J_{0}(p^{2}N)_{p}^{0} is certainly not faithful. The solution to this problem is not to work over Q, but over a suitably ramified (at p) extension over which X_{0}(p^{2}N) has stable reduction (above p). In the next chapter we compute the stable reduction at p of X_{0}(p^{2}N) together with the action of the inertia subgroup of the Galois group of the extension on it. This computation is quite standard for p > 3, because then the extension of Q that we need is only tamely ramified at p. This explains why we exclude the primes 2 and 3, and also why we cannot deal with levels p^{n}N with n > 2; because of wild ramification we do not know how to compute the stable reduction in these cases. Probably, the results of Chapter 2 will have more applications than those given in this thesis.

In Chapter 3 we study the action of the Hecke algebra on the stable reduction J_{t}^{0} of J_{0}(p^{2}N) at p. First we show that the action of the Hecke algebra on J_{t}^{0} is induced by correspondences mod p. Then we start decomposing J_{t}^{0} into parts that are stable under the Hecke action. The first decomposition is into a multiplicative, a horizontal and a vertical part. The horizontal part and the vertical part correspond to the two kinds of irreducible components of the stable reduction at p of X_{0}(p^{2}N). In terms of representation theory, this decomposition into three parts is the decomposition of admissible representations of GL_{2}(Q_{p}) into the special, the cuspidal and the principal series representations. Each of these three parts is then decomposed with respect to the inertia action. The rest of Chapter 3 consists of the computation of the Hecke action on the horizontal part. The result is a “graph algorithm” very similar to that of Mestre and Oesterlé. The matrices we get are of the same size, but whereas the coefficients of their matrices are just non-negative integers (sums of 1’s) ours are sums of p + 1-th roots of unity. We end the chapter by giving a very explicit algorithm for computing the action of T_{2} on J_{0}(p^{2}), plus some examples.

It would be very interesting to obtain some kind of graph algorithm for the vertical part too. Another interesting problem is to do a similar computation for higher weights. It should also be remarked that Birch has recently produced a graph algorithm, using ternary quadratic forms. His algorithm is very fast, but gives only “half” of each S_{2}(M) (the −1 eigenspace of the Atkin-Lehner involution W_{M}) and cannot treat square levels. It would be interesting to have a geometric interpretation of his algorithm.

In Chapter 4 we study Weil curves. By definition, these are the elliptic curves occurring in a decomposition (up to isogeny, over Q) of some J_{0}(M)_{Q}. According to the Taniyama-Weil conjecture, every elliptic curve over Q should be a Weil curve. Let the elliptic curve E be an isogeny factor of J_{0}(M)_{Q}. Composing the canonical injection of X_{0}(M)_{Q} into J_{0}(M)_{Q} (use the cusp “∞”) with the projection J_{0}(M)_{Q} → E gives a non-constant map X_{0}(M)_{Q} → E. Such a non-constant map is called a Weil parametrization. These parametrizations provide a very powerful tool for the study of the arithmetic of Weil curves (for example in the recent work of Kolyvagin). We study Weil parametrizations by looking at their reductions mod p.

Let E be an elliptic curve inside J_{0}(M)_{Q} that does not occur in any J_{0}(M′)_{Q} for proper divisors M′ of M. The resulting parametrization ϕ :

X_{0}(M)_{Q} → E is then called a strong Weil parametrization and according to work of Carayol, M is the conductor of E. This implies that the primes occurring in M are exactly the primes where E has bad reduction. If E has multiplicative reduction at a prime p, then p divides M exactly once. If E has additive reduction at a prime p ≠ 2, 3, then p divides M exactly twice. This means that our knowledge of the stable reduction at p of X_{0}(p^{2}N) is sufficient to study all Weil parametrizations at primes p ≠ 2, 3.

First we study stable models for elliptic curves. In the second section we study the reduction mod p (p > 3) of a strong Weil parametrization, using stable models. Probably the computation concerning the inseparability degree of this parametrization mod p is the most technical part of this thesis. The rest of the chapter deals with strong Weil curves that have bad, but potentially good reduction at a prime p > 3. It turns out that there are two kinds of these, called horizontal or vertical, according to the kind of irreducible components that parametrize them mod p. We study them seperately. In the fourth section we give some examples of horizontal curves of conductor p^{2} (these are the Weil curves that can be detected by the graph algorithm of Chapter 3).

Finally, in the last section, we derive some results concerning the constant “c” attached to a strong Weil curve E. Manin conjectured that c = 1. It is known that c is a positive integer and Mazur proved that only 2 and primes where E has additive reduction can divide c. Our methods show that primes p > 7 where E has additive reduction divide c at most once, and in fact, for most of the possible reduction types (= Kodaira symbols), not (Theorem 4.6.3 and the remarks following this theorem). It might well be that a computation involving the period lattices of normalized newforms can solve the problem in the case of potentially good, ordinary reduction of type II, III or IV. It should also be tried to get bounds on the exponents of 2, 3, 5 and 7 in c.

The stable reduction at p of X_{0}(p^{2}N).

Let p > 3 be a prime, and let N be a positive integer not divisible by p. The first part of this chapter is a review of the construction, given in [Ed 1], of the stable reduction mod p of X_{0}(p^{2}N). In order to obtain this stable model we have to perform a ramified base change. After adjoining some roots of unity, this base change is Galois. The inertia subgroup of the Galois group then acts on the special fibre of the stable model. The second part of this chapter deals with the computation of this action.

2.1 Construction of the stable model.

In order to explain the construction of the stable model of X_{0}(p^{2}N) at p, it is better to consider the following more general situation. Let S be the spectrum of an excellent discrete valuation ring with perfect residue field. (We demand the ring to be excellent because we want the operations “normalization” and “completion” to commute, cf. [Gro 1] IV 7.8.2 and 7.8.3.1(vii)). Let s be the closed point of S and η be the generic point. Let C ⟶ S be a curve: C ⟶ S is proper, flat, of relative dimension one, C_{η} is smooth over η and geometrically irreducible. In addition we suppose C to be regular.

Under the hypotheses above, the irreducible components of C_{s} are Cartier divisors on C. By repeated blow ups in closed points of C_{s} we can achieve that C_{s} is a Cartier divisor on C with normal crossings. Let n be the least common multiple of the multiplicities of the irreducible components of C_{s}, and let π_{0} be a uniformizer on S. Let T be S[π], with π^{n} = π_{0}, thus we have T ⟶ S totally ramified of degree n. Now we consider C̃_{T}, the normalization of the pullback of C to T. Let t be the closed point of T.

Proposition 2.1.1 If n is invertible on S, then the geometric fibre C̃_{T}_{,t̄} of C̃_{T}/T is a reduced, connected curve whose singularities are ordinary double points.

Proof. By replacing S by the completion of its strict henselization, we reduce to the case that S is complete and k(s) algebraically closed. We will check that C̃_{T}_{,t} is reduced and has only ordinary double points. In order to do this we compute the complete local rings of C̃_{T} at closed points. Let x ∈ C(s), then the normalization of Ô_{C,x} ⊗_{O_{S}} O_{T} is given by:

Ô_{C,x} ˜⊗_{O_{S}}O_{T} = ⨁_{y↦x} Ô_{_{C_{T}},y}

where the summation is over the y ∈ C̃_{T}(t) mapping to x. The next step is to describe Ô_{C,x} explicitly by equations. There are two cases to be distinguished. In the first case only one branch of C_{s} passes through x. Let a be its multiplicity and let x, y ∈ Ô_{C,x} be a set of parameters such that x is an equation for this branch. Then in Ô_{C,x} we have: π_{0} = x^{a}u, with u a unit. Since a is invertible on S we can change x by an a-th root of u in order to

obtain: π_{0} = x^{a}. It follows that in this case: Ô_{C,x} \begin{matrix}∼ \\ = \end{matrix} O_{S}[[x, y]]/(x^{a} − π_{0}). In the second case there are two branches passing (transversally) through x, let the multiplicities be a and b and choose equations x and y. This time we

arrive at: Ô_{C,x} \begin{matrix}∼ \\ = \end{matrix} O_{S}[[x, y]]/(x^{a}y^{b} − π_{0}). The tensor product Ô_{C,x} ⊗_{O_{S}} O_{T} is given by:

O_{T} [[x, y]]/(x^{a} − π^{n}) resp. O_{T} [[x, y]]/(x^{a}y^{b} − π^{n}).

The normalizations of these rings can be easily computed, see for example [Ed 1], 2.2. The result of these computations show that indeed C̃_{T}_{,t} is reduced and has only ordinary double points as singularities.

Remarks.

1. If the genus of C_{η} is at least 2, one gets the stable model (in the sense of [De-Mu]) of C_{η} over T by contracting the projective lines in C̃_{T}_{,t̄} intersecting the rest of C̃_{T}_{,t̄} in less than 3 points. By contracting and blowing up one also obtains the minimal model of C over T (n.b. C̃_{T} itself may not be regular).

2. The computations of the normalizations give the morphism C̃_{T} ⟶ C_{T} on the complete local rings. In particular, one knows the ramification structure of C̃_{T}_{,t} ⟶ C_{s,red}.

3. The choice of the uniformizing element π_{0} on S is unimportant, since all totally ramified extensions of degree n of S are isomorphic over the strict henselization of S.

4. On the contrary, if n is not invertible on S, there are lots of nonisomorphic (wildly ramified) extensions of the same degree. If one knows the acion of the inertia subgroup of Gal(η̄/η) on H^{1}(Cη̄, Q_{ℓ}) one can pick the right extension. In the case of modular curves (of arbitrary level) this action is known. The problem is the description of the rings Ô_{C,x}.

2.1.2 The curve X_{0}(p^{2}N).

We return to the modular curves X_{0}(p^{2}N). To be precise, for any positive integer M we define X_{0}(M) to be the compactified coarse moduli scheme over Z associated to the moduli problem [Γ_{0}(M)], see [Ka-Ma] 8.6. Let S be the spectrum of the Witt vectors of \overline{F_{p}}. In order to apply Proposition 2.1.1 we need a regular model over S, but X_{0}(p^{2}N)_{S} is not necessarily regular: it can have some quotient singularities. There are two ways to get around this problem. The first one is to use the minimal resolution of X_{0}(p^{2}N)_{S} (described in [Ed 1]1). The second one is to introduce a suitable extra level structure (e.g. [Γ(3)^{can}]), to apply Proposition 2.1.1 to the associated moduli scheme, and to obtain the stable model of X_{0}(p^{2}N)_{S} as a quotient. The second method is the one followed in [Ed 1] 2. The details of the computations can be found in loc. cit. The multiplicities of the components in the special fibre of the normal crossings model are 1, p − 1 and p + 1, so n = \frac{p^{2}−1}{2} and T is obtained by extracting the n-th root of p: p = π^{n}.

The next problem is to determine the irreducible components of the stable reduction. This problem is of a more global nature, and considering only complete local rings will not suffice. At this point we must distinguish between the “horizontal” and “vertical” components. The components of the stable reduction originating from components in the normal crossings model having multiplicity p + 1 are called horizontal, the others vertical. The vertical components originating from multiplicity p − 1 components are called central, the others outer. The reason for all this terminology is clear from the pictures in [Ed 1] 2.

2.1.3 Horizontal components.

In this section we redo the computation of the horizontal components. The reason for this is that we want to get better coordinates for these components than those given in [Ed 1] 2.2.4. Let x be a supersingular point in X_{0}(p^{2}N)(s). We interpret the moduli problem [Γ_{0}(p^{2}N)] as classifying pairs (ϕ : E_{1} → E_{2}, G), where ϕ is a cyclic isogeny of degree p^{2} and G a cyclic subgroup scheme of order N of E_{1}. Then x corresponds to a pair ([p] : E → E, G), with E/\overline{F_{p}} supersingular. We choose a coordinate t of the universal formal deformation space of (E, G) in [Γ_{0}(N)]. We consider the map [Γ_{0}(p^{2}N)] → [Γ_{0}(N)] × [Γ_{0}(N)] given by (ϕ : E_{1} → E_{2}, G) ↦ ((E_{1}, G), (E_{2}, ϕG)). According to [Ka-Ma] Thm.13.4.7, this map is a formally closed immersion at x, and if we let x and y be t on the first and second factor of [Γ_{0}(N)] × [Γ_{0}(N)], the image of [Γ_{0}(p^{2}N)] is given by an equation f = f_{0} + pf_{1} in W [[x, y]], with f_{0} = (x − y^{p^{2}})(x − y)^{p−1}(x^{p^{2}} − y) and f_{1} a unit. Here W =O_{S}(S)= the ring of Witt vectors of \overline{F_{p}}. The universal formal deformation space of x is thus given by W [[x, y]]/(f). Clearly, its special fibre does not have normal crossings, so we blow up in x. An affine open part of the result is W [v][[x]]/(f̃_{0} + pf̃_{1}), with f̃_{0} := f_{0}(x, vx), f̃_{1} := f_{1}(x, vx). Now we set W_{1} := W [π_{1}], with π_{1}^{p+1} = p. After extension of scalars to W_{1} we have: W_{1}[v][[x]]/(f̃_{0} + π_{1}^{p+1}f̃_{1}). We blow this up along the ideal (x, π_{1}), i.e. we write x := uπ_{1}. This gives:

Ŵ_{1}[v, u]/(u^{p+1}((uπ_{1})^{p^{2}−1} − v)(1 − v)^{p−1}(1 − v^{p^{2}}(uπ_{1})^{p^{2}−1}) + ˜ f̃_{1})

where the completion is with respect to the principal ideal (uπ_{1}), and ˜ f̃_{1} = f̃_{1}(v, uπ_{1}) = f_{1}(uπ_{1}, vuπ_{1}). The curve over \overline{F_{p}} we are looking for is now given by π_{1} = 0. Substituting π_{1} = 0 in the equation above yields:

u^{p+1}(−v)(1 − v)^{p−1} + f_{1}(0, 0) = 0.

This is an affine, singular model of the (smooth) horizontal component (at x). The next thing to do is to get better coordinates for this curve.

We set a := f_{1}(0, 0) ∈ \overline{F_{p}}^{∗}. We leave it to the reader to check that after the following substitutions: v := 1 − v_{1}, u := u_{1}v_{1}, a := \frac{1}{2}a_{1}^{−1}, v_{1} := v_{2} − a_{1}u_{1}^{p+1}, v_{2} := v_{3}a_{1}u1^{\frac{p+1}{2}} we obtain the equation:

v_{3}^{2} = u_{1}^{p+1} + 4a.  (2.1.3.1)

This shows that the horizontal component is a double cover of P^{1}, ramified in p + 1 points. We will compute the various actions on the horizontal components in these coordinates.

As a first example, let us compute the action of Aut(x), since the horizontal component in the stable model of X_{0}(p^{2}N) is the quotient by this action. First we compute the action of Aut(E) on its deformation space. Since the automorphism [−1] lifts to every deformation, its action on the deformation space is trivial. There are only two elliptic curves over \overline{F_{p}} with automorphisms other than [±1].

1. j(E) = 0, E is given by the equation:y^{2} = x^{3} − 1, we have:

μ_{6}(\overline{F_{p}}) ˜→Aut(E) : ζ ↦ \{ \begin{matrix}x & ↦ & ζ^{−2}x \\ y & ↦ & ζ^{3}y \end{matrix} (note: ζ^{∗}(\frac{dx}{y}) = ζ \frac{dx}{y})

The universal formal deformation is given by: y^{2} = x^{3} + tx − 1, and

ζ ∈ μ_{6}(\overline{F_{p}}) acts by: \{ \begin{matrix}y & ↦ & ζ^{3}y \\ x & ↦ & ζ^{−2}x \\ t & ↦ & ζ^{2}t, \end{matrix} since this preserves the equation.

