BSD Summer School, Paris, July 4 and 5, 2002

Bas Edixhoven

July 21, 2002

Parallel to these four lectures, there were four lectures by Joseph Oesterlé giving an introduction on abelian varieties and the statement of the Birch and Swinnerton-Dyer conjecture.

1 Modular parametrisations I (1 hour)

At this conference/Summer school, modular parametrisations (whatever they are) are used to study the arithmetic of elliptic curves, and, more generally, of abelian varieties of GL_{2}-type, over ℚ. In particular, one does not ask too much where these modular parametrisations come from. Let me nevertheless say a few words about this.

elliptic curves over ℚ  E/ℚ
Galois representations  \underleftarrow{lim}_{n} E(\overline{ℚ})[l^{n}]
L-functions  L(E, s) = ∑_{n≥1} a_{n}n^{−s}  (ℜ(s) > 3/2)
modular forms  f_{E} = ∑_{n≥1} a_{n}q^{n}  (Wiles et al.)
elliptic curves  ℤ ⊗_{𝕋} J_{0}(N_{E}) ∼ E  (Eichler-Shimura-Faltings)

With all this, the morphism:

\xymatrix{X_{0}(N_{E}) \ar[r] & J_{0}(N_{E}) \ar@{->>}[r] & ℤ ⊗_{𝕋} J_{0}(N_{E}) \ar@{->>}[r] & E \\ P \ar@{|->}[r] & [P] − [∞]}

is a modular parametrisation of E, unique up to sign if we ask the kernel of ℤ ⊗_{𝕋} J_{0}(N_{E}) → E to be cyclic. The morphism from X_{0}(N_{E}) to ℤ ⊗_{𝕋} J_{0}(N_{E}) is called a strong modular parametrisation.

In this lecture, I want to explain the construction of the strong modular parametrisation, first over ℂ, then over ℚ.

For N ≥ 1, we let Γ_{0}(N) be the subgroup SL_{2}(ℤ) consisting of the elements ( \begin{matrix}a & b \\ c & d \end{matrix}) with c ≡ 0 modulo N. We recall that GL_{2}(ℂ) acts on ℙ^{1}(ℂ), hence GL_{2}(ℝ) on ℙ^{1}(ℂ) − ℙ^{1}(ℝ), which is the same as ℂ − ℝ and hence the union of the upper and lower half planes. Hence GL_{2}(ℝ)^{+} acts on the upper half plane ℍ, hence its subgroup SL_{2}(ℤ) too. We let Y_{0}(N)(ℂ) be the quotient Γ_{0}(N)\ℍ, as a complex analytic curve (which is not compact). We let X_{0}(N)(ℂ) be the union of Y_{0}(N)(ℂ) with the (finite) set Γ_{0}(N)\ℙ^{1}(ℤ) of cusps. Then X_{0}(N)(ℂ) is a compact nonsingular complex analytic curve, and we let X_{0}(N)_{ℂ} be the corresponding complex algebraic projective curve (by GAGA).

As an example we mention the j-map from ℍ to ℂ, sending τ to 1/q + 744 + 196884q + ⋯ (with q = exp(2πiτ)) which identifies ℂ with Y_{0}(1)(ℂ). It induces an isomorphism X_{0}(N)_{ℂ} → ℙ_{ℂ}^{1}. The ramification of X_{0}(N)_{ℂ} → X_{0}(1)_{ℂ} is not hard to compute, and leads to a formula for the genus of X_{0}(N)_{ℂ}.

The next item in our list of things to explain is the jacobian J_{0}(N)(ℂ) associated to X_{0}(N)(ℂ). We have:

J_{0}(N)(ℂ) := H^{0}(X_{0}(N)_{ℂ}, Ω^{1})^{∨}/H_{1}(X_{0}(N)(ℂ), ℤ),

where γ in H_{1}(X_{0}(N)(ℂ), ℤ) is sent to the map ω ↦ ∫_{γ} ω. This quotient is a complex torus, and even an abelian variety. The corresponding complex algebraic variety is denoted J_{0}(N)_{ℂ}.

We will refer to H^{0}(X_{0}(N)_{ℂ}, Ω^{1}) as the space of complex cusp forms of weight 2 on Γ_{0}(N), and denote it by S(Γ_{0}(N), 2)_{ℂ} (in order to relate this with forms of other weights one needs the Kodaira-Spencer morphism). One can view S(Γ_{0}(N), 2)_{ℂ} as a space of certain functions ∑ a_{n}q^{n} on ℍ: pullback via the morphism ℍ → X_{0}(N)(ℂ) gives a map

ω ↦ (∑_{n≥1} a_{n}(ω)q^{n}) \frac{dq}{q} ,

which is called the q-expansion map at the standard cusp ∞.

This cusp ∞ is also used to define the following map:

X_{0}(N)(ℂ) ⟶ J_{0}(N)(ℂ), P ↦ [ω ↦ ∫_{∞}^{P} ω] .

Now the Hecke algebra. It comes from the action of GL_{2}(ℚ)^{+} on ℍ. For Γ_{1} and Γ_{2} of finite index in SL_{2}(ℤ), and g in GL_{2}(ℚ)^{+}, one has (with Γ_{3} := Γ_{1} ∩ g^{−1}Γ_{2}g):

[figure: ℍ; Γ_{1}; Γ_{3}; ∼; g·; ℍ; Γ_{2}; Γ_{1}\ℍ; Γ_{2}\ℍ; Γ_{3}\ℍ; Γ_{1}\ℍ; Γ_{2}\ℍ; X(Γ_{3})_{ℂ}; s_{g}; t_{g}; X(Γ_{1})_{ℂ}; X(Γ_{2})_{ℂ}]

The last of these three diagrams is refered to as the Hecke correspondence T_{g}. It induces maps, called Hecke operators:

T_{g}^{∗}: H^{0}(X(Γ_{2})_{ℂ}, Ω^{1}) ⟶ H^{0}(X(Γ_{1})_{ℂ}, Ω^{1}), s_{g∗} ∘ t_{g}^{∗}
T_{g,∗}: J(Γ_{1})_{ℂ} ⟶ J(Γ_{2})_{ℂ}, t_{g∗} ∘ s_{g}^{∗}

A study of Γ_{0}(N)\GL_{2}(ℚ)^{+}/Γ_{0}(N) leads to Hecke operators T_{n}: J_{0}(N)_{ℂ} → J_{0}(N)_{ℂ}, for all n ≥ 1. To describe these, it is convenient to use the moduli interpretation of Y_{0}(N)(ℂ): it is the set of isomorphism classes of pairs (E/ℂ, G), with E a complex elliptic curve and G ⊂ E(ℂ) a cyclic subgroup of order N. To be precise, a point τ of ℍ is sent to the pair (ℂ/(ℤτ + ℤ), μ_{N}). With this description, the correspondence T_{n} (say on the level of divisors) is given by:

T_{n}: [(E, G)] ↦ ∑_{H}[(E/H, \overline{G})],

where H runs through the set of subgroups of order n of E(ℂ) (not necessarily cyclic) such that H ∩ G = \{0\}, and \overline{G} denotes the image of G under E ↦ E/H. These correspondences T_{n} commute with each other, and satisfy the relations (as correspondences; no equivalence relation is necessary) encoded in the following equality of formal Dirichlet series:

∑_{n≥1} T_{n}n^{−s} = ∏_{p|N}(1 − T_{p}p^{−s})^{−1} ∏_{p∤N}(1 − T_{p}p^{−s} + p^{1−2s})^{−1}.

In terms of q-expansion at ∞ one has the identities:

a_{n}(T_{m}(ω)) = ∑_{(d,N)=1^{d|(n,m)}} d a_{nm/d^{2}}(ω).

We let 𝕋_{N} be the subring of End(J_{0}(N)_{ℂ}) generated by the T_{n}. This ring 𝕋_{N} is called the Hecke algebra of level N; it is a commutative ring, free as ℤ-module of rank g(X_{0}(N)_{ℂ}). In fact, if we define:

S(Γ_{0}(N), 2)_{ℤ} := {ω ∈ S(Γ_{0}(N), 2)_{ℂ} | a_{n}(ω) ∈ ℤ for all n},

then:

𝕋_{n} × S(Γ_{0}(N), 2)_{ℤ} ⟶ ℤ, (t, ω) ↦ a_{1}(tω)

is a perfect pairing of ℤ-modules, in the usual sense that each side is identified with the dual of the other.

Eigenforms. Suppose that ω in §(Γ_{0}(N), 2)_{ℂ} is a common eigenform for the T_{n}:

T_{n}ω = λ_{n}ω

for all n, with suitable λ_{n} in ℂ. Then one has:

a_{n}(ω) = λ_{n} a_{1}(ω)

for all n; hence an eigenform is determined by the eigenvalues and its first Fourier coefficient. An eigenform form ω is called normalized if a_{1}(ω) = 1. We also note that the non-zero qcommon eigenspaces for 𝕋_{N} acting on S(Γ_{0}(N), 2)_{ℂ} are one-dimensional (and generated by the normalized form that they contain). We note that for a normalized eigenform ω the subring ℤ[{a_{n}(ω)}] of ℂ is an order in a number field, as it is the image of 𝕋_{N} under a morphism of rings 𝕋_{N} → ℂ. We remark that S(Γ_{0}(N), 2)_{ℂ} can be seen as the ℂ-vector space Hom(𝕋_{N}, ℂ) of ℤ-module morphisms from T T_{N} to ℂ, and that the eigenforms correspond exactly to the morphisms of rings.

A normalized eigenform ω in S(Γ_{0}(N), 2)_{ℂ} is called a newform if its system of eigenvalues a_{p}(ω) for p ∤ N does not occur in any S(Γ_{0}(M), 2)_{ℂ} with M a proper divisor of N.

Let now ω be a newform of level N, and let O_{ω} be the subring ℤ[{a_{n}(ω)}] of ℂ. Then we define:

A_{ω} := O_{ω} ⊗_{𝕋_{N}} J_{0}(N)_{ℂ} = J_{0}(N)_{ℂ}/I_{ω}J_{0}(N)_{ℂ},

where I_{ω} is the kernel of the morphism of rings 𝕋_{N} → ℂ corresponding to ω. (By definition, I_{ω}J_{0}(N)_{ℂ} is the smallest abelian subvariety of J_{0}(N)_{ℂ} containing the images of all elements of I_{ω}.) The quotient J_{0}(N)_{ℂ} → A_{ω} is called the optimal quotient associated to ω (or more precisely, to the Galois orbit of ω, as it depends only on I_{ω}). The reason to call it optimal is that its kernel is connected, which is equivalent to the condition that the dual morphism: A_{ω}^{∨} → JJ_{0}(N)_{ℂ} be injective. Yet another way equivalent condition for a quotient of complex abelian varieties A → B to be optimal is that the induced map H_{1}(A, ℤ) → H_{1}(B, ℤ) be surjective.

Of course, A_{ω} is an elliptic curve if and only if O_{ω} = ℤ. More generally, the dimension of A_{ω} is that of the ℤ-module O_{ω}. We remark that the theory of newforms by Atkin-Lehner gives a description of J_{0}(N)_{ℂ} up to isogeny as product of A_{ω}’s where ω runs through the set of newforms of level dividing N.

We note the following analytic description of A_{ω}:

A_{ω} = \frac{S(Γ_{0}(N), 2)_{ℂ}^{∨}/I_{ω}S(Γ_{0}(N), 2)_{ℂ}^{∨}}{H_{1}(X_{0}(N)(ℂ), ℤ)/I_{ω}H_{1}(X_{0}(N)(ℂ), ℤ)} = \frac{S(Γ_{0}(N), 2)_{ℂ}[I_{ω}]^{∨}}{H_{1}(X_{0}(N)(ℂ), ℤ)} ,

and S(Γ_{0}(N), 2)_{ℂ}[I_{ω}] has as ℂ-basis the set of Galois conjugates of ω.

