## 2017

### Mini-workshop arithmetic & moduli of K3 surfaces

31 January, UvA Amsterdam. Programme. Registration is compulsary.- Martin Orr Imperial College,
- Tony Várilly-Alvarado Rice University,
- Alexei Skorobogatov Imperial College,
- Bianca Viray University of Washington, Seattle,

### Intercity Number Theory Seminar

10 March, Leiden. Snellius building, room 407/409- 13:30–14:30
- Carlo Pagano Uiversiteit Leiden, Distribution of ray class groups: 4-ranks and general model
In 1983 Cohen and Lenstra provided a probabilistic model to guess correctly statistical properties of the class group of quadratic number fields, viewed as an abelian group. In 2016 Ila Varma computed the average 3-torsion of ray class groups (of fixed integral conductor) of quadratic number fields. She asked whether it was possible to explain her results by a generalization of the Cohen-Lenstra model. In this talk I will explain how to construct a model to guess correctly statistical properties of ray class groups (of fixed integral conductor) of imaginary quadratic number fields, viewed as short exact sequences of Galois modules. This model agrees with Varma's results for imaginary quadratics. Then I will explain the main steps of a proof of this new conjecture for an exact sequence related to the 4-rank of the ray class group and the class group, when the discriminants are coprime to the conductor. As a cruder corollary one obtains the joint distribution of the 4-ranks of the two groups. The methods and the results are a natural extension of the ones of Fouvry and Kluners. This is joint work with Efthymios Sofos.
- 14:45–15:45
- Peter Koymans Universiteit Leiden, On the equation
*x*+*y*= 1 in finitely generated groups in positive characteristicLet*K*be a field of characteristic*p > 0*and let*G*be a subgroup of*K*^{×}×*K*^{×}with dim_{Q}(*G*⊗_{Z}**Q**) =*r*finite. Then Voloch proved that the equation*ax*+*by*= 1 in (*x*,*y*) ∈*G*for given*a*,*b*∈*K*^{×}has at most*p*^{r}(*p*^{r}+*p*- 2)/(*p*- 1) solutions (*x*,*y*) ∈*G*, unless (*a*,*b*)^{n}∈*G*for some*n*≥ 1. Voloch also conjectured that this upper bound can be replaced by one depending only on*r*. Our main theorem answers this conjecture positively. We prove that there are at most 31 ·19^{r + 1}solutions (*x*,*y*) unless (*a*,*b*)^{n}∈*G*for some*n*≥ 1 with (*n*,*p*) = 1. During the proof of our main theorem we generalize the work of Beukers and Schlickewei to positive characteristic, which heavily relies on diophantine approximation methods. This is a surprising feat on its own, since usually these methods can not be transferred to positive characteristic. This is joint work with Carlo Pagano. - 16:00–17:00
- Robin de Jong Universiteit Leiden, New results of effective Bogomolov-type for cycles on jacobians
Let
*A*be an abelian variety over the field of algebraic numbers, and let*L*be a symmetric ample line bundle on*A*. To every subvariety*Z*of*A*one associates two non-negative real numbers: the Néron-Tate height*h*_{L}(*Z*) of*Z*, and the essential minimum*e*_{L}(*Z*) of*Z*. The Néron-Tate height of*Z*generalizes the Néron-Tate height of a point, as it occurs in for example the Birch and Swinnerton-Dyer conjectures, and measures in an intrinsic way the arithmetic complexity of*Z*. The essential minimum of*Z*is, roughly speaking, the liminf of the Néron-Tate heights of the points lying on*Z*.The Bogomolov conjecture, first proved by E. Ullmo and S. Zhang, states that the subvarieties

*Z*that have vanishing*e*_{L}(*Z*) are precisely the translates, by a torsion point, of abelian subvarieties of*A*. For*Z*not of this form, a result of effective Bogomolov-type is an explicit positive lower bound for*e*_{L}(*Z*). It can be proved that*e*_{L}(*Z*) is bounded below by*h*_{L}(*Z*).In this talk we give new explicit positive lower bounds for

*h*_{L}(*Z*) for several tautological subvarieties of jacobians. These include the difference surface, the Abel-Jacobi images of the curve itself and of its square, and any symmetric theta divisor. Our bounds improve in these cases upon earlier effective results by S. David and P. Philippon.