2. j(E) = 12^{3}, Aut(E) is now isomorphic to μ_{4}(\overline{F_{p}}), and it acts on the universal formal deformation y^{2} = x^{3} − x + t by:

ζ ∈ μ_{4}(\overline{F_{p}}) : \{ \begin{matrix}y & ↦ & ζy \\ x & ↦ & ζ^{2}x \\ t & ↦ & ζ^{2}t. \end{matrix}

Since [Γ_{0}(N)]_{S} is etale over [Ell], the coordinate t of the deformation space of E is a coordinate on [Γ_{0}(N)]_{S} at (E, G). It follows that the action of ζ ∈ Aut(E) on W [[x, y]]/(f) is given by: x ↦ ζ^{2}x, y ↦ ζ^{2}y. Peforming all the following substitutions:

\begin{matrix}v = y/x, & u = x/π_{1}, & v_{1} = 1 − v, & u_{1} = uv_{1}, \\ a_{1} = 1/2a, & v_{2} = v_{1} + a_{1}u_{1}^{p+1}, & v_{3} = v_{2}/a_{1}u1^{\frac{p+1}{2}} \end{matrix} (2.1.3.2)

one finds that the action of ζ on the horizontal component is given by:

\{ \begin{matrix}u_{1} & ↦ & ζ^{2}u_{1} \\ v_{3} & ↦ & v_{3}. \end{matrix}  (2.1.3.3)

Of course, this situation only occurs if such a curve E is supersingular (p ≡ −1(4) for j(E) = 12^{3}, and p ≡ −1(3) for j(E) = 0), and has a G ∈ [Γ_{0}(N)](E) fixed by the automorphism group.

2.1.4 The vertical components.

Since these components come from the multiplicity p − 1 component in X_{0}(p^{2}N)_{s}, it is sufficient to consider the normalization of X_{0}(p^{2}N) after adjoining the p-th roots of unity: W [ζ_{p}] is ramified of degree p − 1 over W. In [Ed 1] 2.3.2 this normalization is obtained as a quotient of

\overline{M}([Γ_{0}(N)], [Γ(p^{2})^{can}]).

It would have been easier to use the isomorphism:

[Γ(p)]/( \begin{matrix}∗ & 0 \\ 0 & ∗ \end{matrix} ) ˜ ⟶[Γ_{0}(p^{2})]

described in [Ka-Ma] 11.3.5. The advantage is that the group one has to divide out has order prime to p, and that the number of components to deal with is smaller. This method works without any problem for arbitrary levels.

2.2 The inertia action.

Let S still denote the spectrum of the Witt vectors of \overline{F_{p}} and let X → S be the normal crossings model of X_{0}(p^{2}N)_{S} (possibly with some extra level structure). The stable model of X was constructed as the normalization X̃_{T} of X_{T}, where T = S[π], π^{n} = p, n = \frac{p^{2}−1}{2}. Since we have the n-th roots of unity on S, the extension T → S is Galois with group μ_{n}(S) = μ_{n}(F_{p^{2}}), where ζ acts by: π ↦ ζπ. We let this group act on X_{T} = X ×_{S} T via its action on T. It follows that μ_{n}(S) = μ_{n}(F_{p^{2}}) also acts on the normalization X̃_{T}, and since the morphism X̃_{T} → T is equivariant, we get a linear action on the special fibre X̃_{T}_{,t} (with “linear” we mean that this action is trivial on the base).

On the horizontal components this action is very easy to compute. Let ζ be in μ_{n}(F_{p^{2}}). In 2.1.3 we had π_{1}^{p+1} = p, so we let π_{1} = π^{\frac{p−1}{2}} . Then ζ acts on π_{1} by: π_{1} ↦ ζ^{\frac{p−1}{2}} π_{1}. In the notation of 2.1.3, ζ acts trivially on the coordinates x and y. Now all one has to do is to go through all the substitutions 2.1.3.2 and see what comes out. The result is:

ζ : \{ \begin{matrix}u_{1} & ↦ & ζ^{− \frac{p−1}{2}} u_{1} \\ v_{3} & ↦ & ζ^{\frac{n}{2}} v_{3} \end{matrix}  (2.2.0.1)

Now we discuss the vertical components. The outer two vertical components have a trivial inertia action, since they were already reduced over S. We want to identify the two central vertical components with

\overline{M}([Γ_{0}(N)], [Ig(p)/ ± 1]).

The problem is that there is no privileged identification: it can always be changed by an element of Aut([Ig(p)/±1]/[Ell]) = μ_{p−1}(F_{p})/±1. We prefer to make the identifications in the following way: for one of the two components we choose an arbitrary one. Now we look for an operator bringing us from this component to the other. If p ≡ −1(4) then the Atkin-Lehner involution W_{p^{2}} does the job (this can be easily seen from the computations with the complete local rings ). If p ≡ 1(4) then we are in trouble, since then W_{p^{2}} stabilizes the two components. In this case we choose a non-square d ∈ F_{p}. Viewed as an element of the inertia group, d interchanges the two components. Using these identifications we get the following inertia action:

p ≡ −1(4) : ζ ∈ μ_{p−1}(F_{p}) : (ζ^{\frac{p+1}{4}} , ζ^{\frac{p−1}{2}} )
W_{p^{2}} :  (1, −1)
p ≡ 1(4) :  ζ ∈ μ_{p−1}(F_{p}) : \{ \begin{matrix}(\sqrt{ζ}, 1) & if ζ is square \\ (\sqrt{ζd^{−1}}, −1) & if ζ is not square \end{matrix}
W_{p^{2}} :  (\sqrt{−1}, 1)

(2.2.0.2)

Here we have written the disjoint union of the two components as:

\overline{M}([Γ_{0}(N)], [Ig(p)/ ± 1]) × μ_{2}(F_{p})  (2.2.0.3)

and the action on the factor μ_{2}(F_{p}) is multiplication.

Remarks.

1. The ambiguity in the square roots in the formulas above is harmless, since the action of −1 on [Ig(p)/ ± 1]) is trivial.

2. From the formulas above we see that in both cases the involution coincides with the element −1 from the inertia group. One can also see this as follows: W_{p^{2}} commutes with the inertia action and acts trivially on the corresponding quotient (the (1,1)-component in X_{0}(p^{2}N)). These two facts force W_{p^{2}} to coincide with some element of the inertia group, but since it is an involution, there is only only one possibility.

2.3 A description of the special fibre.

In this section we suppose that ±1 are the only automorphisms of points in X_{0}(N)(s). If we wouldn’t do this, the number of formulas to write down would be much larger. Instead, we will compute the actions of the automorphism groups of points in X_{0}(M)(s), where M is a divisor of N (in that case we view the complete local rings of X_{0}(N) as deformation spaces of the points in X_{0}(M)(s)). In order to get the formulas in all cases, one then only has to divide out by these actions. The reader is now invited to have a look at the figure on page 18. This figure represents the special fibre of the stable model of X_{0}(p^{2}N)_{S}. Here follows a list of its properties that we have been computing until now.

2.3.1 The curve itself.

The outer two vertical components are the (2,0) and (0,2) components from X_{0}(p^{2}N)_{s}. These curves are both isomorphic to X_{0}(N)_{s}, however, the usual morphism X_{0}(p^{2}N) → X_{0}(N) is an isomorphism on one of them and the composition of two Frobenii on the other. The two central vertical components are both isomorphic (over X_{0}(N)_{s}) to \overline{M}([Γ_{0}(N)], [Ig(p)/ ± 1]). This last curve has degree (p − 1)/2 over X_{0}(N)_{s} and is ramified exactly over the supersingular points. The ramification is total, hence the supersingular points on \overline{M}([Γ_{0}(N)], [Ig(p)/±1]) correspond bijectively to those on X_{0}(N)_{s}. The horizontal components are all isomorphic to the smooth complete hyperelliptic curve given by the equation:

y^{2} = x^{p+1} + 1.  (2.3.1.1)

These coordinates are obtained by scaling the u_{1}, v_{3} of 2.1.3.1. The horizontal components are in bijection with the supersingular points on X_{0}(N)_{s}. These components are contracted by the usual morphisms X̃_{0}(p^{2}N)_{T} → X_{0}(N)_{T}.

The glueing relations are as follows. Let x ∈ X_{0}(N)(s) be a supersingular point. Then the two points at infinity (x, y = ∞) of the corresponding horizontal component must be attached to the points x on the outer components. The two points x = 0, y = ±1 must be attached to the two points on the central vertical components lying over x. Up to automorphisms of the horizontal component this is well defined. Later, when discussing the Hecke action, we will describe the glueing somewhat more canonically.

Figure 2.1: A picture of the stable reduction of X_{0}(p^{2}N) at p.

[figure]

2.3.2 The complete local rings of the stable model.

These 2-dimensional local rings are all regular, except at the points where the horizontal components intersect the outer two vertical components. There the ring has a singularity of type A_{\frac{p−1}{2} −1}, i.e. it is of the type

W [π][[x, y]]/(xy − π^{\frac{p−1}{2}} ).

2.3.3 The inertia action.

The inertia group is μ_{n}(F_{p^{2}}) (n.b. n = \frac{p^{2}−1}{2}). An element ζ acts on the uniformizer π of the base by: π ↦ ζπ. In the sequel we will encounter some other actions, so we need some notation. Let χ : μ_{n}(F_{p^{2}}) → μ_{n}(F_{p^{2}}) be the identity. We can now say that μ_{n}(F_{p^{2}}) acts on π by χ.

The action of an element ζ on the horizontal components (eq. 2.3.1.1) is given by:

\{ \begin{matrix}x & ↦ & ζ^{− \frac{p−1}{2}} x \\ y & ↦ & ζ^{\frac{n}{2}} y. \end{matrix}  (2.3.3.1)

The action of μ_{n}(F_{p^{2}}) on the (co)tangent space to the horizontal component at the points of intersection with the outer components is given by χ^{\frac{p−1}{2}} . Let x be a point of intersection of a horizontal with a central vertical component. The stabilizer group of such a point is μ_{\frac{n}{2}} (F_{p^{2}}). The action of μ_{\frac{n}{2}} (F_{p^{2}}) on the tangent space at x to the horizontal component is by χ^{− \frac{p−1}{2}} (the restriction of χ to μ_{\frac{n}{2}} (F_{p^{2}}) is still denoted by χ).

As in formula 2.2.0.3, we write the disjoint union of the two central vertical components as:

\overline{M}([Γ_{0}(N)], [Ig(p)/ ± 1]) × μ_{2}(F_{p}).

The action of μ_{n}(F_{p^{2}}) on this disjoint union is given by:

\begin{matrix}(χ^{\frac{(p+1)^{2}}{8}} , χ^{\frac{n}{2}} ) & if p ≡ −1(4) \\ ((χ^{\frac{p+1}{2}} d^{\frac{χ \frac{n}{2} −1}{2}} )^{\frac{1}{2}} , χ^{\frac{n}{2}} ) & if p ≡ 1(4), \end{matrix}

where d is some fixed non-square in F_{p}^{∗} (compare 2.2.0.2). The stabilizer group of either of the two components is μ_{\frac{n}{2}} (F_{p^{2}}). Its action is given by χ^{\frac{p+1}{4}} . The action of μ_{\frac{n}{2}} (F_{p^{2}}) on the tangent space at a supersingular point to these vertical components is by χ^{\frac{p+1}{2}} .

2.3.4 The action of W_{p^{2}}.

The operator W_{p^{2}} preserves the horizontal components, in the coordinates 2.3.1.1 its action is given by:

\{ \begin{matrix}x & ↦ & −x \\ y & ↦ & (−1)^{\frac{p−1}{2}} y. \end{matrix}  (2.3.4.1)

One sees that the action of W_{p^{2}} on the horizontal components is not by an element of the inertia group ( −1 in μ_{n}(F_{p^{2}}) acts by (−1)^{\frac{p+1}{2}} on y). This corresponds to the fact that W_{p^{2}} doesn’t act trivially on the horizontal components in the normal crossings model.

The action of W_{p^{2}} on the central vertical components is given by:

\{ \begin{matrix}(1, −1) & if p ≡ −1(4) \\ (\sqrt{−1}, 1) & if p ≡ 1(4). \end{matrix}  (2.3.4.2)

The action of W_{p^{2}} on these vertical components coincides with the action of −1 in μ_{n}(F_{p^{2}}).

2.3.5 The action of Aut(E).

Let x ∈ X_{0}(N)(s) be a supersingular point, corresponding to a pair (E, G). We will now view the complete local ring of X_{0}(N)_{S} at x as the deformation space of E. In this way we get an action of Aut(E) on the formal completion of the stable model of X_{0}(p^{2}N) along the horizontal component corresponding to x. We give this action on the horizontal component itself. Let e be the order of Aut(E). The action of Aut(E) on the tangent space of E gives an isomorphism Aut(E) ˜→μ_{e}(F_{p^{2}}). The action of Aut(E) on the horizontal component is via the following morphism to the inertia group:

\begin{aligned}Aut(E) & ˜→ & μ_{e}(F_{p^{2}}) & → & μ_{n}(F_{p^{2}}) \\ & & ζ & ↦ & ζ^{−2}. \end{aligned}  (2.3.5.1)

The action of the Hecke algebra.

As in Chapter 2, let p > 3 be a prime and N a positive integer not divisible by p. Let X → T be the stable model of X_{0}(p^{2}N) over T as described and constructed in Chapter 2. Let J → T be the Néron model of the jacobian J_{η} of the generic fibre X_{η} of X. In this chapter we study the action of the Hecke algebra on the special fibre J_{t} of J. First we compute J_{t} and the action of T_{m} (m prime to pN) on it in terms of the stable models of X_{0}(p^{2}N) and X_{0}(p^{2}Nm). In the second section we decompose J_{t} with respect to the inertia action. This decomposition is stable under the action of the Hecke algebra. In the third section we compute the action of T_{m} on the horizontal part of J_{t} in terms of the correspondence:

X_{0}(Nm)_{t}
↙  ↘
X_{0}(N)_{t}  X_{0}(N)_{t}

restricted to the first infinitesimal neighborhood of the supersingular locus. We describe the effect of this correspondence on tangent vectors in terms of m-isogenies between supersingular elliptic curves. Then we give an algorithm for computing T_{2} in the case N = 1 (i.e. on J_{0}(p^{2})).

3.1 Reduction to the special fibre.

Let m be prime to pN. Every cyclic isogeny of degree p^{2}Nm between elliptic curves (over an arbitrary base scheme): ϕ_{p^{2}Nm} : E_{1} → E_{2} has unique factorizations (up to isomorphism ):

E_{1} ⟶^{ϕ_{p^{2}N,1}} E_{3} ⟶^{ϕ_{m,2}} E_{2} E_{1} ⟶^{ϕ_{m,1}} E_{4} ⟶^{ϕ_{p^{2}N,2}} E_{2}

where ϕ_{∗,i} has degree ∗ (cf. [Ka-Ma] 6.7). On the level of moduli problems this gives two morphisms:

S : [Γ_{0}(p^{2}Nm)] → [Γ_{0}(p^{2}N)] : S(ϕ_{p^{2}Nm}) = ϕ_{p^{2}N,1}
T : [Γ_{0}(p^{2}Nm)] → [Γ_{0}(p^{2}N)] : T(ϕ_{p^{2}Nm}) = ϕ_{p^{2}N,2},

and an involution:

W_{m} : [Γ_{0}(p^{2}Nm)] → [Γ_{0}(p^{2}Nm)] : W_{m}(ϕ_{p^{2}Nm}) = ϕ_{p^{2}N,1}ϕ_{m,1}^{t}.

By construction we have: T = SW_{m}. We obtain induced morphisms:

S, T : X_{0}(p^{2}Nm) → X_{0}(p^{2}N) W_{m} : X_{0}(p^{2}Nm) → X_{0}(p^{2}Nm)

(the morphisms extend over the cusps because of [De-Ra] IV Prop. 3.16, 3.18,Prop. 3.19, ex. 4.4). Over Q, we get the endomorphism T_{m} = T_{∗}S^{∗} of J_{0}(p^{2}N)_{Q}. Note that for non-squarefree m this T_{m} is not the standard one that has eigenvalue a_{m} when acting on an eigenform ∑ a_{n}q^{n}.

Let X → T, Y → T be the stable models over T of X_{0}(p^{2}N) and X_{0}(p^{2}Nm) respectively. As before, t and η denote the closed and generic point of T. Let J_{η} be Pic_{X_{η}/η}^{0}, and J its Néron model over T. By pullback to η we get T_{m} ∈ End(J_{η}), by the Néron property, T_{m} extends to T_{m} ∈ End_{T} (J). Restricting this T_{m} to the connected component of the special fibre, we arrive at T_{m} ∈ End_{t}(J_{t}^{0}). We want to describe this endomorphism of J_{t}^{0} in terms of X_{t} and Y_{t}.