Now over ℚ. Up to now, everything has been done over ℂ. We want to have everything over ℚ. For that, we use the moduli interpretation. The complex curve X_{0}(N)_{ℂ} has a natural model X_{0}(N)_{ℚ} over ℚ, (i.e., a ℚ-scheme that gives X_{0}(N)_{ℂ} after base change from ℚ to ℂ), namely, the compactified coarse moduli scheme associated to the (contravariant) functor:

Γ_{0}(N): Sch/ℚ ⟶ Sets, S ↦ {(E/S, G)}/ ≅,

where E/S is an elliptic curve over a ℚ-scheme, and where G is a closed finite subgroup scheme of E locally free of rank N, such that for every geometric point x of S the group G(x) is cyclic. This means: in the category of contravariant functors Sch/ℚ → Sets, the morphism Γ_{0}(N) → Y_{0}(N)_{ℚ} is universal for morphisms to representable functors. Classically, one also demands the map Γ_{0}(N)(\overline{ℚ}) → Y_{0}(N)_{ℚ}(\overline{ℚ}) to be bijective, but this is automatically true. We remark that for the congruence subgroups Γ_{1}(N) with N ≥ 5 the situation is much simpler: the corresponding functor (pairs (E/S, i) with i: μ_{N,S} ↪ E) is representable, say by X_{1}(N)_{ℚ}.

It follows that we have a model over ℚ for J_{0}(N)_{ℂ}:

J_{0}(N)_{ℚ} := Pic_{X_{0}(N)_{ℚ}/ℚ}^{0}, A_{ω,ℚ} := O_{ω} ⊗_{𝕋_{N}} J_{0}(N)_{ℚ}.

Galois representations. V_{ω,l} := ℚ ⊗ \underleftarrow{lim}_{n} A_{ω}(\overline{ℚ})[l^{n}] is free of rank 2 over ℚ_{l} ⊗ O_{ω}, hence this gives:

ρ_{ω,l}: G_{ℚ} ⟶ GL_{2}(ℚ_{l} ⊗ O_{ω}).

For every σ: O_{ω} → ℂ we get an L-function:

L_{σ}(ρ_{ω,l}) := ∏_{p} det (1 − p^{−s} Frob_{p} |(V_{ω,l})_{I_{p}})^{−1} ,

where one chooses for each factor a prime number l ≠ p (the subscript I_{p} stands for “coinvariants for the inertia at p”). It is not so hard to show that:

L_{σ}(ρ_{ω,l}) = ∑_{n≥1} σ(a_{n}(ω))n^{−s},

because at the places where ρ_{ω} is much ramified, the Euler factor is trivial. In particular, one has:

L(A_{ω,ℚ}, s) = ∏_{σ: O_{ω}→ℂ} ∑_{n≥1} σ(a_{n}(ω))n^{−s}.

It is a difficult theorem by Eichler-Shimura-Langlands-Deligne-Carayol (simplified by Nyssen) that ρ_{ω,l}|_{D_{p}} corresponds by a suitable normalized local Langlands correspondence to π_{ω,p}, the Euler factor at p of the automorphic representation associated to ω (π_{ω,p} is a smooth irreducible representation of GL_{2}(ℚ_{p})). In particular, local ε-factors match. A consequence of this is that the conductor of ρ_{ω} is N.

2 Modular parametrisations II (1 hour)

2.1 Strong parametrisations; Stevens’s conjecture

Let E/ℚ be an elliptic curve, N its conductor. As E is modular, there exists a unique E′ ↪ J_{0}(N)_{ℚ} with E′ isogeneous to E. Equivalently, we have:

X_{0}(N)_{ℚ} ⟶ J_{0}(N)_{ℚ} ⟶ ⟶^{optimal} E′ ⟶ E,

where the last isogeny can be chosen such that its kernel is cyclic (and then it is unique up to sign).

Question: how to characterise E′ in the isogeny class of E?

Answer: I don’t know.

Example. There are three elliptic curves of conductor 11: 11A = X_{1}(11)_{ℚ}, covering 11B = X_{0}(11)_{ℚ} (quotient for the action of (ℤ/11ℤ)^{∗}/{1, −1}), covering 11C (quotient by the cuspidal group). Among these three, 11A has the smallest Faltings height. Also note that the isogenies are of degree 5 and that their kernels are constant group schemes over ℚ, so that they extend to etale morphisms between Néron models over ℤ.

The main point of this section is to say that Glenn Stevens, in his article in Invent. math. 98 (1989) has formulated a better question. Instead of parametrising with the modular curves X_{0}(N), one should seriously consider parametrisations by arbitrary modular curves, i.e., corresponding to arbitrary congruence subgroups. Stevens has shown that “stabilisation takes place at Γ_{1}(N)”, so that it makes better sense to replace X_{0}(N)_{ℚ} by X_{1}(N)_{ℚ}. We recall that in these lectures X_{1}(N)_{ℚ} classifies elliptic curves with embeddings of μ_{N} (i.e., not of ℤ/Nℤ); this is necessary for example for the cusp ∞ to be ℚ-rational. In view of what follows, one is tempted to say that considering parametrisations by X_{0}(N)_{ℚ} is somehow an historical error.

Better question (Stevens): what is the unique E′′ inside J_{1}(N)_{ℚ} isogeneous to E?

Conjectural answer (Stevens): E′′ is the one with the smallest Faltings height in the isogeny class of E.

2.1.1 Theorem. (Stevens) Each isogeny class of elliptic curves over ℚ contains a unique curve of smallest Faltings height, and that one admits étale isogenies to all others (one Néron models over ℤ).