### intercity Number Theory Seminar

24 March, Utrecht. Marinus Ruppertgebouw (Leuvenlaan 21, 3584 CE Utrecht), room: Paars- 13:00–13:50
- Lin Weng Kyushu University / MPIM Bonn, Non-Abelian Zeta Functions And Their Zeros
Non-abelian zeta functions are defined as integrations over moduli spaces of semi-stable lattices. They satisfy standard zeta properties, particularly, functional equation, and are naturally related with integrations of some spacial Eisenstein series associated to maximal parabolic subgroups P(n-1,1) of SL(n) over moduli spaces of lattices with fixed volumes. Hence, Langlands' theory of Eisenstein system can be applied to write down explicit this functions using relative trace formula technique. This then further leads to a more general type zeta functions associated to pairs (P,G) consisting of general reductive groups G and their maximal parabolic subgroups P. We can now show that a weak Riemann Hypothesis for these zeta functions holds provided the rank of these group is at least 1. That is, all but finitely many zeros of these zeta functions are on the central line. In addition, there are two levels of structures for distributions of these zeta zeros: First one for standard pair correlations gives Dirac distributions, and secondary one is conjecturally to be that of GUE.
- 14:00–14:50
- Hatice Boylan İstanbul Üniversitesi / MPIM Bonn, Fourier coefficients of Jacobi Eisenstein series over number fields
In recent work we computed, for any totally real number field
*K*with ring of integers*o*, the Fourier coefficients of the Jacobi Eisenstein series of integral weight and lattice index of rank one and with modified level one on*SL*(2,*o*) attached to the cusp at infinity. This result has a number of important consequences: it provides the first concrete example for the expected lift from Jacobi forms over*K*to Hilbert modular forms, it shows that a Waldspurger type formula holds true in this concrete case (as also expected for the general lifting), and finally it gives us a clue for the Hecke theory still to be developed by giving a concrete example for the action of Hecke operators on Fourier coefficients.In this talk we recall the basic notions of the theory of Jacobi forms over number fields as developed in [BoBo], discuss the general theory of Jacobi Eisenstein series over number fields, and explain in more detail those points in the deduction of our formulas which are not straight forward and require some new ideas. Finally we discuss the indicated implications concerning the arithmetic theory of Jacobi forms over number fields.

References: [BoBo] Boylan, H., "Jacobi forms, finite quadratic modules and Weil representations over number fields", Lecture Notes in Mathematics, volume 2130, Springer International Publishing 2015.

- 15:15–16:05
- Cyril Demarche Paris 6 - ENS, Local-global principles for homogeneous spaces
Given a homogeneous space X of a linear algebraic group G over a number field k, we are interested in the Hasse principle and weak approximation on X, and more precisely in the Brauer-Manin obstruction to those local-global principles. A classical theorem due to Borovoi states that those obtructions are the only ones for homogeneous spaces with connected stabilizers. However, The general case, which can be seen as a generalization of the inverse Galois problem, is still wide open. We will mention recent partial results about the case of homogeneous spaces with finite stabilizers. This is joint work with Danny Neftin and Giancarlo Lucchini-Arteche.
- 16:15–17:05
- Jehanne Dousse Universität Zürich, Refinement of partition identities and the method of weighted words
A partition of a positive integer n is a non-increasing sequence of positive integers whose sum is n. A Rogers-Ramanujan type identity is a theorem stating that for all n, the number of partitions of n satisfying some difference conditions equals the number of partitions of n satisfying some congruence conditions. Alladi and Gordon introduced the method of weighted words in 1993 to find refinements of several Rogers-Ramanujan type identities: Schur's theorem, Göllnitz' theorem and Capparelli's theorem (a partition identity which arose in the study of Lie algebras). Their method relies on q-series identities. After explaining the classical method of weighted words, we will present a new version using q-difference equations and recurrences. It allows one to prove refinements of identities with intricate difference conditions for which the classical method is difficult to apply, such as a conjectural identity of Primc coming from crystal base theory and an identity of Siladic coming from representation theory.