Since X_{t} is reduced and connected, X → T is cohomologically flat ([Ra 1] 1.4) and satisfies the property (N)^{∗} ([Ra 1] 6.1.4, 6.1.6). By [Ra 1] Thm. 8.2.1 Pic_{X/T}^{0} is represented by a smooth group scheme over T. Its special fibre Pic_{X_{t}/t}^{0} is semiabelian (this results from the fact that the only singularities of X_{t} are ordinary double points, cf. [Sz] I Prop. 5.7). It follows that Pic_{X/T}^{0} is the connected component J^{0} of J (cf. [Sz] I Prop. 5.4).

Proposition 3.1.1 J^{0} = Pic_{X/T}^{0} , and J_{t}^{0} = Pic_{X_{t}/t}^{0}.

By construction, the morphisms S, W_{m} and T over Z induce morphisms S, W_{m} and T over T. We want to show that the endomorphism T_{m} of J_{t}^{0} is given by the correspondence induced by S and T over t. In order to do this we try to extend T_{m} to an endomorphism of the functor Pic_{X/T}^{0} . The pullback construction of line bundles gives S^{∗} : Pic_{X/T}^{0} → Pic_{Y/T}^{0} . If T : Y → X is finite and flat, then the norm construction ([Gro 1] II 6.5) of line bundles gives T_{∗} : Pic_{Y/T}^{0} → Pic_{X/T}^{0} and T_{m} would be given by T_{∗}S^{∗}. Since T = SW_{m} and W_{m} is an isomorphism, T is finite and flat if and only if S is. The finiteness of S is a consequence of its construction. However, S is only flat at points y in Y (t) with Aut(y) = Aut(S(y)) (then the morphism on complete local rings is an isomorphism). We have now proved the following proposition.

Proposition 3.1.2 If Aut(x) = {±1} for all x ∈ X_{0}(N)(t), then the endomorphism T_{m} of J_{t}^{0} is given by the correspondence S, T : Y_{t} → X_{t}. The morphisms S and T in this correspondence are finite and flat.

In the case that there are extra automorphisms we can change X and Y by some blow ups in order to get finite flat morphisms as in [Ed 2] 4.1, but this is a bit messy to write down, and moreover, this would introduce non-reduced special fibres.

Let Z be the stable model over T of \overline{M}([Γ_{0}(p^{2}N)], [Γ(m)]). Then we have a diagram:

\begin{matrix}Z \\ ↓ & ↘ \\ Y & →^{S} & X \end{matrix}

where the diagonal arrow is the quotient by the action of G := GL_{2}(Z/m) on Z, and the vertical arrow the quotient by the upper triangular subgroup B of G. These quotients commute with arbitrary base changes U → T, since the possible orders of stabilizers of points in Z(t) (these are 2,3 and divisors of m) are prime to p. We will now define S_{∗} : Pic_{Y/T} → Pic_{X/T}. Let therefore U → T be a morphism of schemes. According to [Gro 1] IV 21.8.2, we have Pic(Y_{U}) = H^{1}(X_{U}, S_{∗}O_{Y_{U}}^{∗}). We get a “norm” morphism:

S_{∗}O_{Y_{U}}^{∗} → O_{X_{U}}^{∗} : f ↦ ∏_{g∈G/B} g^{#}(f)  (3.1.2.1)

where we view f as a function on (some open part of) Z_{U}. By taking the H^{1} we obtain a morphism Pic(Y_{U}) → Pic(X_{U}), varying functorially with U.

Proposition 3.1.3 The endomorphism T_{m} of J_{t}^{0} is given by the correspondence S, T : Y_{t} → X_{t}, where T_{∗} : Pic_{Y_{t}/t}^{0} → Pic_{X_{t}/t}^{0} is defined as S_{∗}W_{m}_{∗}, S_{∗} being defined by the norm map 3.1.2.1.

3.2 The inertia decomposition.

In this section we decompose J_{t}^{0} with respect to the inertia action. By construction, the Hecke algebra commutes with the inertia action, hence this decomposition of J_{t}^{0} is Hecke stable.

First, we consider the usual decomposition of J_{t}^{0} as an extension of an abelian variety by a torus. Since J_{t}^{0} is the jacobian of the stable curve X_{t}, this decomposition is given as follows. Let X̃_{t} → X_{t} be the normalization map: X̃_{t} is the disjoint union of the irreducible components of X_{t}. It follows that Pic_{_{X_{t}}/t}^{0} = ⨁ Pic_{C/t}^{0}, where C runs through the irreducible components of X_{t}. In particular, Pic_{_{X_{t}}/t}^{0} is an abelian variety. The pullback morphism Pic_{X_{t}/t}^{0} → Pic_{_{X_{t}}/t}^{0} is surjective and its kernel is a torus whose character group is the H_{1} of the dual graph associated to X_{t}. We conclude that this torus is the toric part of J_{t}^{0} and Pic_{_{X_{t}}/t}^{0} its abelian variety part. The Hecke algebra acts on these two parts.

The action of the Hecke algebra on the toric part can be seen from the figure on page 18. The “holes” in this picture give a basis for the character group (to be sure: the number of holes is 3(s − 1), where s is the number of supersingular points in X_{0}(N)(t)). The inertia group interchanges the two central vertical components and stabilizes all the others. It follows that the inertia action is by {±1} on the quotient torus given by the central holes. On the remaining part, the inertia action is trivial. In this way we find that the toric part of J_{t}^{0} consists of three copies of the toric part of J_{0}(pN)_{p}^{0}, one of which is twisted by −1. As in [Me-Oe] the action of the Hecke algebra on these tori is by Brandt matrices, or equivalently, in terms of isogenies of supersingular elliptic curves over \overline{F_{p}}. We will not go further into this.

We have already seen that the abelian variety part of J_{t}^{0} is given by Pic_{_{X_{t}}/t}^{0} = ⨁ Pic_{C/t}^{0}, where C ranges through the irreducible components of X_{t}. The action of T_{m} (m prime to pN) on it is given by the correspondence S, T : Ỹ_{t} → X̃_{t} (cf. Prop. 3.1.1 and Prop. 3.1.2.1, note that these S and T are finite flat, simply because the source and target are smooth curves). Since these correspondences don’t mix horizontal with vertical components, we get a further Hecke stable decomposition:

Pic_{_{X_{t}}/t}^{0} = ⨁_{hor} Pic_{C/t}^{0} ⊕ ⨁_{ver} Pic_{C/t}^{0}.

One can prove (using the description of the vertical part in terms of X(p)) that T_{m} fixes the outer two vertical components, and interchanges the central two components precisely when (\frac{m}{p}) = −1. The inertia action on the central parts is through μ_{p−1}(F_{p}). One can show (by the Lefschetz formula) that, apart from some small values of pN, all characters of μ_{p−1}(F_{p}) occur in the Tate module, or equivalently, in the H^{1}(∗, \overline{Q}_{ℓ}), of the vertical part. The horizontal components are mixed by the T_{m} according to the graph of cyclic m-isogenies between supersingular elliptic curves (as in the graph method by [Me-Oe]), but the question is, how? Before going into this we split up the horizontal part with respect to the inertia action.

Let C be the horizontal component corresponding to a supersingular point without extra automorphisms. The inertia group μ_{n}(F_{p^{2}}) acts on C through μ_{p+1}(F_{p^{2}}). Let ℓ ≠ p be a prime and χ_{ℓ} : μ_{p+1}(F_{p^{2}}) → \overline{Q}_{ℓ}^{∗} a character of order p + 1. Proposition 3.2.1 As μ_{p+1}(F_{p^{2}}) representations we have:

H^{1}(C, \overline{Q}_{ℓ}) = ⨁ ; \begin{matrix}1 ≤ a ≤ p \\ a ≠ \frac{p+1}{2} \end{matrix} χ_{ℓ}^{a}.

Proof. Let χ_{r} be the character of μ_{p+1}(F_{p^{2}}) on H^{r}(C, Q_{ℓ}). According to the Lefschetz formula ([Mi] V Thm. 2.5) we have:

∀ζ ≠ 1 χ_{0}(ζ) − χ_{1}(ζ) + χ_{2}(ζ) = number of fixed points of ζ

From our description 2.3.1, 2.3.3 we see that this number of fixed points is 4 if ζ is a square (in μ_{p+1}(F_{p^{2}})) and 2 if not. We obtain:

χ_{1}(ζ) = \{ \begin{matrix}−2 & if 1 ≠ ζ = square \\ 0 & if ζ ≠ square. \end{matrix}

By the Hurwitz formula the genus of C equals \frac{p−1}{2}, this gives χ_{1}(1) = p − 1. Let a ∈ Z/(p + 1), then the multiplicity of χ_{ℓ}^{a} in χ_{1} is given by:

⟨χ_{1}|χ_{ℓ}^{a}⟩ = \frac{1}{p + 1} ∑_{ζ} χ_{1}(ζ)χ_{ℓ}(ζ)^{−a} = \frac{1}{p + 1}(p − 1 − 2 ∑_{1≠ζ=square} χ_{ℓ}(ζ)^{−a}) =

= \frac{1}{p + 1}(p + 1 − 2 ∑_{ζ=square} χ_{ℓ}(ζ)^{−a}) = \{ \begin{matrix}1 & if 2a ≠ 0 in Z/(p + 1) \\ 0 & if a = 0, \frac{p+1}{2}. \end{matrix}

Let J(C) be the jacobian of C, and let d divide p + 1. Then we define J(C)_{(d)} to be the largest abelian subvariety of J(C) on which the inertia action is of order d. By this we mean that J(C)_{(d)} has no non-zero abelian subvarieties on which the inertia action has order less than d. We set

P_{d}(x) = ∏_{d≠d′|p+1} Φ_{d′}(x) = (x^{p+1} − 1)/Φ_{d}(x),

where Φ_{m}(x) is the cyclotomic polynomial of order m. Then J(C)_{(d)} is the image of the endomorphism P_{d}(ζ^{∗}) of J(C), if ζ generates μ_{p+1}(F_{p^{2}}). By Proposition 3.2.1 we have:

dim J(C)_{(d)} = \{ \begin{matrix}0 & if d = 1, 2 \\ \frac{1}{2}ϕ(d) & otherwise. \end{matrix}

Here ϕ(d) denotes the Euler ϕ-function.

For future use we state the following proposition.

Proposition 3.2.2 As μ_{p+1}(F_{p^{2}}) representations we have:

H_{0}(C, Ω_{C}) = ⨁_{1≤a≤ \frac{p−1}{2}} χ^{a}

where χ is the identity on μ_{p+1}(F_{p^{2}}).

Proof. From equation 2.3.1.1 we see that \{ω_{m} = x^{m} \frac{dx}{y} ; 0 ≤ m < \frac{p−1}{2}\} is a basis of H_{0}(C, Ω_{C}). The equations for the inertia action 2.3.3.1 then give: ζ^{∗}ω_{m} = ζ^{\frac{p−1}{2}} ω_{m}.

3.2.3 The horizontal 3,4 and 6 parts.

Let C be the horizontal component corresponding to a supersingular point without extra automorphisms. We will now give an explicit description of J(C)_{(d)} for d ∈ {3, 4, 6}. Note that for these values of d, J(C)_{(d)} is an elliptic curve if d|p + 1, and J(C)_{(d)} = 0 otherwise.

First we treat the case d ∈ {3, 6}, supposing p ≡ −1(3). The curve C is given by: y^{2} = x^{p+1} + 1 and ζ ∈ μ_{p+1}(F_{p^{2}}) acts by: x ↦ ζ^{−1}x, y ↦ ζ^{\frac{p+1}{2}} y. It follows that the quotient of C by the action of μ_{\frac{p+1}{6}} (F_{p^{2}}) is given by:

C ⟶ C′ : (x, y) ↦ (x^{\frac{p+1}{6}} , y)

where C′ is given by the equation: y^{2} = x_{1}^{6} + 1. The image of the pullback map J(C′) → J(C) is the sum of J(C)_{(3)} and J(C)_{(6)}, since the kernel is finite and the dimensions correspond. We claim that in fact this map is a closed immersion: since the degree of C → C′ is \frac{p+1}{6}, the kernel is contained in the \frac{p+1}{6}-torsion, secondly: since C → C′ is totally ramified over some point in C′ there is no such torsion in the kernel. Let C_{(6)} be the curve: y^{2} = x_{2}^{3} + 1, then we have a morphism:

C′ ⟶ C_{(6)} : (x_{1}, y) ↦ (x_{1}^{2}, y).

One can now easily check that J(C)_{(6)} is the image of the closed immersion J(C_{(6)}) → J(C) corresponding to the composition C → C′ → C_{(6)}. If we let C_{(3)} be the quotient of C by μ_{\frac{p+1}{3}} (F_{p^{2}}) we obtain J(C)_{(3)}. The case d = 4 is similar to the case d = 3. We summarize the situation in the following proposition.

Proposition 3.2.4 Let d ∈ {3, 4, 6} and let p ≡ −1(d). Then C → C′ induces a closed immersion J(C_{(d)}) → J(C) with image J(C)_{(d)}. The various actions on C_{(d)} are:

1. \{ \begin{matrix}d = 4, p ≡ −1(4) & C_{(4)} : & y^{2} = x_{1}^{4} + 1 \\ & C → C_{(4)} : & x_{1} = x^{\frac{p+1}{4}} \\ ζ ∈ μ_{n}(F_{p^{2}}) : & x_{1} ↦ ζ^{\frac{n}{4}} x_{1} & y ↦ ζ^{\frac{n}{2}} y \\ W_{p^{2}} : & x_{1} ↦ (−1)^{\frac{p+1}{4}} x_{1} & y ↦ −y \end{matrix}

2. \{ \begin{matrix}d = 3, p ≡ −1(3) & C_{(3)} : & y_{1}^{2} = x_{2}^{4} + x_{2} \\ & C → C_{(4)} : & y_{1} = x^{\frac{p+1}{6}} y, x_{2} = x^{\frac{p+1}{3}} \\ ζ ∈ μ_{n}(F_{p^{2}}) : & x_{2} ↦ ζ^{− \frac{n}{6}} x_{2} & y_{1} ↦ ζ^{\frac{n}{3}} y_{1} \\ W_{p^{2}} : & x_{2} ↦ x_{2} & y_{1} ↦ −y_{1} \end{matrix}

3. \{ \begin{matrix}d = 6, p ≡ −1(3) & C_{(6)} : & y^{2} = x_{2}^{3} + 1 \\ & C → C_{(6)} : & x_{2} = x^{\frac{p+1}{3}} \\ ζ ∈ μ_{n}(F_{p^{2}}) : & x_{2} ↦ ζ^{− \frac{n}{6}} x_{2} & y ↦ ζ^{\frac{n}{2}} y \\ W_{p^{2}} : & x_{2} ↦ x_{2} & y_{1} ↦ (−1)^{\frac{p−1}{2}} y_{1} \end{matrix}

Let x be a supersingular point in X_{0}(N)(s) having extra automorphisms. Then the horizontal component C_{x} at x is the quotient of C by the action of Aut(x), and J(C_{x})_{(d)} is the jacobian of C_{x(d)} := C_{(d)}/Aut(x). The action of Aut(x) on C_{(d)} can be easily computed from the formulas above and 2.3.5.1. If this action is not trivial then C_{(d)}/Aut(x) is a P^{1} and its jacobian is 0. One finds the following table.

Table 3.2.5 The dimension of J(C_{x})_{(d)}.

#Aut(x) d p  dim J(C_{x})_{(d)}
4  4 p ≡ −1(8)  1
p ≡ 3(8)  0
3, 6 p ≡ −1(4)  1
6  4 p ≡ −1(3)  1
3, 6 p ≡ −1(9)  1
p ≡ 2, 5(9)  0

In the proposition above we have described the action of W_{p^{2}} on C_{(d)}. The action on J(C)_{(d)} is given in the following table.