Some explanation is in order here. Let A/ℚ be an abelian variety, and A_{ℤ} its Néron model over ℤ. One puts:

\underline{ω}_{A_{ℤ}/ℤ} := ⋀^{dim(A)} 0^{∗}Ω_{A_{ℤ}/ℤ}^{1};

it is a free ℤ-module of rank 1. Integration over A(ℂ) equips \underline{ω}_{A_{ℤ}/ℤ} with a hermitian metric, and one defines the Faltings height of A to be the Arakelov degree:

h(A) := deg_{Ar}(\underline{ω}_{A_{ℤ}/ℤ}) = −\frac{1}{2} log ((\frac{i}{2})^{d} ∫_{A(ℂ)} ω_{1}\overline{ω_{1}} ⋯ ω_{d}\overline{ω_{d}}) ,

where (ω_{1}, …, ω_{d}) is a ℤ-basis of 0^{∗}Ω_{A_{ℤ}/ℤ}^{1}. In words: the Faltings height of A is minus one half of the logarithm of the covolume of the period lattice of A(ℂ) with respect to a basis of Néron differentials.

For E/ℚ an elliptic curve, one has 0^{∗}Ω_{E_{ℤ}/ℤ}^{1} = ℤ·ω_{E}, with ω_{E} unique up to sign; these are called the Néron differentials of E. If y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6} is a globally minimal Weierstrass equation for E, then:

ω_{E} = ±\frac{dx}{2y + a_{1}x}.

An isogeny ϕ: E → E′ of elliptic curves over ℚ is étale if ϕ^{∗}ω_{E′} = ±ω_{E}.

2.1.2 Theorem. (Vatsal, preprint Febr. 2002) If E is semistable, then Stevens’s conjecture is true up to an isogeny of degree a power of 2.

Let us mention the ingredients of his proof: geometric class field theory, a theorem of Ihara (that we recall in a moment), Heegner points of conductor p^{n} with n → ∞, work of Rubin and Hida involving special values of L-functions.

Ihara’s result alluded to is the following. For N ≥ 1, we let Σ_{N} be the kernel of J_{0}(N_{ℚ}) → J_{1}(N)_{ℚ}; it is called the Shimura subgroup of J_{0}(N)_{ℚ}. Equivalently, the Cartier dual Σ_{N}^{D} is the Galois group of the largest unramified cover of X_{1}(N)_{ℚ} in X_{1}(N)_{ℚ} → X_{0}(N)_{ℚ} (which is Galois with group (ℤ/Nℤ)^{∗}/1, −1). Then Ihara shows that the sequence (obtained by reducing modulo a prime p that does not divide N):

ℤ[X_{0}(N)(𝔽_{p^{2}})^{s.s.}]_{0} ⟶ Pic(X_{0}(N)_{𝔽_{p^{2}}}) ⟶ Σ_{N}^{D} ⟶ 0

is exact. The superscript “s.s.” stands for “supersingular”, and the subscript zero for the kernel of the sum map to ℤ.

In the same preprint, Vatsal proves the following theorem, generalising work of Mazur for N prime (using very different methods).

2.1.3 Theorem. (Vatsal) For N square free, 2^{∞}Σ_{N} is the largest μ-type subgroup of J_{0}(N)_{ℚ}.

The recent results mentioned in this talk show perhaps that the use of L-functions (and especially p-adic ones) looks very promising.

2.2 Manin constants

Let ϕ: X_{0}(N)_{ℚ} → E be a strong modular parametrisation, f_{E} dq/q the normalized newform on X_{0}(N)_{ℚ} corresponding to E, and ω_{E} a Néron differential on E. Then ϕ^{∗}ω_{E} = c_{E}·f_{E} dq/q with c_{e} in ℚ^{∗} (use multiplicity one). The number c_{E}, defined up to sign, is called the Manin constant of E. It is of interest for the Birch and Swinnerton-Dyer conjecture, because ω_{E} plays a rôle in it, but in practice one has to work with f_{E}, so that it is important to know the relation between the two. (This is especially so for optimal quotients of higher dimension.)

2.2.1 Conjecture. (Manin) c_{E} = ±1.

Question. What does this conjecture mean? Geometrically, say.

Answer. I don’t know. The parametrisation ϕ being strong means that from the point of view of etale cohomology one has taken ϕ to be optimal (sujective on H_{1}, even with torsion coefficients). Then one would ask: is ϕ optimal for de Rham cohomology (over ℤ)? But, first of all, it is not that easy to make sense out of this, because of bad reduction, and, secondly, the conjecture asks if ϕ^{∗}ω_{E} is a generator of Ω_{X_{0}(N)/ℤ}^{1} at a given point, namely the cusp ∞. So even if one knows quite a lot about this conjecture, and that there seems little doubt that it is true, I claim not to understand what the conjecture really means. Anyway, the Stevens version of Manin’s conjecture (same article as mentioned above) that we will state in a moment is a cleaner one, but suffers from the same problem.

2.2.2 Conjecture. (Stevens) Every elliptic curve E/ℚ admits a parametrisation ϕ: X_{1}(N_{E})_{ℚ} → E such that ϕ^{∗}ω_{E} = ±f_{E} dq/q.

Let us mention the results about these conjectures obtained so far (in a non-chronological order).

1. For all parametrisations ϕ (of elliptic curves over ℚ), by any X_{1}(N)_{ℚ} or X_{0}(N)_{ℚ}, the corresponding Manin constant is in ℤ. The proof of this just uses that the completion of the modular curves X_{0}(N)_{ℤ} and X_{1}(N)_{ℤ} over ℤ (defined by extending the moduli problems to arbitrary schemes) along the cusp ∞ correspond to ℤ[[q]], via the Tate curve, and that ϕ extends to a neighborhood of ∞ because to the Néron property of E_{ℤ}. Proofs of this can be found Stevens’s article and in an article by Edixhoven in the 1989 Texel proceedings. A nice consequence of this result is as follows. Let f be a newform in S(Γ_{0}(N), 2)_{ℚ}. Then one has the elliptic curve ℂ moduly the period lattice of f dq/q. The standard Weierstrass equation of this elliptic curve (associated to the differential dz and the period lattice in ℂ) gives coefficients c_{4} and c_{6} that are in ℤ. They correspond to the c_{4} and c_{6} of a minimal Weierstrass equation if and only if Manin’s conjecture is true for the strong curve isogeneous to E. The same type of result is true for newforms in S(Γ_{1}(N), 2)_{ℚ}. In particular, this makes it possible first to compute the elliptic curve ℤ ⊗_{𝕋} X_{0}(N), and secondly to verify the conjecture about the Manin constant for that curve. All curves in Cremona’s tables satisfy Manin’s conjecture. Stevens also verified his version in some cases. Now that we know that Manin constants are integers, the question becomes: what primes can divide them?