### Intercity Number Theory Seminar

7 April, Groningen. Bernoulliborg 105- 12:00–13:00
- Mark Jeeninga Groningen, Lenstra's epsilon: A curious periodicity in RevLex field extensions of degree
*p*over*F*_{p}.In the 70's, J.H. Conway introduced an algorithm for constructing the algebraic closure of*F*_{2}, by use of games. H.W. Lenstra continued to study algebraic properties of Conway's algorithm, and discovered a curious sequence in*F*_{2}. In his 1978 paper "Nim Multiplication", he poses the problem how the sequence will develop and whether or not the sequence is periodic.In this talk we generalize the problem to arbitrary positive characteristic and prove that this sequence is indeed periodic. We do so by constructing the

*p*-closure of*F*_{p}by means of an Artin-Schreier tower of fields over*F*_{p}, while forcing a 'natural' ordering on the elements in this algebraic structure.This talk is based on my Master's thesis "On a tower of fields related on

*On*_{p}" (2015). - 13:30–14:30
- Ricardo Buring Groningen, Relations among Kontsevich graph weight integrals.
The Kontsevich graph weights are period integrals whose values make Kontsevich's star-product associative for any Poisson structure. These weights of graphs are not all independent: they satisfy algebraic relations. We review such relations (e.g. the associativity constraint, decomposition into prime graphs, cyclic relations) and show (using software) to what extent they determine the values of the weights. Up to the order 4 in ℏ we express all the weights in terms of 10 parameters (6 parameters modulo gauge-equivalence), and we verify pictorially that the star-product expansion is associative modulo ō(ℏ⁴) for every value of the 10 parameters. This is joint work with Arthemy Kiselev.
- 14:45–15:45
- Jan Steffen Müller Oldenurg, Computing canonical heights on elliptic curves in quasi-linear time.
I will discuss an algorithm that can be used to compute the canonical height of a point on an elliptic curve over the rationals in quasi-linear time. As in most previous algorithms, the idea is to decompose the difference between the canonical and the naive height into an archimedean term and a sum of non-archimedean terms. The main innovation is an algorithm for the computation of the latter sum that requires no integer factorization and runs in quasi-linear time. This is joint work with Michael Stoll.
- 16:00–17:00
- Ulrich Derenthal Hannover, Manin's conjecture for a family of nonsplit del Pezzo surfaces
Manin's conjecture predicts the asymptotic behavior of the number of rational points of bounded height on Fano varieties over number fields. We prove this conjecture for a family of nonsplit singular quartic del Pezzo surfaces over the rationals. The proof uses a nonuniversal torsor. This is joint work in progress with Marta Pieropan.

### Intercity Number Theory Seminar

21 April, Nijmegen. Linnaeusgebouw, Heyendaalseweg 137, 6525 AJ, Nijmegen - room LIN7- 11:30–12:30
- Lars Halle Copenhagen, Motivic zeta functions of degenerating Calabi-Yau varieties
Let
*K*=*C*((*t*)), and let*X*be a smooth projective*K*-variety with trivial canonical sheaf. To*X*, we can associate an invariant*Z*_{X}(*T*), called the motivic zeta function of*X*. This is a formal power series in*T*, with coefficients in a suitable Grothendieck ring of*C*-varieties. This series encodes the asymptotic behaviour of the set of rational points of*X*under ramified extension of*K*, and its properties are closely related to the behaviour of*X*under degeneration.I will discuss recent joint work with J. Nicaise, investigating the case where

*X*admits a particularly nice type of model (called equivariant Kulikov model), after some suitable base change. Under this assumption, we show that*Z*_{X}(*T*) has a unique pole. - 13:45–14:45
- Giuseppe Ancona Strasbourg, Standard conjectures for abelian fourfolds
Let
*X*be a smooth projective variety and*V*the finite-dimensional*Q*-vector space of algebraic cycles on*X*modulo numerical equivalence. Grothendieck defined a quadratic form on*V*(basically using the intersection product) and conjectured that it is positive definite. This conjecture is a formal consequence of Hodge theory in characteristic zero, but almost nothing is known in positive characteristic.Instead of studying the quadratic form at archimedean place (the signature), we will study it at the