Table 3.2.6 The action of W_{p^{2}} on C_{(d)}.

d p  W_{p^{2}} on J(C)_{(d)}
4 p ≡ −1(8)  −1
p ≡ 3(8)  1
3 p ≡ −1(3)  −1
6 p ≡ −1(12)  −1
p ≡ 5(12)  1

3.3 Computation of T_{m} on the horizontal part.

The horizontal part of J_{t}^{0} was defined as ⨁_{x} J(C_{x}), where the sum is over s.s.X_{0}(N)(s), (the supersingular points in X_{0}(N)(s)), and C_{x} is the horizontal component of X_{t} at x. From Propositions 3.1.2 and 3.1.3 it follows that the action of T_{m} (m prime to pN) is given by the correspondence:

∐_{y} C_{y}
S ↙  ↘ T
∐_{x} C_{x}  ∐_{x} C_{x}

where the disjoint unions are labeled by the x ∈ s.s.X_{0}(N)(s) and the y ∈ s.s.X_{0}(Nm)(s). We will now compute the contribution to T_{m} of one component C_{y}. Let x = Sy and x′ = Ty. We suppose that Aut(x) = Aut(x′) = Aut(y) = ±1. Let x ∼ (E, G), y ∼ (ϕ : E → E′, G) and x′ ∼ (E′, ϕG), where ϕ : E → E′ is a cyclic m-isogeny between supersingular elliptic curves, and G a cyclic subgroup of order N of E. We choose coordinates t and t′ of the deformation spaces of E and E′ : these spaces are W[[t]] and W[[t′]] respectively. Since [Γ_{0}(N)]_{S} is etale over [Ell]_{S}, t and t′ are coordinates at x and x′. The morphisms S, T: [Γ_{0}(Nm)] → [Γ_{0}(N)] give isomorphisms between the deformation spaces at x, y and x′:

TS^{−1} : [Γ_{0}(N)]_{x} ˜→[Γ_{0}(Nm)]_{y} ˜→[Γ_{0}(N)]_{x′}.

We write (TS^{−1})^{#}t′ = a_{0}t + pa_{p}(t) + t^{2}a_{1}(t), with a_{0} ∈ W^{∗}, a_{p}(t) and a_{1}(t) in W[[t]]. Since x is supersingular there is a unique point (also denoted by x) in [Γ_{0}(p^{2}N)] over x: x ∼ ([p] : E → E, G). The same holds for y and x′: y ∼ (pϕ : E → E′, G). The two morphisms S,T: [Γ_{0}(p^{2}Nm)] → [Γ_{0}(p^{2}N)] give an isomorphism: TS^{−1} : [Γ_{0}(p^{2}N)]_{x} → [Γ_{0}(p^{2}N)]_{x′}. The commutativity of:

\begin{matrix}E & →^{p} & E \\ ϕ ↓ & & ϕ ↓ \\ E′ & →^{p} & E′ \end{matrix}

implies the commutativity of:

[Γ_{0}(p^{2}N)]_{x} → [Γ_{0}(N)]_{x} × [Γ_{0}(N)]_{x}
TS^{−1} ↓  TS^{−1} ↓  TS^{−1} ↓
[Γ_{0}(p^{2}N)]_{x′} → [Γ_{0}(N)]_{x′} × [Γ_{0}(N)]_{x′}

where the horizontal arrows are the closed immersions composed of the source and target morphisms relative to the degree p^{2} isogeny. Written out in coordinates as in Section 2.1.3 this means:

TS^{−1}^{#} : \{ \begin{matrix}x′ ↦ a_{0}x + pa_{p}(x) + x^{2}a_{1}(x) \\ y′ ↦ a_{0}y + pa_{p}(y) + y^{2}a_{1}(y). \end{matrix}

Going through the substitutions 2.1.3.2 and reducing mod π_{1} gives the map on horizontal components:

v′_{3} ↦ a0^{\frac{p+1}{2}} v_{3}  u′_{1} ↦ a_{0}u_{1}  (3.3.0.1)

where C_{x} and C_{x′} are given by the equations:

v_{3}^{2} = u_{1}^{p+1} + 4a (v′_{3})^{2} = (u′_{1})^{p+1} + 4a′.

We also get he information: a′ = a_{0}^{p+1}a. Specializing to the case that ϕ is an endomorphism ϕ : E → E with ϕG = G, we get the following proposition.

Proposition 3.3.1 Let x ∈ X_{0}(N)(s) be a supersingular point with Aut(x) = ±1 and let ϕ ∈ End(x) be an endomorphism of degree m prime to pN. Let a_{0} ∈ \overline{F_{p}} be the automorphism induced by ϕ (as above) on the tangent space to X_{0}(N)_{s} at x. Then a_{0}^{p+1} = 1 and the contribution of ϕ to T_{m} is the automorphism:

\{ \begin{matrix}x ↦ a_{0}x \\ y ↦ a0^{\frac{p+1}{2}} y \end{matrix}

of C_{x}.

Remark. Note that the automorphism induced by ϕ on C_{x} is in the inertia group (compare 2.3.3.1).

We will now compute the “a_{0}” in the proposition above in terms of ϕ itself. We do this computation in the context of abelian varieties because that’s just as easy.

Let A be an abelian variety over a field k, then the deformations of A over k[ε] (ε^{2} = 0) correspond to H^{1}(A, T_{A}), where T_{A} is the tangent sheaf Der_{k}(O_{A}) on A. This works as follows ([Gro 2] exp. III, or [Oo 2] Proposition 2.2.5). Given t ∈ H^{1}(A, T_{A}), represent it by a 1-cocycle D_{ij} ∈ T_{A}U_{ij}, where U_{i} is an affine open cover of A. Let Ũ_{i} := U_{i} ×_{k} k[ε], the constant deformation of U_{i}. Define “glueing” isomorphisms:

ψ_{ij} : Ũ_{i} ⊃ Ũ_{ij} ˜→Ũ_{ij} ⊂ Ũ_{j}  (3.3.1.1)

by:

ψ_{ij}^{#}(f_{0} + εf_{1}) = f_{0} + ε(f_{1} + D_{ij}f_{0}).

Glueing the Ũ_{i} by the ψ_{ij} then gives the deformation Ã corresponding to t. Since A is a group, we have T_{A} = T_{A,0} ⊗_{k} O_{A}, whence:

H^{1}(A, T_{A}) = T_{A,0} ⊗_{k} H^{1}(A, O_{A}) = T_{A,0} ⊗_{k} T_{A^{t},0},

where A^{t} is the dual abelian variety Pic_{A/k}^{0} of A.

Let π : A_{1} → A_{2} be an isogeny of abelian varieties over k, such that deg π is invertible in k. Deformations of A_{1} correspond bijectively to deformations of A_{2} by demanding that π can be extended (note that ker π and ker π^{t} are etale over k).

Proposition 3.3.2 The isomorphism: T_{A_{1},0} ⊗ T_{A_{1}^{t},0} ˜→T_{A_{2},0} ⊗ T_{A_{2}^{t},0} induced by π as above is given by T_{π,0} ⊗ T_{π^{t},0}^{−1}.

Proof. We start with a deformation over k[ε] of A_{2}: let A_{2} = ∪U_{i}^{2} and let D_{ij}^{2} ∈ Der_{k}(O_{A_{2}}U_{ij}^{2}) be a 1-cocycle. We put U_{i}^{1} := π^{−1}U_{i}^{2}, then we have A_{1} = ∪U_{i}^{1}. We get π^{#}D_{ij}^{2} in Der_{k}(O_{A_{2}}U_{ij}^{2}, O_{A_{1}}U_{ij}^{1}). Since π is etale, this can be extended in exactly one way to a derivation D_{ij}^{1} in Der_{k}(O_{A_{1}}U_{ij}^{1}). These D_{ij}^{1} give a deformation of A_{1}. We claim that π extends to an isogeny π̃, and that π̃_{i} := π̃|Ũ_{i}^{1} is constant:

π̃_{i}^{#}(f_{0} + εf_{1}) = π^{#}f_{0} + επ^{#}f_{1}.

The only thing to be checked is that these π̃_{i} are compatible with the glueing isomorphisms:

ψ_{ij}^{2}π̃_{i} = π̃_{j}ψ_{ij}^{1},

where the ψ_{ij}^{•} are defined as in 3.3.1.1. This is a trivial verification. The map D_{ij}^{2} ↦ D_{ij}^{1} is given by a morphism of sheaves:

π_{∗}Der_{k}(O_{A_{1}}) ˜→Der_{k}(O_{A_{2}}, π_{∗}O_{A_{1}}) ← Der_{k}(O_{A_{2}}).

Taking H^{1}(A_{2}, −) and making the right identifications then proves the proposition.

Corollary 3.3.3 The number a_{0} of Proposition 3.3.2 is given by:

a_{0} = T_{ϕ,0} ⊗ T_{ϕ^{t},0}^{−1} = T_{ϕ,0}^{2}/ deg(ϕ).

3.4 An algorithm for T_{2} on J_{0}(p^{2}).

We will now present an algorithm for the computation of T_{2} on the horizontal 3,4 and 6-parts of J_{0}(p^{2}). This algorithm is very similar to the graph algorithm of [Me-Oe]. It uses explicit equations for the correspondence on the j-line given by X_{0}(2). In order to describe and compute the isomorphisms between the horizontal components (the numbers “a_{0}” from proposition 3.3.2 ) we must equip the supersingular j-values with tangent vectors. This means that we will work with F_{p^{2}}[ε] valued points of Spec(F_{p}[j]), where ε^{2} = 0, (usually). The j-invariants 0 and 12^{3} cause some problems, because the correspondence given by X_{0}(2) is ramified over these points. We start with some formulas.

The correspondence on X_{0}(1) induced by X_{0}(2) is given by:

\{ \begin{matrix}xy = 2^{12} \\ j = \frac{(x+16)^{3}}{x} \\ W_{2} : x ⟷ y \end{matrix}  (3.4.0.1)

These formulas are copied from [Me]. Next we give the number of automorphisms of points j on the j-line and points x on the x-line X_{0}(2).

e(j) := \frac{1}{2}#Aut(j) = \{ \begin{matrix}3 & if j = 0 \\ 2 & if j = 12^{3} \\ 1 & otherwise \end{matrix}  (3.4.0.2)

e(x) := \frac{1}{2}#Aut(x) = \{ \begin{matrix}2 & if x = −64 \\ 1 & otherwise \end{matrix}

Let d ∈ {3, 4, 6}. The horizontal d-part is 0 if p ≢ −1(d), so we suppose p ≡ −1(d).

Step 1. Find a supersingular j-value j_{0} in F_{p^{2}}. This is very easy: since p ≡ −1(d) we can take j_{0} = 0 if d ∈ {3, 6} and j_{0} = 12^{3} if d = 4.

Step 2. Build the graph of supersingular 2-isogenies. The vertices of this directed graph are the supersingular j-values (these lie in F_{p^{2}} ), the edges are the supersingular x-values (these lie in F_{p^{2}} too). The edge x has source j = (x + 16)^{3}/x and target j′ = (y + 16)^{3}/y, where xy = 2^{12} (note that an edge can be a loop). The connectedness of this graph ([Me-Oe]) ensures that it can be built starting with j_{0}. In order to find the edges with source j one has to solve the equation: (x + 16)^{3} − jx = 0. Since one of the three roots is already known (after the first step) one only has to solve a quadratic equation, which is very easy.

In the next step we will compute a suitable system of “tangent vectors” at the supersingular points on the j-line. Such a tangent vector at the point j will be an F_{p^{2}}[ε] valued point of the j-line of the form: j + ε^{e(j)}v^{e(j)}, where ε^{e(j)+1} = 0 and v^{e(j)} ∈ F_{p^{2}} (for e(j), see 3.4.0.2). In practice this means that we put a number v^{e(j)} ∈ F_{p^{2}} at every vertex j of the graph. We do not want nor need to compute the numbers v themselves, writing v^{e(j)} is just a matter of notation.

Step 3. Choose a vertex j such that j ∈ F_{p}, j ≠ 0, 12^{3} (if p > 11 there is always such a j: a neighbor of j_{0}). Let the number v = v^{e(j)} at this vertex be 1 (this is an arbitrary choice). Now compute one tangent vector at every vertex by transporting them along the edges as follows. Suppose we have j + ε^{e(j)}v^{e(j)} at j = (x + 16)^{3}/x. Solve w^{e(x)} in the equation:

j + ε^{e(j)}v^{e(j)} = \frac{(x + ε^{e(x)}w^{e(x)} + 16)^{3}}{x + ε^{e(x)}w^{e(x)}} ( (mod ε^{e(j)+1})).

The right hand side of this equation is j + ε^{e(j)}w^{e(j)}a for a certain non-zero a ∈ F_{p^{2}}, hence we must have: w^{e(x)} = v^{e(x)}a^{\frac{e(x)}{e(j)}} . Since j = 0, 12^{3} are always endpoints of the graph, we may suppose that e(j) = e(x) = 1. We have now a tangent vector x + ε^{e(x)}w^{e(x)} at x. Solve xy = 2^{12}. Solve:

(x + ε^{e(x)}w^{e(x)})(y + ε^{e(y)}t^{e(y)}) = 2^{12}.

Noting that e(y) = e(x) one finds: t^{e(y)} = −\frac{y}{x}w^{e(x)}. Compute j′ and u^{e(j′)} in:

((y + ε^{e(y)}t^{e(y)} + 16)^{3}/(y + ε^{e(y)}t^{e(y)}) = j′ + ε^{e(j′)}u^{e(j′)}.

Note that now e(j′) can be 1,2 or 3. Let j′ + ε^{e(j′)}v^{′e(j′)} denote the tangent vector at j′ that we want in our “suitable system”. If j′ ∉ F_{p} then we take (v′)^{e(j′)} := u^{e(j′)} at j′, and \overline{u^{e(j′)}} at \overline{j′}, where the bar denotes Galois conjugation over F_{p}. Suppose j′ ∈ F_{p}. If u^{e(j′)} ∈ F_{p} then we take (v′)^{e(j′)} := u^{e(j′)}. If u^{e(j′)} ∉ F_{p}, we take (v′)^{e(j′)} := (u^{e(j′)})^{\frac{p+1}{2}} .

We will now explain the use of this system of tangent vectors. Remember that in the computation of the horizontal components (2.1.3) we had to choose a coordinate t. Taking only t that coincide up to first order with our tangent vectors then gives the same equation v_{3}^{2} = u_{1}^{p+1} + 4a for every horizontal component. Note that a ∈ F_{p}^{∗}, and that F_{p}^{∗} acts on our system of tangent vectors. It follows that a can be changed by a square. It is thus interesting to know whether a is a square or not. One can show that a is a square.

Anyhow, we are now able to compare every horizontal component to a fixed one. Let ϕ : E → E′ be an isogeny sending v at j(E) to a_{0}v′ at j(E′). Then formula 3.3.0.1 tells us that ϕ acts on our fixed curve as the element a_{0}^{−1} in the inertia group μ_{p+1}(F_{p^{2}}). For the moment, we did forget the points with extra automorphisms. These will cause some trouble in the next steps.

Step 4. If d ∈ {3, 6} and p ≢ −1(9) then delete the vertex 0 and the edges having this vertex as an endpoint. If d = 4 and p ≢ −1(8) then delete the vertex 12^{3} and the edges having 12^{3} as an endpoint. This step is the deletion of the horizontal components having no d-part (cf. table 3.2.5).

The d-part of J_{0}(p^{2}) is now given by: ⊕_{vertices}E, where E is an elliptic curve over F_{p} having μ_{d} ⊂ Aut_{E/F_{p}} (one can compute which one of the two F_{p}-forms it is). Note that E is supersingular. We will write E_{j} for the summand E at the vertex j. Then we have:

End_{F_{p^{2}}} (⊕E_{j}) = ⨁_{j,j′} hom(E_{j′}, E_{j}).

This means that we write endomorhisms as matrices, where the rows are labeled by j and the columns by j′. The next step computes the matrix of T_{2}. The coefficients are sums of d-th roots of unity in End_{F_{p^{2}}} (E). We view these roots of unity as elements of μ_{d}(F_{p^{2}}), the identification is the action of End_{F_{p^{2}}} (E) on the tangent space at 0 of E.

Step 5. Let j be a vertex. Let x_{1}, x_{2}, x_{3} be the roots of (x + 16)^{3}/x = j. The row of T_{2} labeled by j is the sum of the contributions from the x_{i}. Let x be one of the x_{i}. Consider the equation:

j + ε^{e(j)}v^{e(j)} = \frac{(x + ε^{e(x)}w^{e(x)} + 16)^{3}}{x + ε^{e(x)}w^{e(x)}} ( (mod ε^{e(j)+1})).