2. The two conjectures are related:

[figure: J_{1}(N_{E})_{ℚ}; E′; J_{0}(N_{E})_{ℚ}; E; Σ_{N_{E}}; E ∩ Σ_{N_{E}}]

Now (E ∩ Σ_{N_{E}})^{D} is a constant group scheme over ℚ, in E, and hence, by Mazur’s work, of the form ℤ/2ℤ × ℤ/2nℤ with 1 ≤ n ≤ 4, or ℤ/nℤ with 1 ≤ n ≤ 10 or n = 12. We conclude that the two conjectures on Manin constants are equivalent as far as primes numbers p > 7 are concerned.

3. Now consider a strong parametrisation ϕ: X_{0}(N_{E})_{ℚ} → E. Then we have the following results.

(a) p ∤ N_{E} implies that p ∤ c_{E} (Mazur for p > 2, Abbes-Ullmo plus Raynaud for p = 2).

(b) p^{2} ∤ N_{E} together with p > 2 imply that p ∤ c_{E} (Mazur).

(c) 2^{2} ∤ N_{E} implies that 2^{2} ∤ c_{E} (Raynaud, see Abbes-Ullmo).

(d) p > 7 implies that p ∤ c_{E} (Edixhoven (1992), details still not yet published (shame on me!)).

The tools used for these results: geometry over ℤ_{p}, plus everything else if necessary. The last result also uses that Stevens has shown that if his conjecture holds for E, then it also holds for twists of E over quadratic extensions of ℚ that are unramified at all p whose square divide N_{E}.

For example, let us prove, after Mazur, that p^{2} ∤ N_{E} together with p > 2 imply that p ∤ c_{E}. Then J_{0}(N_{E})_{ℚ} has semistable reduction at p. We have an exact sequence:

0 ⟶ A ⟶ J_{0}(N_{E})_{ℚ} ⟶ E ⟶ 0,

which induces a complex of Néron models:

0 ⟶ A_{ℤ_{p}} ⟶ J_{0}(N_{E})_{ℤ_{p}} ⟶ E_{ℤ_{p}} ⟶ 0.

This complex is exact at all terms except possibly E_{ℤ_{p}}. The induced complex on cotangent spaces:

0 ⟵ Cot_{0}(A_{𝔽_{p}}) ⟵ Cot_{0}(J_{0}(N_{E})_{𝔽_{p}}) ⟵ Cot_{0}(E_{𝔽_{p}}) ⟵ 0

is exact (this uses that p > 2 because of the usual condition “e < p−1”). It follows that ϕ^{∗}ω_{E}|_{E_{𝔽p}} is not zero, etc.. . .

Other tools that are used.

1. More complicated geometry (pass to finite extensions of ℚ_{p} over which one has stable reduction).

2. Analytic tools: deg(ϕ) Vol(E(ℂ), ω) = ∥f_{E}∥^{2}c_{E}^{2}, and:

∥f_{E}∥^{2} = C[SL_{2}(ℤ) : Γ_{0}(N_{E})]Res_{2} ∑_{n≥1} |a_{n}(f_{E})|^{2}n^{−s}

from an article by Zagier on degrees of modular paramatrisations, and:

\frac{L(Sym^{2}E, 2)}{πiΩ} = \frac{deg(ϕ)}{Nc_{E}^{2}} ∏_{p^{2}|N} U_{p}(2)

by Shimura (1976) (used by Flach, see recent work by Mark Watkins from Penn State for computational issues). The factor U_{p} reflects the difference between the two symmetric square L-functions that one may define: the usual one from the symmetric square of the l-adic representation, the other one (more naively) purely in terms of the local factors of L(E, s).

It is interesting to note that the Euler factors at p of L(Sym^{2}E, 2) gets a geometrical interpretation using reduction mod p: the factors p are powers of Frobenius, etc.

3 Non-triviality of Heegner points I; André-Oort conjecture (1 hour)

Let E/ℂ be an elliptic curve. Then End(E) = ℤ or End(E) ≅ O_{K,c} = ℤ + cO_{K} (c ≥ 1) with ℚ → K an imaginary quadratic extension. In the second case we say that E has no complex multiplications (CM), by the order O_{K,c} of conductor c. Just to show that there are many (but only countably) elliptic curves with CM: for τ in ℍ, ℂ/ℤτ + ℤ has CM if and only if ℚ → ℚ(τ) is a quadratic extension.

3.1 Galois action

CM elliptic curves are defined over \overline{ℚ}, for the simple reason that if E/ℂ has CM, then for every automorphism σ of ℂ, the conjugate E^{σ} has CM, and we have just seen that there are only countably many.

Let K ⊂ \overline{ℚ} quadratic imaginary, and c ≥ 1. Then we define a set:

S_{K,c} := \{(E/\overline{ℚ}, α) | α: O_{K,c} →^{∼} End(E) inducing K → \overline{ℚ} via Lie(E)\}

The group G_{K} = Gal(\overline{ℚ}/K) acts on S_{K,c}. But also the group Pic(O_{K,c}):

(L, E) ↦ L ⊗_{O_{K,c}} E,

where L ⊗_{O_{K,c}} E is the representable functor on K-schemes that sends a K-scheme S to L ⊗_{O_{K,c}} E(S). Indeed, one can realise L as the kernel of an idempotent p in M_{2}(O_{K,c}) (choose two generators of L, and a splitting of O_{K,c}^{2} → L), which shows that L ⊗_{O_{K,c}} E is the same as ker(p: E^{2} → E^{2}). For E = ℂ/Λ and α: O_{K,c} →^{∼} End(E), one sees that Λ is an invertible O_{K,c}-module (use that O_{K,c} is of the form ℤ[x]/(g), hence is Gorenstein). This implies the following important fact:

S_{K,c} is a Pic(O_{K,c})-torsor.