*p*-adic places. It turns out that this question is more tractable thanks to*p*-adic comparison theorems and the Shimura-Taniyama formula. Moreover, using a classical product formula for quadratic forms, the*p*-adic information will give us non-trivial information on the archimedean place. For instance, we will prove the original conjecture when*X*is an abelian variety of dimension (up to) 4. - 15:15–16:15
- Yohan Brunebarbe Zürich, Hyperbolicity of moduli spaces of abelian varieties
For any positive integers
*g*and*n*, let*A*_{g}(*n*) be the moduli space of principally polarized abelian varieties with a level-*n*structure; it is a smooth quasi-projective variety for*n*>2. Building on work of Nadel and Noguchi, Hwang and To have shown that the minimal genus of a curve contained in*A*_{g}(*n*) grows with*n*. We will explain a generalization of this result dealing with subvarieties of any dimension. In particular, we show that all subvarieties of*A*_{g}(*n*) are of general type when*n*> 6*g*. Similar results are true more generally for quotients of bounded symmetric domains by lattices. - 16:30–17:30
- Olivier Wittenberg Paris, Zero-cycles on homogeneous spaces of linear algebraic groups
(Joint work with Yonatan Harpaz.) The Brauer-Manin obstruction is conjectured to control the existence of rational points on homogeneous spaces of linear algebraic groups over number fields (a far-reaching generalisation of the inverse Galois problem). We establish the zero-cycle variant of this statement.

### Intercity Number Theory Seminar

19 May, UvA and VU Amsterdam. All lectures are in room A1.10, at Science Park 904.- 11:00–12:00
- Ana Caraiani Bonn, Galois representations and torsion classes
I will describe joint work in progress with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne on potential modularity for elliptic curves over imaginary quadratic fields. The key ingredients are the Calegari-Geraghty method and a result on torsion in the cohomology of Shimura varieties that is joint with Scholze. I will aim to explain how these ingredients fit together in my talk.
- 13:00–14:00
- Arne Smeets Njmegen, Pseudo-split fibres and the Ax-Kochen theorem
In 1965 Ax and Kochen proved a famous theorem concerning the existence of p-adic points on hypersurfaces of sufficiently small degree defined over number fields. This theorem was originally proved using tools from model theory. Denef, following a strategy suggested by Colliot-Thélène, recently found a purely geometric proof that gives more general results. In this talk we build upon Denef's work and give a criterion that characterizes those classes of varieties for which an analogue of the Ax-Kochen theorem holds. This work is joint with Dan Loughran and Alexei Skorobogatov.
- 14:30–15:30
- Maxim Mornev Leiden & Amsterdam, Shtuka cohomology and special values of Drinfeld modules
The motivic Tamagawa number conjecture gives a formula for special values of L-functions in terms of motivic cohomology. Among other things it integrates the analytic class number formula for number fields and the BSD conjecture in a single framework. While this conjecture is exceptionally hard, it has a tractable analogue in positive characteristic as discovered by Lenny Taelman. Here the usual L-functions are replaced by Goss L-functions which take values in a positive characteristic field, and one is interested in Goss L-functions attached to Drinfeld modules. In the talk I will explain how special values of these L-functions can be expressed in terms of shtuka cohomology which plays the role of motivic cohomology in this story.
- 16:00–17:00
- Jeroen Sijsling Ulm, Computing endomorphisms of Jacobians
Let
*C*be a curve over a number field, with Jacobian*J*, and let End (*J*) be the endomorphism ring of*J*. The ring End (*J*) is typically isomorphic with, but the cases where it is larger are interesting for many reasons, most of all because the corresponding curves can then often be matched with relatively simple modular forms. We give a provably correct algorithm to verify the existence of additional endomorphisms on a Jacobian, which to our knowledge is the first such algorithm. Conversely, we also describe how to get upper bounds on the rank of End (**Z***J*). Together, these methods make it possible to completely and explicitly determine the endomorphism ring End(*J*) starting from an equation for*C*, with acceptable running time when the genus of*C*is small. This is joint work with Edgar Costa, Nicolas Mascot, and John Voight.