As in step 3, the RHS can be written as j + ε^{e(j)}w^{e(j)}a for some a ∈ F_{p^{2}}^{∗}, giving: w^{e(j)} = v^{e(j)}a^{−1}. We do not need to know w^{e(x)} since in the final formula for the coefficient of the matrix only powers of w^{e(j)} will occur. However, this would force us to do the rest of the computation formally, treating w^{e(x)} as a symbol. Let us therefore take w^{e(x)} to be one of the (e(x)/e(j))-th roots of w^{e(j)}. Let y := 2^{12}x^{−1}. Let t^{e(y)} := −\frac{y}{x}w^{e(x)}. Compute j′ and u^{e(j′)} in:

((y + ε^{e(y)}t^{e(y)} + 16)^{3}/(y + ε^{e(y)}t^{e(y)}) = j′ + ε^{e(j′)}u^{e(j′)}.

Remember that we have a “standard” tangent vector j′ + ε^{e(j′)}(v′)^{e(j′)}. The contribution of x to a_{jj′} is:

(\frac{v′}{u})^{\frac{p+1}{d}} = (\frac{(v′)^{e(j′)}}{u^{e(j′)}} )^{\frac{p+1}{de(j′)}}  (3.4.0.3)

Remark. One can compute the matrices of T_{l} for l = 3, 5, 7 and 13 by repeating step 5, with the formulas for X_{0}(2) replaced by those for X_{0}(l) (see [Me]).

3.5 Some examples.

3.5.1 p = 7.

X_{0}(7^{2}) is the elliptic curve 49A of [Bi-Ku] table 1. According to table 3 of [Bi-Ku] the operator T_{2} has eigenvalue 1 on J_{0}(7^{2}). We will now check this, using the algorithm of the preceding section. The graph of 2-isogenies has only one vertex: j = −1, with e(−1) = 2. There are two edges: x = −1, 1, with e(−1) = 2 and e(1) = 1. In a picture, this graph looks as follows:

[figure]

Note that this is a picture of an undirected graph. When drawing pictures like this, we will always identify the edges x and y with xy = 2^{12}. In other words, an edge corresponds to an isogeny together with its transpose. At j = −1 we choose the tangent vector −1 + ε^{2}. Now we consider the 4-part of J_{0}(p^{2}), this is J_{0}(p^{2}) itself. The equation −1 = j = (x + 16)^{3}/x has the three solutions x = −1, 1, 1. At x = −1 we find the tangent vector −1 − 2ε^{2}. Solving xy = 2^{12} gives the tangent vector −1 + 2ε^{2} at −1. At j′ = −1 we find −1 − ε^{2}. This implies that the contribution of the loop x = −1 equals (−1)^{\frac{7+1}{4·2}} = −1. At x = 1 we get 1 + 2ε (in the computation one has to solve w^{2} = 4, so we could also take 1 − 2ε). At y = 1 we find 1 − 2ε. We arrive at j′ = −1 with −1 + ε^{2}. The contribution of x = 1 is thus 1. The total contribution is −1 + 1 + 1 = 1, giving the right eigenvalue for T_{2}.

Now we do the same computation directly in terms of the moduli interpretation. The vertex corresponds to the elliptic curve E with automorphism group μ_{4}(F_{p^{2}}). Let i be a generator of this group. The vertex x = −1 corresponds to the endomorphism 1 + i. By corollary 3.3.3, 1 + i acts by \frac{1+i}{1−i} = i on the deformation space of E. It follows that the action on the 4-part is by i^{\frac{7+1}{4}} = −1. The edge x = 1 corresponds to the endomorphism \frac{1+F}{2} (F is Frobenius). Since F acts by 0 on the tangent space of E, we get a trivial action of \frac{1+F}{2} on the deformation space of E. On the 4-part we find 1.

3.5.2 p = 11.

The special fibre of the stable model of X_{0}(11^{2}) looks as follows:

1 2 0  0  0  0

The numbers denote the genera of the components. The genus 1 component corresponds to the supersingular point j = 0. The other supersingular jvalue is 12^{3} = 1. The graph of 2-isogenies looks as follows:

[figure]

The graph of the 4-part is:

Since there are no edges, we get T_{2}=0. From the tables of [Bi-Ku] we see that the 4-part is the elliptic curve 121D. The graph of the 3 and 6-parts is:

[figure]

The loop again represents the endomorphism 1 + i. For the action on the 3 and 6-parts one finds: i^{\frac{11+1}{3}} = 1 and i^{\frac{11+1}{6}} = −1. The 3 and 6-parts must be the curves 121F and 121H.

3.5.3 p = 43.

We get the graph of 2-isogenies:

[figure]

The second graph is the graph for the 4-part. The matrix for T_{2} on the 4-part is:

( \begin{matrix}0 & 1 & 1 \\ 1 & 0 & 2i \\ 1 & −2i & 0 \end{matrix} ) .

The characteristic polynomial of this matrix is: λ(λ^{2} − 6). The eigenvalue λ = 0 corresponds to the complex multiplication curve of conductor 43^{2} (cf. [Gr]).

40 3. HECKE ACTION

Weil curves.

Let E be an elliptic curve over Q and let M be its conductor. Conjecturally (Taniyama-Weil), E is an isogeny factor of the new part of the jacobian variety J_{0}(M)_{Q} of X_{0}(M)_{Q}. Elliptic curves satisfying this conjecture are called Weil curves. First we will study the stable models of elliptic curves (outside characteristics 2 and 3). In the rest of this chapter we use our knowledge of stable models to study Weil curves. We give some examples of Weil curves of conductor p^{2}. In the last section we derive some results concerning the Manin constant c attached to a Weil curve.

4.1 Stable models of elliptic curves.

In this section we apply the method for constructing stable models, as described in Section 2.1, to elliptic curves. We compute the inertia action on the stable models and on their differential forms.

Let S be the spectrum of a complete discrete valuation ring with algebraically closed residue field. In order to satisfy the condition “n is invertible on S” of Proposition 2.1.1 we suppose that 6 is invertible on S. Let s be the closed point of S and η the generic point. Let E be an elliptic curve over η, and E its minimal model over S (as a surface, not as an abelian variety). The special fibre of E is of one of the following types: I_{0}, I_{ν}, II, III, IV, I_{0}^{∗}, I_{ν}^{∗}, IV^{∗}, III^{∗}, II^{∗}. These types are described in [Ta]. Let E_{S} denote the model of E obtained by blowing up E until its special fibre has normal crossings. In all cases except II, III and IV, we do not need any blow ups and E_{S} is just E itself. In the remaining cases, we need 3, 2 and 1 blow up(s) respectively. A picture of this blowing up process can be found in Figure 4.1.

Let π_{0} be a uniformizer on S. We will now describe which base change T → S of the form: T = S[π], π^{n} = π_{0} we need in order to get a stable model of E (over T). If E is of type I_{0} or I_{ν} (ν > 0) then E is already stable over S.

Suppose that E is of type I_{ν}^{∗} (ν > 0). Then the l.c.m. of the multiplicities of the components of E_{s} is 2, so we must take n = 2. From the picture in [Ta] we see that E has reduction type I_{2ν} over T.

In the remaining cases E has bad but potentially good reduction. There is a unique component in E_{s} (:= the special fibre of E_{S}) that intersects more than two other components. This is the component that gives rise to the elliptic curve in the stable model. Let n be its multiplicity. Then E acquires good reduction over T.

Next we compute the inertia action on the stable reduction. The base change T → S is Galois, with group μ_{n}(S): ζ ∈ μ_{n}(S) acts by ζ^{#}(π) = ζπ. Let Ẽ_{T} be the Néron model of E over T. Then μ_{n}(S) acts on Ẽ_{T}/T. Let t be the closed point of T. We get a linear action of μ_{n}(S) on Ẽ_{T} _{,t}^{0}/t. Since n is chosen minimal, this action is faithful. The group Aut_{t}(Ẽ_{T} _{,t}^{0}) acts faithfully on the tangent space at 0 of Ẽ_{T} _{,t}^{0}, or equivalently, on the translation invariant differential forms. The inertia action of μ_{n}(S) on Ẽ_{T} _{,t}^{0} is thus described by the character χ_{ω} : μ_{n}(S) → μ_{n}(S), giving the action of μ_{n}(S) on the translation invariant differential forms on Ẽ_{T} _{,t}^{0}. The computation of these characters is very easy and is therefore omitted (in fact, they can be read off from the pictures of the normal crossings model directly). The characters χ_{ω} can be found in the next proposition.

Proposition 4.1.1 Let S be the spectrum of a complete discrete valuation ring, with algebraically closed residue field not of characteristic 2 or 3. Let E be an elliptic curve over the generic point of S. We attach a number n and a character χ_{ω} : μ_{n}(S) → μ_{n}(S) to E according to its reduction type as Figure 4.1: Construction of the normal crossings model. follows (χ : μ_{n}(S) → μ_{n}(S) denotes the identity):

[figure: II:; 1; ⇒; 2; 1; ⇒; 1; 2; 3; ⇒; 6; 1; 2; 3; III:; 1; 1; ⇒; 2; 1; 1; ⇒; 4; 1; 2; 1; IV:; 1; 1; 1; 3; 1; 1; 1]

\begin{matrix}type & I_{0} & I_{ν} & II & III & IV & I_{0}^{∗} & I_{ν}^{∗} & IV^{∗} & III^{∗} & II^{∗} \\ n & 1 & 1 & 6 & 4 & 3 & 2 & 2 & 3 & 4 & 6 \\ χ_{ω} & χ & χ & χ & χ & χ & χ & χ & χ^{−1} & χ^{−1} & χ^{−1} \end{matrix}

Let π_{0} be a uniformizer on S. Then E acquires stable reduction over T := S[π], where π^{n} = π_{0}. The Galois group of T over S is identified with μ_{n}(S) by: ζ^{#}π = ζπ for all ζ ∈ μ_{n}(S). Let t be the closed point of T and Ẽ_{T} the Néron model of E_{T}. The the action of μ_{n}(S) on the tangent space at 0 of Ẽ_{T}_{,t} is given by χ_{ω}.

Let S and E be as before. Let E denote the Néron model of E over S. Let T → S be the base change as given in Proposition 4.1.1, and let Ẽ_{T} be the Néron model of E_{T}. Since Ẽ_{T} has the Néron property, we get a (unique) morphism f : E_{T} → Ẽ_{T} extending the identity on the generic fibres. Note that the kernel of f contains E_{s}^{0}, if E has additive reduction. On the cotangent spaces along the zero sections f induces an injection

f^{∗} : ω_{_{E_{T}}/T} ⟶ ω_{E_{T} /T} = ω_{E_{S}/S} ⊗_{O_{S}} O_{T} .

Let ω_{S} be a generator of ω_{E_{S}/S}: ω_{E_{S}/S} = O_{S} · ω_{S}.

Proposition 4.1.2 Let ω_{E_{S}/S} = O_{S} · ω_{S} and let a be as follows:

\begin{matrix}type & I_{0} & I_{ν} & II & III & IV & I_{0}^{∗} & I_{ν}^{∗} & IV^{∗} & III^{∗} & II^{∗} \\ a & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 2 & 3 & 5 \end{matrix}

Let ω_{T} := π^{a}ω_{S}. Then ω_{_{E_{T}}/T} = O_{T} · ω_{T}.

Proof. This is done by explicit computation using minimal Weierstrass equations (see [Ta]). We only give the computation for reduction type II^{∗}, the computations for the other cases are almost the same.

The minimal Weierstrass equation for E (over S) is:

y^{2} = x^{3} + ax + b

where a = \frac{−c_{4}}{48} and b = \frac{−c_{6}}{864} , hence v_{π_{0}}(a) = 4 and v_{π_{0}}(b) = 5. We take ω_{S} := \frac{dx}{2y}. We have π_{0} = π^{6}. We rewrite the Weierstrass equation:

y^{2} = x^{3} + ax + b = x^{3} + π_{0}^{4}a_{4}x + π_{0}^{5}b_{5} = x^{3} + π^{24}a_{4}x + π^{30}b_{5}.

We set x := π^{10}u, y := π^{15}v and find the new equation:

v^{2} = u^{3} + π^{4}a_{4}u + b_{5}.

The generating differential form for this model is \frac{du}{2v} = π^{5} \frac{dx}{2y}.

4.2 Reduction of Weil parametrizations.

Let p > 3 be a prime and N an integer not divisible by p. Let E be an elliptic curve over Q with a strong Weil parametrization ϕ : X_{0}(p^{2}N)_{Q} → E ([Ma 1], [Ma 2], [Ma-SD], [Bi-SD], [Ca]). In this section we study the reduction of ϕ mod p using stable models. We will be mainly interested in curves E having potentially good reduction at p. This means that most of the time we will exclude the case where E has reduction type I_{ν}^{∗} (ν > 0) at p (from [Ca] we know that the conductor of E is p^{2}N, hence the types I_{0} and I_{ν} are already excluded)^{1}.

Let S := Spec(W (\overline{F_{p}})), where W (\overline{F_{p}}) is the ring of Witt vectors of \overline{F_{p}}. Let n := \frac{p^{2}−1}{2} and U := S[π] with π^{n} = p. Both E and X_{0}(p^{2}N) acquire stable reduction over U (cf. Proposition 4.1.1 and Section 2.1.2). Let Ẽ_{U} be the Néron model of E over U and let X̃_{U} be the stable model of X := X_{0}(p^{2}N)_{S}. The Weil parametrization ϕ : X_{0}(p^{2}N)_{Q} → E, when pulled back to U, gives a morphism (still denoted ϕ) between the generic fibres of X̃_{U} and Ẽ_{U}. Let u be the closed point of U.

Proposition 4.2.1 Suppose that E has potentially good reduction at p. Then the morphism ϕ extends to a morphism ϕ : X̃_{U} → Ẽ_{U}.

Proof. Let X̃_{U}^{0} be the open subscheme of X̃_{U} where the morphism X̃_{U} → U is smooth (its complement is the set of double points in X̃_{U}_{,u}). By the Néron property of Ẽ_{U} we get ϕ : X̃_{U}^{0} → Ẽ_{U}. It remains to extend this ϕ over the double points. It follows from [Li] Thm.26.1 or Thm.4.1. that there is a composition f : X′ → X̃_{U} of blow ups in closed points of X̃_{U} such that ϕ can be extended to a morphism ϕ′ : X′ → Ẽ_{U} (the normalizations in [Li] Thm. 26.1 do nothing in this case). Since Ẽ_{U}_{,u} is an elliptic curve, ϕ′ contracts the exceptional divisor of X′. It follows that ϕ′ factorizes through f, hence that ϕ extends over the double points.

1It is not difficult to prove that I_{0} and I_{ν} are excluded using the description by [Ka-Ma] of X_{0}(p^{2}N)

We still suppose that E has potentially good reduction. Then we have two μ_{n}(S)-equivariant diagrams:

\begin{aligned}X̃_{U} & ⟶^{ϕ} & Ẽ_{U} \\ ↘ & & ↙ \\ & U \end{aligned}  \begin{aligned}X̃_{U}_{,u} & ⟶^{ϕ} & Ẽ_{U}_{,u} \\ ↘ & & ↙ \\ & u \end{aligned}

It may be useful to note that by dividing out by part of the action of μ_{n}(S), one gets other versions of Proposition 4.2.1. Comparing the inertia actions on X̃_{U}_{,u} (Section 2.3.3, Proposition 3.2.1, Proposition 3.2.2 and Proposition 3.2.4) and on Ẽ_{U}_{,u} (Proposition 4.1.1) gives lots of information. For the moment we only ask ourselves on which components ϕ has to be constant.

Proposition 4.2.2 Suppose that E has potentially good reduction at p. Let ϕ : X̃_{U}_{,u} → Ẽ_{U}_{,u} be the Weil parametrization mod p (Proposition 4.2.1). Then ϕ is constant on the outer two vertical components. On the other components we have:

type

p hor. components central vert. comps.