The actions of Pic(O_{K,c}) and G_{K} commute, which means that G_{K} acts on S_{K,c} as Pic(O_{K,c})- torsor. But, as Pic(O_{K,c}) is commutative, the automorphism group of a Pic(O_{K,c})-torsor is just Pic(O_{K,c}) itself (it acts by right-translations, after one identifies S_{K,c} with Pic(O_{K,c})). It follows that G_{K} acts on S_{K,c} via a morphism G_{K} → Pic(O_{K,c}). The main result of complex multiplication theory for elliptic curves says:

G_{K} acts on S_{K,c} via a morphism G_{K} → Pic(O_{K,c}); this morphism is unramified outside c; for m ⊂ O_{K} a maximal ideal not containing m, the morphism sends Frob_{m} to the class [m]^{−1} of Pic(O_{K,c}).

In order to formulate the conjecture of André-Oort in the context of elliptic curves, we need to define (or rather make explicit) the notion of special subvariety of products of modular curves.

3.2 Definition. Let n ≥ 0, and Γ_{i} (1 ≤ i ≤ n) be congruence subgroups of SL_{2}(ℤ). Let X_{i} := Γ_{i}\ℍ be the (affine) complex modular curve associated to Γ_{i}, and X := ∏_{i} X_{i}. Let Z ⊂ X be a closed irreducible subvariety of X. Then Z is called special if there exists a partition of {1, …, n} into subsets S_{1}, …, S_{r} such that Z = ∏_{1≤j≤r} Z_{j} with each Z_{j} ⊂ ∏_{i∈S_{j}} X_{i} of one of the forms:

1. |S_{j}| = 1 and Z_{i} is a CM-point;

2. the image of ℍ → ∏_{i∈S_{j}} X_{i} under τ ↦ (i ↦ [g_{i}·τ]) for certain g_{i} in GL_{2}(ℚ)^{+}.

3.3 Conjecture. (Special case of André-Oort) Let Σ be a set of CM-points in X (notation as above). Then all irreducible components of the Zariski closure of Σ are special.

3.4 Theorem. The conjecture is true, if one accepts the generalised Riemann hypothesis for imaginary quadratic fields (Edixhoven, 1999, not yet published). For n = 2 is has been proved unconditionally by Yves André (Crelle), and conditionally by Edixhoven (published! (Compositio)).

We remark that Florian Breuer has proven analogous statements for rank two Drinfel’d modules (see his thesis). For the next lecture we want to give a relatively simple proof of the following weaker version.

3.5 Theorem. Let K be imaginary quadratic, and c ≥ 1. Let Σ ⊂ X be a set of CM-points such that for every x in Σ, and for every i, one has End(E_{x_{i}}) = ℤ + c_{x,i}O_{K} with c_{x,i}|c^{∞}. Then all irreducible components of Σ^{Zar} are special.

This case follows from a much more general result by Ben Moonen (Compositio, 1998), but in this simpler case we can give a much simpler proof. Actually, my intention was to adapt Moonen’s proof for this lecture, but it still stays quite technical in this situation (and one has to consider base changes from the p-adic numbers to the complex numbers, for instance, with all kinds of different notions of convergence).

Proof. Let Z be an irreducible component of Σ^{Zar}. We replace Σ by Σ ∩ Z. Consider the projections pr_{i}: X → X_{i}. If pr_{i}Z is a point, then it is a CM-point, and we can replace X by the product of the X_{j} with j ≠ i. So we may (and do) assume that all pr_{i} are dominant. Suppose now that i ≠ j and pr_{i,j}: Z → X_{i} × X_{j} is not dominant. Then, admitting the theorem for n = 2 for the moment, the closure of pr_{i,j}Z is the graph of a Hecke correspondence in X_{i} × X_{j}, call it T, itself of the form Γ\ℍ for some Γ. So we can replace the factor X_{i} × X_{j} by T. So we may (and do) assume that all pr_{i,j} (with i ≠ j) are dominant, and we have to prove that Z = X.

Now consider the map X → ℂ^{n}, where we view ℂ as SL_{2}(ℤ)\ℍ. We must show that the image of Z is ℂ^{n}, so we replace Σ by its image in ℂ^{n}, and must show that Z = ℂ^{n}.

We use induction on n: we may suppose that all pr_{I}: ℂ^{n} → ℂ^{n−1} are dominant. As Σ consists of \overline{ℚ}-points, Z is defined over \overline{ℚ}, hence over a finite extension F of ℚ. Take l a prime number such that l ≥ 13, l ≥ deg(pr_{I}) for all I, l split in O_{K,c} and in F.

3.5.1 Lemma. T_{l}Z ⊂ Z.

Proof. For all x in Σ, and for all σ in G_{F}, we have σ(x) ∈ Z. Now take σ = Frob_{m}, with m a maximal ideal of O_{F} containing l. Then we see: σ(x) ∈ T_{l}x, hence σ(x) ∈ T_{l}Z, hence x ∈ T_{l}Z. □

3.5.2 Lemma. T_{l}Z is absolutely irreducible if n ≥ 3.

Proof. We work over ℂ. Let Γ_{l} be the kernel of the (surjective) morphism of groups SL_{2}(ℤ) → SL_{2}(𝔽_{l}), and let X denote Γ_{l}\ℍ. Then X → ℂ is Galois, with group SL_{2}(𝔽_{l})/{1, −1}. We consider the following diagram:

[figure: X^{n}; G^{n}; π_{n}; □; π_{n}^{−1}Z; G^{n}; ℂ^{n}; Z]

As T_{l}Z is an image of π_{n}^{−1}Z, it suffices to show that π_{n}^{−1}Z is irreducible. Let V be an irreducible component of π_{n}^{−1}Z, and H its stabilizer in G^{n}. All that we have to show is that H = G^{n}. Now consider:

[figure: V; H; π_{n}^{−1}Z; G^{n}; X^{n}; G^{n}; ^{pr^{I}}; X^{n−1}; G^{n−1}; Z; ℂ^{n}; _{pr_{I}}; ℂ^{n−1}; P_{I}; G^{n−1}; □; X^{n−1}; G^{n−1}; Z; ℂ^{n−1}]

Because of the hypotheses l ≥ deg(pr_{I}) and l ≥ 13 the fibered product P_{I} is irreducible. It follows that the projection V → P_{I} is dominant, hence that pr_{I}H = G^{n−1}. The next wel known lemma (proof left as exercise) finishes the proof of the lemma. □

3.5.3 Lemma. (Kolchin?) Let G be a non-commutative simple group, n ≥ 2, H ⊂ G^{n} a subgroup such that pr_{I}H = G^{2} for all I ⊂ {1, …, n} with |I| = 2. Then H = G^{n}.

So we have:

Z = T_{l}Z.

But all T_{l}-orbits in ℂ^{n} are dense (even for the archimedian topology): the subgroup of GL_{2}(ℤ[1/l]) generated by SL_{2}(ℤ) and ( \begin{matrix}l & 0 \\ 0 & 1 \end{matrix} ) contains SL_{2}(ℤ[1/l]), and this last group is dense in SL_{2}(ℝ) (as SL_{2} is generated by additive groups).

It only remains to prove the theorem for n = 2. The reader is refered to my article in Compositio Math. 114 (1998) for that. □

4 Non-triviality of Heegner points II (1 hour)

Reference: C. Cornut, “Non-trivialité des points de Heegner”, CRAS, 2002.

Let E/ℚ be an elliptic curve, N its conductor, π: X_{0}(N)_{ℚ} → E a modular parametrisation, ℚ → K a quadratic extension such that O_{K}/NO_{K} is isomorphic to (ℤ/Nℤ)^{2} (as rings), N ⊂ O_{K} an ideal such that O_{K}/N = ℤ/Nℤ. We note that there are 2^{r} choices for N if N is composed of r distinct primes. For c ≥ 1 we put:

N_{c} := N ∩ O_{K,c}, x_{c} := [ℂ/O_{K} → ℂ/N_{c}] ∈ X_{0}(N)(ℂ).

The point x_{c} is called a Heegner point of conductor c. In fact, x_{c} is in X_{0}(N)(K[c]), where K → K[c] is the abelian extension unramfied outside c that corresponds by class field theory to the quotient Pic(O_{K,c}) of (K ⊗ ℤ̂)^{∗}. As K is fixed in this talk, we drop it from the notation from now on. We are going to take a somewhat closer look at this group Pic(O_{c}).

The first thing we note is how to understand O_{c} “geometrically”:

[figure: O; □; O/cO; O_{c}; ℤ/cℤ] .

This means that X_{c} := Spec(O_{c}) is obtained from X := Spec(O) by “pinching” the closed subscheme O/cO into ℤ/cℤ. Let f denote the morphism X → X_{c}. Then we have a short exact sequence:

0 ⟶ O_{X_{c}}^{∗} ⟶ f_{∗}O_{X}^{∗} ⟶ Q_{c} ⟶ 0,

with Q_{c} a skyscraper sheaf supported on ℤ/cℤ. The long exact cohomology sequence reads:

0 ⟶ O_{c}^{∗} ⟶ O^{∗} ⟶ \frac{(O/cO)^{∗}}{(ℤ/cℤ)^{∗}} ⟶ Pic(O_{c}) ⟶ Pic(O) ⟶ 0.

Now we fix a prime number p that does not divide N, and we define K[p^{∞}] := ∪_{n≥0}K[p^{n}], and x_{n} will denote the previously defined x_{p^{n}}. We let G := Gal(Kp^{∞}/K), and we have:

0 ⟶ ℤ^{∗} ⟶ O^{∗} ⟶ \frac{(ℤ_{p} ⊗ O)^{∗}}{ℤ_{p}^{∗}} ⟶ G ⟶ Pic(O) ⟶ 0.

Using this, one sees that Hom(G, ℤ_{p}) is isomorphic to ℤ_{p}. It follows that there is a unique ℤ_{p}-extension:

\xymatrix{K[p^{∞}] \\  & H_{∞} \ar[ul]_{G_{0}=G_{tors}} \\ K \ar[uu]_{G} \ar[ur]_{≅ℤ_{p}}}

with G_{0} a finite group.

4.1 Theorem. (Cornut) For almost all n ≥ 0 the Heegner point trace_{G_{0}}(π(x_{n})) in E(H_{∞}) is of infinite order.

Before giving the proof, we want to mention that a previous proof was given, first in the case where ℚ → K is ramified at only one prime by Vatsal, and in the general case by Cornut, using a theorem in ergodic theory by Marina Ratner. That tool has been replaced in this proof (following Cornut) by the special case of the conjecture of André and Oort of the previous talk. In any case, the proof uses some miraculous properties of the extension K → K[p^{∞}]: it has certain finite residue fields, the Galois group G has certain elements of order 2. These properties follow immediately from the description as inverse limit of the Pic(O_{p^{n}}) that we have given above.