### Intercity Number Theory Seminar

3 November, UvA and VU Amsterdam. All talks will be in the main building (Hoofdgebouw) at the VU Campus. The first talk in room HG-05A24 followed by two talks in room HG-15A16. Lunch will be provided in the latter room before the start of the second talk.- 12:05–12:50
- Julie Desjardins Bonn, Density of rational points on elliptic surfaces
Let X be an algebraic surface. We are interested in the set of rational points X(ℚ). Is it non-empty ? Is it infinite ? Is it Zariski-dense ? We are concerned with elliptic surfaces, i.e. 1-parameter families of elliptic curves. The density of rational points is not well known in general. When the surface is isotrivial, one shows the density in most cases when it is also rational. The rational elliptic surfaces are particularly interesting since they always have a minimal model which is a conic bundle or a del Pezzo surface. Moreover, by studying the variation of the root number of the fibers, one predicts the density on non-isotrivial elliptic surfaces conditionally to some conjectures (parity conjecture, squarefree conjecture, Chowla's conjecture). The last two conjectures impose a restriction on the degree of the factors of the discriminant. We manage to avoid the squarefree conjecture in certain cases, and thus show unconditionally the variation of the root number, without imposing a bound for the degree of the irreducible factors.
- 13:15–14:00
- Jan Tuitman Leuven, An update on effective Chabauty
The method of Chabauty-Coleman often allows one to find the rational points on higher genus curves over the rationals, but has a lot of limitations. On a theoretical level, the Mordell-Weil rank of the Jacobian of the curve has to be strictly smaller than its genus. In practice, even when this condition is satisfied, the relevant Coleman integrals can usually only be computed for hyperelliptic curves. We will report on recent work of ours (with different combinations of collaborators) on extending the method to more general curves.
- 14:05–14:50
- Giulio Orecchia Leiden, A monodromy criterion for existence of Neron models
Neron models are central to the study of the reduction of an abelian variety defined over a number field at primes of the ring of integers. Only recently, people have become interested in studying Neron models of abelian schemes over bases of dimension higher than 1. The main reason why Neron models are harder to study in this setting, is that they often fail to exist. In this talk I will show that the existence of a Neron model is closely related to a condition (called toric-additivity) on the action of monodromy on the l-adic Tate module of the generic fibre, for a prime l invertible on the base.

### Intercity Number Theory Seminar

17 November, Leiden. Snellius, room 312. (The Snellius restaurant will be closed; the restaurants in the Huygens and Gorlaeus buildings are open.)- 13:00–14:00
- Reinier Bröker Brown University, Lower bounds for Hilbert class polynomials
Hilbert class polynomials (minimal polynomials of j-invariants of CM elliptic curves) have various algorithmic applications. There are several algorithms that compute them in time proportional to proven upper bounds on their sizes. Based on examples, it is widely believed that the proven upper bounds are close to the truth. In this talk we address whether we can rigorously prove any lower bounds on their sizes. As we will see, the answer depends on what we are willing to assume.
- 14:30–15:30
- Christopher Lazda Universiteit van Amsterdam, A Néron-Ogg-Shafarevich criterion for K3 surfaces
The classical Neron-Ogg Shafarevich criterion gives a criterion for good reduction of abelian varieties over a p-adic field K in terms of the Galois action on their Tate module. A similar criterion exists for curves by working instead with the unipotent fundamental group. For K3 surfaces this criterion fails in a rather surprising way: Liedtke and Matsumoto constructed examples of K3 surfaces over
*Q*(p ≥ 5) which only admit good reduction over the unique unramified quadratic extension of_{p}*Q*_{p}. Assuming potential semi-stable reduction, however, they showed that this is the only way it can fail. Namely, if X/K is a K3 surface and the inertia group I_{K}acts trivially on the etale cohomology of X, then X admits good reduction over a finite and unramified extension of K. Still assuming potential semi-stable reduction, we explain how to improve on this by showing that a K3 surface X/K has good reduction if and only if its second etale cohomology group is unramified, and moreover coincides (as a Galois representation) with the etale cohomology of a certain “canonical reduction” of X. This is joint work with B. Chiarellotto and C. Liedtke. - 15:45–16:45
- Nils Bruin Simon Fraser University, Genus 2 Jacobians with full level 3 structure
The family of genus 2 curves with fully marked 3-torsion on their Jacobians has been studied extensively in a classical setting. However, a few arithmetic questions remain. We prove that the family can be parametrized over Q and give a hyperelliptic model of the family, including the marking of divisors that mark the 3-torsion. This is joint work with Brett Nasserden.