I_{0}^{∗}  contracted
II, IV, IV^{∗}, II^{∗} p ≡ 1(3) contracted
p ≡ −1(3)  contracted
III, III^{∗}  p ≡ 1(4) contracted
p ≡ −1(4)  contracted

The next thing we discuss is the degree of the Weil parametrization. We still suppose that E has potentially good reduction. Over the complement of a finite set of points of Ẽ_{U}(u), the morphism ϕ : X̃_{U} → Ẽ_{U} is finite and flat. This gives the following proposition.

Proposition 4.2.3 Suppose that E has potentially good reduction at p. Then the degree of ϕ : X_{0}(p^{2}N)_{Q} → E is the sum over the irreducible components C of X̃_{U}_{,u} of the degrees of the ϕ|_{C} : C → Ẽ_{U}_{,u} (the degree of a constant map is zero).

In the next section we will need some kind of “upper bound” on ϕ. This upper bound is a consequence of the injectivity of

ϕ^{∗} : E ⟶ J_{0}(p^{2}N)_{Q}.

Proposition 4.2.4 Let G be the kernel of ϕ^{∗} : Ẽ_{U}_{,u} ⟶ J̃_{U}_{,u}. Then G is a finite group scheme, annihilated by some power of p. Suppose that p > 7, then we have:

1. if Ẽ_{U} _{,u}^{0} \begin{matrix}∼ \\ = \end{matrix} G_{m,u} then G ∩ Ẽ_{U} _{,u}^{0} = 0.

2. if Ẽ_{U}_{,u} is supersingular then either G = 0 or G = ker F : Ẽ_{U}_{,u} → Ẽ_{U}_{,u} (where F is the Frobenius morphism of degree p).

3. if Ẽ_{U}_{,u} is ordinary then either G = 0 or G = ker V : Ẽ_{U}_{,u} → Ẽ_{U}_{,u} (where V is the Verschiebung of degree p).

Proof. The morphism ϕ^{∗} : Ẽ_{U} ⟶ J̃_{U} is injective on the generic fibre. This implies that ϕ^{∗} is injective on the prime to p torsion in Ẽ_{U}_{,u}. It follows that G is a finite group scheme consisting of p-power torsion.

Let T → S be the minimal base change such that E acquires stable reduction over T (this base change is given in Proposition 4.1.1). Then T → S is totally ramified of degree 2,3,4 or 6. Let X̃_{T}/T be the quotient of X̃_{U}/U by the action of Gal(U/T) ⊂ μ_{n}(S). Then X̃_{T}/T satisfies property N^{∗} of [Ra 1](6.1.4) and (6.1.6) since X̃_{T} is normal (being the quotient of X̃_{U} with X̃_{U} normal) and H^{0}(X̃_{T}, O_{X_{T}}) = O_{T} (O_{T} = Gal(U/T)-invariants in O_{U} = H^{0}(X̃_{U}, O_{X_{U}})). By [Ra 1] Theorem 8.2.1, Proposition 8.0.1 and Proposition 8.1.2(ii) Pic_{_{X_{T}} /T}^{0} is representable by a smooth group scheme J̃_{T}^{0} (we do not know whether J̃_{T}^{0} is the connected component of the Néron model of J_{0}(p^{2}N)_{Q} over T: X̃_{T} can have some singularities). If E has potentially multiplicative reduction at p, then T → S has degree 2, X̃_{T} is regular and J̃_{T}^{0} is the connected component of its Néron model. If E has potentially good reduction, then (by taking the quotient by Gal(U/T)) we have ϕ : X̃_{T} → Ẽ_{T}. In both cases we get a morphism ϕ^{∗} : Ẽ_{T}^{0} → J̃_{T}^{0}. Since the absolute ramification index of T is less than p − 1 we can apply [Ra 2] Corollaire 3.3.6.1 (see also [Ma 1] Proposition 1.1). It follows that ϕ^{∗} : Ẽ_{T}^{0} → J̃_{T}^{0} is injective. We have Ẽ_{U}^{0} = Ẽ_{T} _{,U}^{0} because Ẽ_{T} is stable. The morphism ϕ^{∗} : Ẽ_{U}^{0} → J̃_{U}^{0} is the composition:

Ẽ_{T} _{,U}^{0} ⟶ J̃_{T} _{,U}^{0} ⟶ J̃_{U}^{0}

and we know that the first one of these two is an injection. We conclude that:

ker(ϕ^{∗} : Ẽ_{U} _{,u}^{0} → J̃_{U} _{,u}^{0}) = Ẽ_{T} _{,t}^{0} ∩ ker(J̃_{T} _{,t}^{0} → J̃_{U} _{,u}^{0}).

Claim. The kernel of J̃_{T} _{,t}^{0} → J̃_{U} _{,u}^{0} is purely additive: it is isomorphic to a direct sum of G_{a,t}’s.

Proof (of the claim). For the moment, let X denote X̃_{U}_{,u} and let Y denote X̃_{T}_{,t}. Then J̃_{T} _{,t}^{0} = Pic_{Y/t}^{0} and J̃_{U} _{,u}^{0} = Pic_{X/u}^{0}, the morphism between them is the pullback along X → Y. Since X is reduced, this morphism has the facorization X → Y_{red} → Y. On Y we have the exact sequence of sheaves:

0 → I → O_{Y} → O_{Y_{red}} → 0.

The multiplicities of the components of Y are at most \frac{p+1}{2}, hence less than p. This means that we get an exact sequence:

0 → I →^{exp} O_{Y}^{∗} → O_{Y_{red}}^{∗} → 1

where exp is given by the usual power series (note that for f a local section of I we have f^{p} = 0). Taking H^{1}(Y, −) gives an exact sequence:

0 → H^{1}(Y, I) → Pic(Y) → Pic(Y_{red}) → 0.

We conclude that ker(Pic_{Y/t}^{0} → Pic_{Y_{red}/t}^{0}) is purely additive. Let X̃ and Ỹ_{red} denote the normalizations of X and Y_{red}. Then we have a diagram with exact rows:

0 → T_{Y} → Pic_{Y_{red}/t}^{0} → Pic_{_{Y_{red}}/t}^{0} → 0
↓  ↓  ↓
0 → T_{X} → Pic_{X/u}^{0} → Pic_{_{X}/u}^{0} → 0

It is not difficult to check that the morphisms T_{Y} → T_{X} and Pic_{_{Y_{red}}/t}^{0} → Pic_{_{X}/u}^{0} are injective (for the latter case one can use the fact that on every component of Ỹ_{red} there are points over which the ramification is total). This completes the proof of the claim.

It remains to find out what the intersection of Ẽ_{T} _{,t}^{0} with a direct sum of

additive groups can be. If Ẽ_{T} _{,t}^{0} \begin{matrix}∼ \\ = \end{matrix} G_{m,t} then this intersection is 0, because μ_{p,t} cannot be embedded in a direct sum of copies of G_{a,t}. If Ẽ_{T} _{,t}^{0} is an ordinary elliptic curve over t, then its p^{m}-torsion is isomorphic to μ_{p^{m},t} × Z/p^{m} and it follows that the intersection is at most Z/p, the kernel of V. If Ẽ_{T} _{,t}^{0} is a supersingular elliptic curve, then its p-torsion subgroup is a group scheme called M_{2} (see [Oo 1](15.5)). This M_{2} cannot be embedded in a direct sum of additive groups, hence the intersection is at most the Frobenius kernel (which is isomorphic to α_{p,t}). This completes the proof of the proposition.

Remark. The condition “p > 7” is a bit stronger than we need: the proof also works without change if :

1. E is of type I_{ν}^{∗} or I_{0}^{∗} and p > 3,

2. E is of type III or III^{∗} and p > 5,

3. E is of type IV or IV^{∗} and p > 3.

If E has type I_{ν}^{∗} the proposition is even true for all p. One can show for p = 5, 7 that G is contained in ker F ∗ 2 or in ker V ∗ 2.

4.3 Horizontal Weil curves.

Let p and N be as in the preceding section. In this section we investigate the strong Weil curves of conductor p^{2}N that are parametrized by the horizontal components. Proposition 4.2.2 tells us that these are the strong Weil curves with reduction type (at p): II, IV, IV^{∗} or II^{∗} if p ≡ −1(3) and III or III^{∗} if p ≡ −1(4). Equivalently, they are the strong Weil curves with potentially good, supersingular reduction at p, that are not of type I_{0}^{∗}. Let ϕ : X → E be the strong Weil parametrization over \overline{F_{p}} between the stable reductions of X_{0}(p^{2}N) and the Weil curve (Proposition 4.2.1). Let d be the order of the inertia action on E (this is the number “n” in Proposition 4.1.1), hence d ∈ {3, 4, 6} and p ≡ −1(d). Let C_{x} be the horizontal component in X corresponding to a supersingular point x ∈ X_{0}(N)(\overline{F_{p}}). Then ϕ|_{C_{x}} : C_{x} → E factorizes as:

C_{x} ⟶ C_{x(d)} ⟶^{ϕ_{x}} E

(for C_{x} → C_{x(d)}, see Proposition 3.2.4 and Table 3.2.5). If C_{x(d)} happens to have genus 0, then ϕ_{x} = 0 (of course). In the other cases, C_{x(d)} is isomorphic to E. For every such x, we choose an isomorphism : E ˜→C_{x(d)}. The ϕ_{x} are now elements of End\overline{F_{p}}(E). Since E is supersingular, its ring of endomorphisms is a maximal order in a quaternion algebra. Let ζ_{d} ∈ F_{p^{2}} be a primitive d-th root of 1 and let F : E → E be the \overline{F_{p}}-linear Frobenius of degree p (note that indeed E^{(p)} = E). Then in End\overline{F_{p}}(E) we have the commutation rule: Fζ_{d} = ζ_{d}^{−1}F.

Proposition 4.3.1 1. If the strong Weil curve has reduction type II, III or IV, then ϕ_{x} ∈ Z[ζ_{d}], let ψ_{x} := ϕ_{x}.

2. In the other cases (reduction types IV^{∗}, III^{∗} and II^{∗}) we have ϕ_{x} ∈ FZ[ζ_{d}], we write ϕ_{x} = Fψ_{x}, with ψ_{x} ∈ Z[ζ_{d}].

3. for all x, x′:

ψ_{x} ≡ ψ_{x′} mod \{ \begin{matrix}1 − ζ_{3} & if & d = 3 \\ 2 & if & d = 4 \\ 1 + ζ_{6} & if & d = 6. \end{matrix}

4.

g.c.d.((ψ_{x})_{x}) ∈ \{ \begin{matrix}\{1, 1 − ζ_{3}\} & if & d = 3 \\ \{1, 1 + i, 2\} & if & d = 4 \\ \{1, 1 + ζ_{6}\} & if & d = 6. \end{matrix}

Proof. Comparing the inertia actions on X (Section 2.3.3, Propositions 3.2.1, 3.2.2 and 3.2.4) and on E (Proposition 4.1.1) gives (1) and (2). We give the proof for parts (3) and (4) for d = 4 only. The other 2 cases can be done in the same way. The 4 intersection points of the horizontal component C_{x} with the vertical components map to 4 points on the elliptic curve C_{x(4)}. The cusp “∞” lies on one of the outer two vertical components, choose the corresponding point on C_{x(4)} as origin. Then the other 3 points are the 2-torsion points (one of them is fixed by i, the other two by i^{2}). Part (3) of the proposition now follows from the fact that ϕ contracts the vertical components.

Let X → X_{(4)} be the quotient by the action of μ_{\frac{n}{4}} (F_{p^{2}}) on X. Then ϕ : X → E factorizes as:

X ⟶ X_{(4)} ⟶^{ϕ_{(4)}} E.

From Proposition 4.2.4 we get: ker(ϕ_{(4)}^{∗} : E → Pic_{X_{(4)}/t}^{0}) is either 0 or ker F. We will now construct a commutative diagram:

X_{(4)} ⟶ Y
↓ ϕ_{(4)}  ↓
E  ⟶^{2} E

The components of Y are: one copy of X_{0}(N)_{t} and the horizontal components C_{x(4)} of X_{(4)}. The origin of each C_{x(4)} is glued to x ∈ X_{0}(N)(\overline{F_{p}}). The vertical components of X_{(4)} are canonically isomorphic to X_{0}(N)_{t}. The morphism X_{(4)} → Y is multiplication by two on the horizontal components and the canonical isomorphism on the vertical components. The morphism Y → E contracts the vertical component and is ϕ_{x} on the component C_{x(4)}. Taking Pic^{0} we obtain a commutative diagram:

Pic_{X_{(4)}/t}^{0} ⟵ ⊕_{x}C_{x(4)}
↑ ϕ_{(4)}^{∗}  ↑ ⊕ϕ_{x}^{∗}
E  ⟵^{2} E

Part (4) of the proposition is now a consequence of the fact that ker(2ϕ_{(4)}^{∗}) is either E[2] or E[2F].

Remarks.

1. It follows from (3) and (4) that usually we will have to expect that the g.c.d. of the ψ_{x} is maximal.

2. If one of the C_{x(d)} has genus 0 then the g.c.d. of the ψ_{x} is maximal.

It is known that twists of Weil curves are again Weil curves. The quadratic twist of the horizontal curves we consider over the unique quadratic extension of Q ramified only at p is again a Weil curve of conductor p^{2}N. This twist acts on the reduction types as follows:

original type II III IV IV^{∗} III^{∗} II^{∗}
type of twist IV^{∗} III^{∗} III^{∗} II III IV

(4.3.1.1)

We see that the twist is again a horizontal Weil curve. We will now discuss the relation between the two strong Weil parametrizations.

Theorem 4.3.2 Let E be a strong, horizontal Weil curve. Let ϕ denote its (strong) Weil parametrization. Let Ẽ denote the strong Weil curve that is isogenous to the twist of E over the quadratic extension of Q that is ramified only at p. Let ϕ̃ denote its strong Weil parametrization. Then we have deg(ϕ̃) = \frac{a}{b}p^{ε} deg(ϕ), with a, b ∈ \{1, 2, 4\} if d = 4, a, b ∈ \{1, 3\} if d = 3, 6 and ε ∈ {−1, 0, 1}. If E has reduction type II, III or IV (at p) then ε ∈ {0, 1}. If E has reduction type IV^{∗}, III^{∗} or II^{∗} (at p) then ε ∈ {0, −1}.

Proof. The proof gives much more information than the theorem itself: we will describe ϕ̃ in terms of ϕ (up to a small factor). Let ψ := (ψ_{x})_{x} be the vector with coefficients in Z[ζ_{d}] associated to ϕ and let ψ̃ be the same for ϕ̃. We will show that deg(ψ) and deg(ψ̃) difer only by one of the factors listed in Proposition 4.3.1(4). The factor p is then explained by the Frobenius occurring in Proposition 4.3.1(1) and (2).

In the proof we will use some properties of the Hecke operators T_{m} on the horizontal part. We describe these properties in terms of the matrix of T_{m}, but then we have to be a bit careful with the choice of isomorphisms between the horizontal components. Let x and x′ be supersingular points in X_{0}(N)(\overline{F_{p}}). Let A and A′ be the supersingular elliptic curves associated to x and x′. Then an isogeny A → A′ of degree prime to p induces an isomorphism C_{x} ˜→C_{x′} (see Section 3.4, between step 3 and step 4). We demand that for all x and x′ the identification C_{x} ˜→C_{x′} is induced by an isogeny A → A′ whose degree is a square in F_{p}^{∗}. Now let φ : x → x′ be an isogeny of degree m. Then it is not hard to see that the contribution of φ to T_{m}(this is an element of μ_{p+1}(F_{p^{2}}), see Sections 3.3 and 3.4) is in μ_{\frac{p+1}{2}} if and only if (\frac{m}{p}) = 1, where (\frac{m}{p}) denotes the Legendre symbol.