4.2 Lemma. E(H_{∞})_{tors} is finite.

Proof. For q prime, different from p and inert in K, K → K_{[}p^{∞}] is completely decomposed at q: Frob_{q} in Gal(K[p^{n}]/K) = Pic(O_{p^{n}}) corresponds to the class of the ideal qO_{p^{n}}, hence is trivial. So we have morphisms O_{p^{∞}} → 𝔽_{q^{2}} for all such q. As prime-to-q torsion specializes injectively, and the image is in the finite group E(𝔽_{q^{2}}), it suffices to take two different q’s. □

To prove the thereom, it is enough to prove that the fibres of the following map f are finite:

[figure: X_{0}(N)^{G_{0}}; π; E; ∑; E; n; (σ ↦ σ(x_{n})); trace_{G_{0}}(π(x_{n})); ℕ; f; E(H_{∞})]

Now the idea is to distinguish in G_{0} the “geometric part” and the “chaotic part”. Let q_{1}, …, q_{g} be the primes that are ramified in K, other than p, and let Q_{1}, …, Q_{g} be the maximal ideal of O_{K} over the q_{i}. Then Frob_{Q_{i}} has order 2 in G, because its image in each Pic(O_{K,p^{n}}) corresponds to the class of Q_{i}, and we have Q_{i}^{2} = q_{i}O_{K,p^{n}}. We let G_{1} be the subgroup of G_{0} that is generated by the Frob_{Q_{i}}. We have:

G ⊃ G_{1} = ⊕_{i}𝔽_{2}· Frob_{Q_{i}} .

This subgroup G_{1} accounts for the “geometric part” of the action of G_{0}. We define:

N′ := Nq_{1} ⋯ q_{g}, N′ := N Q_{1} ⋯ Q_{g}, x′_{n} := [ℂ/O_{p^{n}} → ℂ/N′_{p^{n}}] ∈ X_{0}(N′)(K[p^{n}]).

We note that N′ is square free and prime to p.

For 1 ≤ i ≤ g, we have the Atkin-Lehner involution W_{q_{i}} of X_{0}(N′); it sends (F, G, H) (with F an elliptic curve, G a subgroup of order q_{i} and H a cyclic subgroup of order N′/q_{i}) to (F/G, F [q_{i}]/G, \overline{H}). This gives 2^{g} degeneracy maps from X_{0}(N′) to X_{0}(N), indexed by G_{1}. We let δ denote their product: δ: X_{0}(N′) → X_{0}(N)^{G_{1}}. We define a new parametrisation π′: X_{0}(N′) → E by:

[figure: π′: X_{0}(N′); δ; X_{0}(N)^{G_{1}}; π; E^{G_{1}}; ∑; E; x′_{n}; (σ ↦ σ(x_{n})); trace_{G_{1}}(π(x_{n}))]

We note that π′ is dominant because its derivative at ∞ is non-zero (all degeneracy maps making up δ are ramified at ∞ except one)^{1}.

Let now R be a set of representatives for G_{1} ⊂ G_{0}, and consider:

\xymatrix{ & X_{0}(N′)^{R} \ar[r]^{π′} & E^{R} \ar[r]^{σ} & E \\ n \ar@{|->}[r] & H(n) := (σ ↦ σ(x′_{n})) \ar@{|->}[rr] &  & trace_{G_{0}}(π(x_{n}))}

4.3 Proposition. For I ⊂ ℕ infinite, H(I) is Zariski dense in X_{0}(N′)^{R}.

Of course, this proves that f has finite fibers, hence proves the theorem.

Proof. Because of the theorem of the previous lecture, if suffices to prve:

Suppose σ_{1}, σ_{2} are in G_{0}, M ≥ 1, such that (j(σ_{1}(x′_{n})), j(σ_{2}(x′_{n}))) is in X_{0}(M) for infinitely many n, then σ_{1}^{−1}σ_{2} is in G_{1}.

1I thank Florian Breuer for pointing out to me that one has to verify that π′ is not constant; the answer that I gave to his question during the lecture was nonsense

4.4 Lemma. Let E_{1} and E_{2} have CM by K: End(E_{i}) = O_{c_{i}}. Let f: E_{1} → E_{2} be a cyclic isogeny, d := deg(f). Then f factors as follows:

[figure: E_{1}; f; f_{1}; E_{2}; O_{c′} ⊗_{O_{c_{1}}} E_{1}; f′; O_{c′} ⊗_{O_{c_{2}}} E_{2}; f_{2}^{∨}]  c′ = \frac{c_{1}}{deg(f_{1})} = \frac{c_{2}}{deg(f_{2})},  (c′, deg(f′)) = 1.

Proof. The problem can be analyzed prime by prime, as it is a question about ℤ-lattices in K. For every prime l, think of the tree of lattices up to ℚ_{l}^{∗} in V_{l}(E) ≅ ℚ_{l} ⊗ K, with E chosen such that End(E) = O. To each lattice one associates the exponent of l in the conductor of the endomorphism ring. Then one gets the following pictures.

l inert There is one point with label 0, and the labels at all points are the distance to this one point.

l split The is a unique path, infinite in two directions, and the labels are the distance to this path.

l ramified There are two neighboring points such that the labels are the distance to these two points.

This shows that cyclic isogenies behave with respect to conductors of endomorphism rings as the lemma claims. □

Now we finish the proof of the proposition. Suppose that σ_{1}, σ_{2} and M are as in that proposition. Then we have infinitely many f_{n}: E_{n} → σ_{1}^{−1}σ_{2}E_{n} of degree M, with End(E_{n}) = O_{p^{n}}. The last lemma gives us infinitely many f′_{n}: E′_{n} → σ_{1}^{−1}σ_{2}E′_{n} of fixed degree M′ prime to p, with End(E′_{n}) = O_{p^{n−a}} (a fixed), and ideals m_{n} ⊂ O of index M′, such that:

[figure: E′_{n}; can; σ_{1}^{−1}σ_{2}E′_{n}; (m_{n} ∩ O_{p^{n−a}})^{−1} ⊗_{O_{p}^{n−a}} E′_{n}]

There are only finitely many possibilities for m_{n}, so we may assume that m_{n} = m, independent of n. Then in G = \underleftarrow{lim}_{n} Pic(O_{p^{n}}) we have:

[m] = σ_{1}σ_{2}^{−2} ∈ G_{0} = G_{tors}.

This means that for some e ≥ 1 the intersection m∩O_{p^{n}} is principal for all n ≥ 0. Some thinking gives that m is then of the form Q_{1}^{e_{1}} ⋯ Q_{g}^{e_{g}}, hence that σ_{1}σ_{2}^{−1} is in G_{1}, as we had to prove. □