All the horizontal components are given by the equation y^{2} = x^{p+1} + 1. On the disjoint union of the horizontal components we define an involution α by:

α^{#} : \{ \begin{matrix}x & ↦ & x^{−1} \\ y & ↦ & yx^{− \frac{p+1}{2}} \end{matrix}

For ζ ∈ μ_{p+1}(F_{p^{2}}) we find (see Formula 2.3.3.1):

(αζα)^{#} : \{ \begin{matrix}x & ↦ & ζx \\ y & ↦ & y \end{matrix}

It follows that on the jacobian of each horizontal curve we have:

(αζα)^{∗} = [ζ^{\frac{p+1}{2}} ]ζ^{−1}^{∗}  (4.3.2.1)

where [ζ^{\frac{p+1}{2}} ] is multiplication by ±1. This formula implies that α^{∗} exchanges the 3- and 6-parts, and maps the 4-part to itself. We have just seen that all contributions to T_{m} are either squares or non-squares (in μ_{p+1}(F_{p^{2}})). It follows that:

α^{∗}T_{m}α^{∗} = (\frac{m}{p})\overline{T_{m}}  (4.3.2.2)

where the bar denotes complex conjugation. We conclude that:

T_{m}α^{∗}ψ̄^{∗} = α^{∗}α^{∗}T_{m}α^{∗}ψ̄^{∗} = (\frac{m}{p})α^{∗}\overline{T_{m}}ψ̄^{∗} = (\frac{m}{p})a_{m}α^{∗}ψ̄^{∗}

if T_{m}ψ^{∗} = a_{m}ψ^{∗}. It follows that up to one of the factors of Proposition 4.3.1(4) we have ψ̃^{∗} = α^{∗}ψ̄^{∗}. The proof is finished.

Remarks.

1. It follows from the proof that if the conditions (3) and (4) of Proposition 4.3.1 force the g.c.d. of the ψ_{x} to be maximal, then they also force the g.c.d. of the ψ̃_{x} to be maximal. The factor \frac{a}{b} in the theorem is then 1.

2. Combining the theorem with a computation similar to one of Zagier ([Za], page 382) one gets the result that the factor \frac{a}{b} is always 1.

4.4 Examples: conductor p^{2}.

In this section we discuss some examples of horizontal Weil curves of conductor p^{2}. These examples were obtained by a computer calculation using the algorithm described in Section 3.4.

For primes p up to 3, 500 the matrices of T_{2} on the horizontal 3,4 and 6-parts of J_{0}(p^{2}) were computed. Then the program computed the dimensions of the eigenspaces with eigenvalues −2, −1, 0, 1, 2. For one dimensional eigenspaces the program computed an eigenvector.

The program was written in Pascal and executed on a (1-megabyte) Atari ST home computer. The reason to stop at 3, 500 was the limitation on the memory: the matrix of T_{2} was implemented as a two dimensional array. If the program would take into account the sparsity of the matrix of T_{2} one could go much further (not quite as far as Mestre and Oesterlé, since their matrices are twice as small).

Another program, written (in C) by drs. A. de Groot, looked for solutions of the equation: c_{4}^{3} − c_{6}^{2} = 12^{3}Δ, where Δ was supposed to ±p^{2}, ±p^{3} or ±p^{4}. For each solution, this program computed the minimal Weierstrass equation (using Tate’s algorithm [Ta]) of the corresponding elliptic curve over Q and checked whether its conductor was p^{2}. The list of elliptic curves of conductor p^{2} found by this program is contained in [E-G-T].

For every Weil curve detected by the first program, the second program found an elliptic curve over Q with the right reduction type. In order to actually prove that these curves are the same (or at least isogenous) one should bound the c_{4} and c_{6} of the Weil curve (as in [Me-Oe]) in terms of p and the degree of the Weil parametrization (which can be computed from the eigenvector) and check that below this bound there is no other elliptic curve with the right discriminant and a_{2}. This has not (yet) been done.

For every elliptic curve found by the search program that should be a horizontal Weil curve (potentially supersingular reduction at p), the eigenspace of T_{2} for the corresponding eigenvalue had the right dimension (the twist over Q(\sqrt{±p}) can be in the same eigenspace!).

4.4.1 Complex multiplication curves.

For p = 7, 11, 19, 43, 67 and 163 the 0-eigenspace for T_{2} acting on the 4-part is one dimensional. These eigenspaces correspond to the complex multiplication curves over Q of conductor p^{2}. Equations for them can be found in [Gr] (24).

1. p = 7. X_{0}(7^{2}) is an elliptic curve, the degree d of the strong Weil parametrization is 1.

2. p = 11. From the description of the stable model of X_{0}(11^{2}) in Section 3.5.2 it follows that d = \frac{11+1}{3} = 4 (according to the tables in [Bi-Ku] the strong Weil curve is of type III).

3. p = 19. The graph for T_{2} on the 4-part is:

[figure]

For the degree d of the strong Weil parametrization we find \frac{19+1}{4} ·4 = 20 (according to Cremona [Cr] the strong Weil curve has reduction type III).

4. p = 43. The matrix of T_{2} acting on the 4-part is:

( \begin{matrix}0 & −i & i \\ i & 0 & −2i \\ −i & 2i & 0 \end{matrix} ) = i · ( \begin{matrix}0 & −1 & 1 \\ 1 & 0 & −2 \\ −1 & 2 & 0 \end{matrix} ) .

This matrix differs from the one in Section 3.5.3 because the identifications between the horizontal components are different. The primitive eigenvector with eigenvalue 0 is now (2, 1, 1). It follows that the vector (ψ_{x})_{x} is (0, 4, 2, 2). The degree d of the Weil parametrization is

\frac{43 + 1}{4} · 4 · (0^{2} + 2^{2} + 1^{2} + 1^{2}) · 43^{ε} = 2^{3} · 3 · 11 · 43^{ε}

where ε = 0 (or 1) if the reduction type of the strong Weil curve is III (or III^{∗}).

5. For p = 67 we find: d = 2^{2} · 17 · 19 · 67^{0,1}.

6. For p = 163 we find: d = 2^{3} · 41 · 181 · 163^{0,1}.

Remark. It follows from work of Stevens ([St] Thm. 6.4) that in these examples of complex multiplication curves the strong curve has reduction type III at p. This implies that the à priori possible factor p in the degree of the parametrizations does not occur.

For all primes p ≡ 3(8) up to 3, 500 (except p = 3) the dimension of the 0-eigenspace of T_{2} acting on the 4-part turned out to be exactly the class number of Q(\sqrt{−p}). The corresponding abelian variety should be the Weil restriction of scalars of an elliptic curve with complex multiplication (cf. [Gr] (20), or [Sh]). This has not been checked.

4.4.2 Another example: p = 47.

Let p = 47. The graph of T_{2} is:

[figure]

The matrix of T_{2} acting on the 3-part is:

( \begin{matrix}1 & 2 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & −1 \end{matrix} )

with characteristic polynomial (x + 1)(x^{3} − 2x^{2} − 3x + 5). The vector (1, −1, 0, 1) is an eigenvector with eigenvector −1. However, this is not the vector (ψ_{x})_{x} describing the Weil parametrization. The vector (ψ_{x})_{x} is an eigenvector of the transposed martix. One gets:

(ψ_{x})_{x} = (1 − ζ_{3}) · (0, 1, −2, 0, 2)

and the degree of the strong Weil parametrization is:

\frac{47 + 1}{3} · 3 · (1^{2} + 2^{2} + 2^{2}) · 47^{ε} = 2^{4} · 3^{3} · 47^{ε}

where ε = 0 (or 1) if the reduction type of the corresponding strong Weil curve is IV (or IV^{∗}). Until now we do not know how to see whether ε = 0 or 1.

4.5 Vertical Weil curves.

Let p and N be as before: p > 7 and p doesn’t divide N. In this section we consider strong Weil curves of conductor p^{2}N that are parametrized (mod p) by the vertical components. We exclude the reduction types I_{0}^{∗} and I_{ν}^{∗}. According to Proposition 4.2.2 we are left with the strong Weil curves with potentially good, ordinary reduction at p, that are not of type I_{0}^{∗}. Let ϕ : X → E be the strong Weil parametrization over \overline{F_{p}} between the stable reductions of X_{0}(p^{2}N) and the Weil curve (cf. Proposition 4.2.1). Let d be the order of the inertia action on E (this is the number “n” in Proposition 4.1.1), hence d ∈ {3, 4, 6} and p ≡ 1(d). Let C_{1} and C_{2} be the central vertical components of X and let ϕ_{i} : C_{i} → E be the restriction of ϕ to C_{i}. Let ζ_{p−1} be a generator of μ_{p−1}(F_{p}). The inertia action of μ_{p−1}(F_{p}) is given in the formulas 2.2.0.2 and 2.2.0.3. We choose an isomorphism:

\overline{M}([Γ_{0}(N)], [Ig(p)/ ± 1]) ˜ ⟶C_{1}.

We identify C_{2} with the Igusa curve by using this isomorphism composed with ζ_{p−1} : C_{1} ˜→C_{2}. Let ζ_{d} be the primitive d-th root of unity (in F_{p}) by which ζ_{p−1} acts on E (see Proposition 4.1.1). Since ϕ commutes with the inertia action, we have ϕζ_{p−1} = ζ_{d}ϕ, giving:

ϕ_{2} = ζ_{d}ϕ_{1}  (4.5.0.1)

Because E is ordinary, we have End\overline{F_{p}}(E) = Z[ζ_{d}]. Let J := Pic_{X/\overline{F_{p}}}^{0}, and let:

0 → T → J → A → 0

be its decomposition in a toric and abelian variety part.

Proposition 4.5.1 ker(ϕ_{1}^{∗}) is a subgroup scheme of ker((ζ_{d}^{2} − 1)V).

Proof. Recall (Proposition 4.2.4) that

G := ker(ϕ^{∗} : E → J) ⊂ ker(V).

It follows that ker(ϕ_{1}^{∗})/G = (ϕ^{∗}E) ∩ T. The inertia action on the torus T has order 2, implying that V ker(ϕ_{1}^{∗}) is contained in ker(ζ_{d}^{2} − 1).

Proposition 4.5.2 Let ϕ̃ : X → Ẽ denote the strong Weil parametrization (reduced mod p) corresponding to the twist of E over the quadratic extension of Q that is ramified only at p. Then:

deg(ϕ̃) = \frac{a}{b}p^{ε} deg(ϕ)

with ε ∈ {−1, 0, 1}, a, b ∈ {1, 2, 4} if d = 4 and a, b ∈ {1, 3} if d = 3 or d = 6. Moreover, if ε = 1 the ϕ is separable and ϕ̃ is inseparable. If ε = −1 then ϕ̃ is separable and ϕ is inseparable.

Proof. Let ψ denote the separable part of ϕ. It follows from Proposition 4.2.4 that ϕ = F^{0,1}ψ. Let ψ_{i} be the restriction of ψ to C_{i}. Put ψ̃_{1} := ψ_{1} and ψ̃_{2} := −ψ_{2}. Let m be prime to p, then we have T_{m}ϕ^{∗} = a_{m}ϕ^{∗} with a_{m} ∈ Z. If (\frac{m}{p}) = 1, then T_{m} preserves the components C_{1} and C_{2}. It follows immediately that in this case T_{m}ψ̃^{∗} = a_{m}ψ̃^{∗}. If (\frac{m}{p}) = −1 then T_{m} interchanges C_{1} and C_{2}. We find ζ_{p−1}^{∗}T_{m}ψ̃_{1}^{∗} = −a_{m}ψ̃_{2}^{∗}, implying that T_{m}ψ̃^{∗} = −a_{m}ψ̃^{∗}. Thus ψ̃ : X → E is an eigenvector for the Hecke operators T_{m} (m prime to pN) and its eigenvalues are those of ϕ, but twisted by (\frac{m}{p}). This implies that ψ̃^{∗} and ϕ̃^{∗} have the same image in J. The proposition now follows from Proposition 4.5.1.

Remark. As in the remarks following Theorem 4.3.2 one can show that the factor \frac{a}{b} is 1. One can also show that ε ≠ 0. It would be very useful (especially in the next section) to know how the seperability of ϕ is related to the reduction type of E.

4.6 The constant c of a Weil curve.

Let ϕ : X_{0}(M)_{Q} → E be a strong Weil parametrization. The relative cotangent space at the zero section of the Néron model of E over Z is a rank 1 (locally) free Z module, let ω_{Z} be one of the two generators. Then ω_{Z} is called a Néron differential on E. The differential form ϕ^{∗}ω_{Z} is an eigenvector for the Hecke algebra, with the same eigenvalues as the normalized newform ∑_{n≥1} a_{n}q^{n} \frac{dq}{q} corresponding to E (normalized means: a_{1} = 1). It follows that we have:

ϕ^{∗}ω_{Z} = c ∑_{n≥1} a_{n}q^{n} \frac{dq}{q}

with c ∈ Q^{∗}. Changing ω_{Z} by a factor ±1 we can achieve that c > 0. Manin has conjectured that c = 1 (c is sometimes called the Manin constant of E).

Proposition 4.6.1 The number c is an integer.

Proof. Let X_{0}(M) be the compactified coarse moduli scheme \overline{M}([Γ_{0}(M)]) (cf. [Ka-Ma] (8.6)). Let X_{0}(M)^{0} be the open part of X_{0}(M) where the projection to Spec(Z) is smooth. Let E be the Néron model of E over Z. By the Néron property ϕ extends to ϕ : X_{0}(M)^{0} ⟶ E. It follows immediately from [Ka-Ma] Thm.8.11.10 that the formal completion of X_{0}(M) along the (unramified) cusp ∞ is Spf(Z[[q]]). We see that ∞ ∈ X_{0}(M)^{0}(Z) and that ϕ^{∗}ω_{Z} is a differential form (without poles!) in a neighborhood of ∞. If c would not be integral, then ϕ^{∗}ω_{Z} would have poles along the fibers of X_{0}(M)^{0} in characteristics dividing the denominator of c.

Remark. This proposition is sometimes attributed to Gabber (not published). The analogous result for parametrizations involving the modular curve X_{1}(M) is proved by Stevens in [St].

Next we cite a result of Mazur ([Ma 1] Corollary 4.1).

Theorem 4.6.2 (Mazur) Let m be the largest square dividing M, then c is a unit in Z[\frac{1}{2m}].

In other words, if p is a prime different from 2 such that E has stable reduction at p, then p does not divide c. Our aim is to prove the following theorem.

Theorem 4.6.3 Let E be a strong Weil curve. Let p > 7 be a prime such that:

1. E has additive reduction at p, not of type I_{0}^{∗} or I_{ν}^{∗},

2. E does not admit an isogeny (over Q) of degree p.

If E has potentially supersingular reduction at p then p does not divide the (Manin) constant c attached to E. Let Ẽ be the strong Weil curve in the isogeny class of the twist of E over the quadratic extension of Q that is ramified only at p. Let c̃ be its Manin constant. If E has potentially ordinary reduction then p divides cc̃ at most once.

Remarks. 1. By the deep theorem [Ma 1] Thm. 7.1 of Mazur on rational isogenies, condition (2) is satisfied for p ∉ {2, 3, 5, 7, 11, 13, 17, 19, 37, 43, 67, 163}. Moreover, the list of elliptic curves over Q admitting an isogeny of degree p > 13 (p a prime) is finite (up to twist).

2. In the beautiful paper [St] Stevens studies parametrizations of Weil curves by the modular curves X_{1}(M). He conjectures ([St] Conj. I) that every Weil curve E (not just the strong ones) admits a parametrization by X_{1}(M) (where M is the conductor of E) with Manin constant equal to 1. He shows this conjecture to be compatible with quadratic twists over fields that are unramified at the primes where E has additive reduction. He also shows, using results of Rubin on special values of L-functions, that a weak form of his conjecture holds for certain elliptic curves with complex multiplication.

It is easy to show that up to primes that can occur as degrees of isogenies over Q (see the list above) his version of Manin’s conjecture is equivalent to Manin’s conjecture. It follows that the Manin constant c of a strong Weil curve E is not divisible by primes p > 163 where E has reduction type I_{0}^{∗} or I_{ν}^{∗}. Combining the results of Stevens on complex multiplication curves with the results in this thesis one can show that elliptic curves over Q with complex multiplication by Q(\sqrt{−p}), p > 3, have Manin constants (in the original sense as well as in the sense of Stevens) equal to 1 or 2.

We will first prove the following lemma.

Lemma 4.6.4 Let E be a strong Weil curve with potentially good reduction at a prime p > 7. Suppose that the stable reduction mod p of its Weil parametrization (Proposition 4.2.1) is not inseparable on all the irreducible components. Then p does not divide the constant c of E.

Proof. We use the notation of Section 4.2: let ϕ : X̃_{U} → Ẽ_{U} be the Weil parametrization extended to the stable models over U (S = Spec(W (\overline{F_{p}})), U = S[π], π^{n} = p, n = \frac{p^{2}−1}{2}). Let ω_{S} be a generator of ω_{E_{S}/S}. Let ω_{U} := π^{a}ω_{S} where a is as follows:

\begin{matrix}type & II & III & IV & I_{0}^{∗} & IV^{∗} & III^{∗} & II^{∗} \\ a & \frac{1}{6} · \frac{p^{2}−1}{2} & \frac{1}{4} · \frac{p^{2}−1}{2} & \frac{1}{3} · \frac{p^{2}−1}{2} & \frac{1}{2} · \frac{p^{2}−1}{2} & \frac{2}{3} · \frac{p^{2}−1}{2} & \frac{3}{4} · \frac{p^{2}−1}{2} & \frac{5}{6} · \frac{p^{2}−1}{2} \end{matrix}

(4.6.4.1)

Then according to Proposition 4.1.2 ω_{U} is a generator of ω_{_{E_{U}}/U}. Its pullback ϕ^{∗}ω_{U} is a differential form on X̃_{U}. The O_{X_{U}} module Ω_{X_{U}} of (Kähler) differentials is invertible outside the double points of X̃_{U}_{,u}. We will investigate the multiplicities of the irreducible components of X̃_{U}_{,u} in the divisor (of zeros) of ϕ^{∗}ω_{U}. We replace the strong Weil parametrization ϕ : X_{0}(p^{2}N)_{Q} → E by a composition:

X_{0}(p^{2}Nm)_{Q} ⟶ X_{0}(p^{2}N)_{Q} ⟶ E

where m is prime to p, X_{0}(p^{2}Nm) → X_{0}(p^{2}N) is the usual morphism and points in X_{0}(p^{2}Nm)(s) have no extra automorphisms. This does not change the multiplicities that we want to know. In order to deal with the double points we use the embedding

0 ⟶ Ω_{X_{U}} ⟶ ω_{X_{U}}

of Ω_{X_{U}} in the invertible sheaf of the socalled “regular” differentials (ω_{X_{U}} is the dualizing sheaf, cf. [De-Ra] I.2).

Suppose now that E is a horizontal Weil curve. Then E has reduction type II, III or IV (see Proposition 4.3.1). Since ϕ is not inseparable mod p, there must be a horizontal component C_{x} such that ϕ^{∗}ω_{U} is not zero on C_{x}. The action of μ_{n}(F_{p^{2}}) on this differential form is by χ^{a} (χ is the identity: μ_{n}(F_{p^{2}}) → μ_{n}(F_{p^{2}})). The action of μ_{n}(F_{p^{2}}) on the cotangent space to C_{x} at the intersection point x of C_{x} with an outer vertical component is by χ^{\frac{p−1}{2}} (see Section 2.3.3). Let z be a local coordinate on C_{x} at x. It follows that ϕ^{∗}ω_{U}|_{C_{x}} is of the form:

(z^{b} + higher terms)\frac{dz}{z}

with b ≡ \frac{2a}{p−1}mod(p + 1). The genus of C_{x} is \frac{p−1}{2}, hence ϕ^{∗}ω_{U} has exactly p − 3 zeros. It follows that b = \frac{2a}{p−1}. This number b is the order of vanishing of ϕ^{∗}ω_{U} at x as a section of the sheaf ω_{X_{U}} restricted to C_{x}. Obviously this puts a bound on the multiplicity of the outer vertical component in the divisor of ϕ^{∗}ω_{U}, but we have to be a bit careful because X̃_{U} has a singularity at x (Section 2.3.2). An easy computation (see Lemma 4.6.5) shows that ϕ^{∗}ω_{U} has a zero of order at most \frac{p−1}{2} \frac{2a}{p−1} = a along the vertical component on which the cusp ∞ lies. But this means that ϕ^{∗}ω_{S} = π^{−a}ϕ^{∗}ω_{U} does not vanish on this vertical component. We conclude that in this case p does not divide c.

Now suppose that E is a vertical Weil curve. Then ϕ^{∗}ω_{U} vanishes on the horizontal and outer vertical components, but doesn’t vanish on the two central vertical components. We pick one of these two components, call it C. The action of μ_{\frac{n}{2}} (F_{p^{2}}) on ϕ^{∗}ω_{U}|_{C} is by χ^{a}. The action of μ_{\frac{n}{2}} (F_{p^{2}}) on the cotangent space at a supersingular point x of C is by χ^{\frac{p+1}{2}} (see Section 2.3.3). Let z be a local coordinate at x. Then ϕ^{∗}ω_{U} is of the form:

(z^{b} + higher terms)\frac{dz}{z}

with b ≡ \frac{2a}{p+1}mod(\frac{p−1}{2}). We will now compute the degree of the canonical divisor of C. On X_{0}(Nm)_{s} we have the Kodaira-Spencer isomorphism:

\underline{ω}^{⊗2} ˜ ⟶Ω(cusps)

(cf. [Ka-Ma] Thm.10.13.11, note that ±1 are the only automorphisms of points in X_{0}(Nm)(s)). There is also the Hasse invariant ([Ka-Ma] Th.12.4.3):

O(s.s.) ˜ ⟶\underline{ω}^{⊗p−1}

Combining these two gives: O(s.s.) ˜→Ω(cusps)^{⊗ \frac{p−1}{2}} . Taking the degree yields:

s = \frac{p − 1}{2}(2g(X_{0}(Nm) − 2 + c)

where s and c are the number of supersingular points and the number of cusps in X_{0}(Nm)(\overline{F_{p}}). The morphism C → X_{0}(Nm)_{s} is ramified exactly over the supersingular points (and there the ramification is total) of degree \frac{p−1}{2}. The Hurwitz formula gives:

2g(C) − 2 = \frac{p − 1}{2}(s − c)  (4.6.4.2)

We are very lucky, since this last formula implies that there exist supersingular points x on C where ϕ^{∗}ω_{U} has a zero of order less than \frac{p−1}{2}. It follows that at such a point, ϕ^{∗}ω_{U}, considered as a regular differential on X̃_{U}_{,u}, has a zero of order r, where r is given in the following table:

\begin{matrix}type & II & III & IV & I_{0}^{∗} & IV^{∗} & III^{∗} & II^{∗} \\ r & \frac{p−1}{6} & \frac{p−1}{4} & \frac{p−1}{3} & \frac{p−1}{2} & \frac{p−1}{6} & \frac{p−1}{4} & \frac{p−1}{3} \end{matrix}

Let now b denote the multiplicity of the horizontal component C_{x} in the divisor of ϕ^{∗}ω_{U}. Then we have 0 < b ≤ r. But since \frac{ϕ^{∗}ω_{U}}{π^{b}} is a non-zero differential form on C_{x} we must have: (χ^{a−b})^{p+1} = 1, or equivalently, a − b must be divisible by \frac{p−1}{2}. Since a ≡ r(\frac{p−1}{2}) we have b = r. Write a−b = \frac{p−1}{2}c, then μ_{n}(F_{p^{2}}) acts on \frac{ϕ^{∗}ω_{U}}{π^{b}} by (χ^{\frac{p−1}{2}} )^{c}. We list the various c:

\begin{matrix}type & II & III & IV & I_{0}^{∗} & IV^{∗} & III^{∗} & II^{∗} \\ c & \frac{p−1}{6} & \frac{p−1}{4} & \frac{p−1}{3} & \frac{p−1}{2} & \frac{2p+1}{3} & \frac{3p+1}{4} & \frac{5p+1}{6} \end{matrix}

As before, we get a congruence mod p + 1 on the orders of the zeros of the differential form \frac{ϕ^{∗}ω_{U}}{π^{b}} |_{C_{x}} at the intersection points of C_{x} with the outer two vertical components. We know that the polar part of the divisor has degree 2(b + 1). One easily checks that at the intersection points of C_{x} with the outer vertical components \frac{ϕ^{∗}ω_{U}}{π^{b}}, considered as a regular differential, has zeros of order c. We conclude (by Lemma 4.6.5) that ϕ^{∗}ω_{U} has a zero of multiplicity at most b + \frac{p−1}{2}c along the outer vertical components. Equivalently, ϕ^{∗}ω_{S} = π^{−a}ϕ^{∗}ω_{U} has valuation at most b + \frac{p−1}{2}c − a along the outer vertical components. A trivial case by case computation shows that ϕ^{∗}ω_{S} does not vanish on the outer vertical components. We conclude that in this case p does not divide the Manin constant either.

Lemma 4.6.5 Let W be a complete discrete valuation ring with uniformizer π. Let X := SpecW [[x, y]]/(xy − π^{n}), with n ∈ Z_{>0}. Suppose that f ∈ O_{X}(X), considered as a function on the subscheme defined by y = 0, has a zero of order k at the point x = 0. Then the multiplicity of the (Weil) divisor x = 0 on X in the divisor of f is at most kn.

Proof. Let f̃ = ∑_{i,j≥0} a_{ij}x^{i}y^{j} ∈ W [[x, y]] be a lift of f. Using the relation xy = π^{n} we can rewrite f̃:

f̃ = a_{0} + ∑_{i>0} a_{i}x^{i} + ∑_{j>0} b_{j}y^{j}.

Modulo y and π we have: f̃ = \overline{a_{0}} + ∑_{i>0} \overline{a_{i}}x^{i}, hence \overline{a_{0}} = … = \overline{a_{k−1}} = 0 and \overline{a_{k}} ≠ 0. Now we localize with respect to y: x = π^{n}y^{−1}, and we get

f̃ = a_{0} + ∑_{i>0} a_{i}π^{ni}y^{−i} + ∑_{j>0} b_{j}y^{j}

(in W [[y, π^{n}y^{−1}]]). The coefficient of y^{−k} is a_{k}π^{nk}, with valuation nk. The proof of Lemma 4.6.4 is now complete.

Theorem 4.6.3 has now been proved for potentially supersingular E with reduction type II, III or IV since we know (Theorem 4.3.2) that their Weil parametrizations are separable. We attack the other cases by considering E and Ẽ at the same time, but first we prove another preliminary lemma.

Lemma 4.6.6 Let ϕ : X_{0}(M)_{Q} → E be a not necessarily strong Weil parametrization. Let ω be a Néron differential on E. Write ϕ^{∗}ω = cf \frac{dq}{q}, where f is the normalized newform corresponding to E. Then we have:

\frac{deg(ϕ)}{c^{2}} = \frac{∥f∥^{2}}{vol(E, ω)}

where

∥f∥^{2} = ∫_{X_{0}(M)(C)} \frac{i}{2}|f|^{2} \frac{dq ∧ dq}{qq̄} and vol(E, ω) = ∫_{E(C)} \frac{i}{2}ω ∧ \overline{ω}.

Proof. Use:

∫_{X_{0}(M)(C)} \frac{i}{2}ϕ^{∗}ω ∧ \overline{ϕ^{∗}ω} = deg(ϕ) ∫_{E(C)} \frac{i}{2}ω ∧ \overline{ω}.

Proof. (of Theorem 4.6.3). We start with a strong Weil parametrization ϕ : X_{0}(p^{2}N)_{Q} → E satisfying hypotheses (1) and (2) of the theorem. Let E′ be the twist of E over Q(\sqrt{±p}), where the sign is such, that there is only ramification at p. According to the hypotheses of the theorem, we have a commutative diagram:

X_{0}(p^{2}N)_{Q} ⟶^{ϕ̃}  Ẽ
ϕ′ ↘  ↙ α
E′

with ϕ̃ strong, α cyclic and deg(α) not divisible by p. It is easy to see (from the inertia action, Prop. 4.1.1) that Ẽ and E′ have the same reduction type at p. The relation between the reduction types of E and E′ is given in (4.3.1.1). By replacing E if necessary by Ẽ we may suppose that E is of type II, III or IV. Using Proposition 4.1.2 it is easy to compute that: vol(E′, ω′) = p · vol(E, ω), where ω′ is a Néron differential on E′. Let f = ∑ a_{n}q^{n} and f̃ = ∑ ã_{n}q^{n} be the normalized newforms corresponding to E and Ẽ. Since we have |a_{n}|^{2} = |ã_{n}|^{2} for all n, the formula:

∥f^{2}∥ = C · [PSL_{2}(Z) : Γ_{0}(p^{2}N)] · Res_{s=2} ∑_{n=1}^{∞} \frac{|a_{n}|^{2}}{n^{s}}

(where C is some fixed constant) from [Za] shows that ∥f∥^{2} = ∥f̃∥^{2}. From Lemma 4.6.6 we get the equalities:

\frac{deg(ϕ′)}{(c′)^{2}} = p\frac{deg(ϕ)}{c^{2}} deg(ϕ̃) = p deg(ϕ)\frac{(c′)^{2}}{c^{2}} \frac{1}{deg(α)}.

Since p does not divide deg(α), we have v_{p}(c̃) = v_{p}(c′), where v_{p} denotes the valuation on Q corresponding to p. We arrive at:

v_{p}(deg ϕ̃) = v_{p}(deg ϕ) + 1 + 2v_{p}(\frac{c̃}{c}).

If E is horizontal (= potentially supersingular) then Theorem 4.3.2 tells us that v_{p}(c̃) equals v_{p}(c), and we already know that v_{p}(c) = 0. If E is vertical, then Proposition 4.5.2 shows that either v_{p}(c̃) = v_{p}(c) and ϕ is not inseparable (hence v_{p}(c) = 0) or v_{p}(\frac{c̃}{c}) = −1, and ϕ̃ is separable and ϕ is inseparable, hence v_{p}(c̃) = 0 and v_{p}(c) = 1.

Remark. Of course, if the last case occurs, Manin’s conjecture would not be true.

Bibliography

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70

Samenvatting.

Het doel van dit proefschrift is het bestuderen van de actie van de Hecke algebra op modulaire vormen met behulp van stabiele modellen van modulaire krommen.

Eerst worden deze modellen geconstrueerd en wordt de actie van de inertie groep op de speciale vezels uitgerekend. Vervolgens wordt de actie van de Hecke algebra op deze vezels bestudeerd. Het resultaat hiervan is een generalisatie van het grafen algoritme van Mestre en Oesterlé.

Het laatste hoofdstuk is gewijd aan Weil krommen. Er wordt een vrij sterk resultaat bewezen betreffende Manin constanten van sterke Weil krommen.

72

Dankwoord.

Allereerst wil ik mijn promotor Frans Oort bedanken voor de algebraische meetkunde die ik van hem heb geleerd in mijn studietijd en voor zijn begeleiding gedurende het promotie onderzoek. Zijn inzicht dat met de mij bekende methoden informatie gekregen kan worden omtrent Manin constanten is juist gebleken.

Veel dank ben ik verschuldigd aan Bert van Geemen. Zijn invloed op mijn onderzoek is van groot belang geweest. Veel van wat ik weet van modulaire krommen heb ik van hem geleerd. Van hem is ook het idee afkomstig om Hecke operatoren mod p te reduceren (onafhankelijk van Mestre en Oesterlé).

Gerard van der Geer wil ik bedanken voor zijn colleges, voor de discussies die wij hadden en voor zijn opmerkingen over een voorlopige versie van dit proefschrift.

Gedurende de laatste vier jaar was mijn gezellige kamergenoot Jaap Top mijn steun en toeverlaat bij vragen over getaltheorie en elliptische krommen.

Verder wil ik Jean-François Mestre, Joseph Oesterlé, Ken Ribet en René Schoof bedanken voor hun belangstelling in mijn werk en voor de gesprekken met hen daarover.

Tenslotte dank aan Shell Nederland B.V. voor het betalen van een reis naar de U.S.A.

74

Curriculum Vitae.

Op 12 maart 1962 ben ik geboren te Leiden. In 1980 behaalde ik het diploma Voorbereidend Wetenschappelijk Onderwijs aan het Erasmus-College te Zoetermeer. Daarna begon ik de studie natuurkunde aan de Rijksuniversiteit te Utrecht. In december 1982 en maart 1983 behaalde ik de kandidaatsexamens natuurkunde met bijvak wiskunde en wiskunde met bijvak natuurkunde (beide cum laude). In augustus 1985 ben ik (cum laude) afgestudeerd in de zuivere wiskunde. Sinds september 1985 werk ik als promovendus bij de vakgroep wiskunde van de Rijksuniversiteit te Utrecht.