
Miniworkshop arithmetic & moduli of K3 surfacesJanuary 31, UvA Amsterdam. Programme. Registration is compulsary. 
Intercity Number Theory SeminarMarch 10, Leiden. Snellius building, room 407/40913:3014:30  Carlo Pagano (Uiversiteit Leiden), Distribution of ray class groups: 4ranks and general model Abstract. 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 3torsion 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 CohenLenstra 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 4rank 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 4ranks 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.  14:4515:45  Peter Koymans (Universiteit Leiden), On the equation x + y = 1 in finitely generated groups in positive characteristic Abstract. Let 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.  16:0017:00  Robin de Jong (Universiteit Leiden), New results of effective Bogomolovtype for cycles on jacobians Abstract. 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 nonnegative real numbers: the NéronTate height h_{L}(Z) of Z, and the essential minimum e_{L}(Z) of Z. The NéronTate height of Z generalizes the NéronTate height of a point, as it occurs in for example the Birch and SwinnertonDyer conjectures, and measures in an intrinsic way the arithmetic complexity of Z. The essential minimum of Z is, roughly speaking, the liminf of the NéronTate 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 Bogomolovtype 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 AbelJacobi 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 SeminarMarch 24, Utrecht. Marinus Ruppertgebouw (Leuvenlaan 21, 3584 CE Utrecht), room: Paars13:0013:50  Lin Weng (Kyushu University / MPIM Bonn), NonAbelian Zeta Functions And Their Zeros Abstract. Nonabelian zeta functions are defined as integrations
over moduli spaces of semistable 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(n1,1) 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.  14:0014:50  Hatice Boylan (İstanbul Üniversitesi / MPIM Bonn), Fourier coefficients of Jacobi Eisenstein series over number fields Abstract. 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.  15:1516:05  Cyril Demarche (Paris 6  ENS), Localglobal principles for homogeneous spaces Abstract. 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 BrauerManin obstruction to those
localglobal 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
LucchiniArteche.  16:1517:05  Jehanne Dousse (Universität Zürich), Refinement of partition identities and the method of weighted words Abstract. A partition of a positive integer n is a nonincreasing sequence of positive integers whose sum is n. A RogersRamanujan 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 RogersRamanujan 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 qseries identities.
After explaining the classical method of weighted words, we will present a new version using qdifference 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 SeminarApril 7, Groningen. Bernoulliborg 10512:0013:00  Mark Jeeninga (Groningen), Lenstra's epsilon: A curious periodicity in RevLex field extensions of degree p over F_{p}. Abstract. 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 pclosure of F_{p} by means of an ArtinSchreier 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).  13:3014:30  Ricardo Buring (Groningen), Relations among Kontsevich graph weight integrals. Abstract. The Kontsevich graph weights are period integrals whose values make Kontsevich's starproduct 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 gaugeequivalence), and we verify pictorially that the starproduct expansion is associative modulo ō(ℏ⁴) for every value of the 10 parameters. This is joint work with Arthemy Kiselev.  14:4515:45  Jan Steffen Müller (Oldenurg), Computing canonical heights on elliptic curves in quasilinear time. Abstract. 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
quasilinear 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 nonarchimedean terms. The main
innovation is an algorithm for the computation of the latter sum that
requires no integer factorization and runs in quasilinear time. This
is joint work with Michael Stoll.  16:0017:00  Ulrich Derenthal (Hannover), Manin's conjecture for a family of nonsplit del Pezzo surfaces Abstract. 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 SeminarApril 21, Nijmegen. Linnaeusgebouw, Heyendaalseweg 137, 6525 AJ, Nijmegen  room LIN7 11:3012:30  Lars Halle (Copenhagen), Motivic zeta functions of degenerating CalabiYau varieties Abstract. Let K = C((t)), and let X be a smooth projective Kvariety 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 Cvarieties. 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.  13:4514:45  Giuseppe Ancona (Strasbourg), Standard conjectures for abelian fourfolds Abstract. Let X be a smooth projective variety and V the finitedimensional Qvector 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 padic places. It turns out that this question is more tractable thanks to padic comparison theorems and the ShimuraTaniyama formula. Moreover, using a classical product formula for quadratic forms, the padic information will give us nontrivial information on the archimedean place. For instance, we will prove the original conjecture when X is an abelian variety of dimension (up to) 4.  15:1516:15  Yohan Brunebarbe (Zürich), Hyperbolicity of moduli spaces of abelian varieties Abstract. For any positive integers g and n, let A_{g}(n) be the moduli space of principally polarized abelian varieties with a leveln structure; it is a smooth quasiprojective 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 > 6g. Similar results are true more generally for quotients of bounded symmetric domains by lattices.  16:3017:30  Olivier Wittenberg (Paris), Zerocycles on homogeneous spaces of linear algebraic groups Abstract. (Joint work with Yonatan Harpaz.) The BrauerManin obstruction is conjectured to control the existence of rational points on homogeneous spaces of linear algebraic groups over number fields (a farreaching generalisation of the inverse Galois problem). We establish the zerocycle variant of this statement. 

Intercity Number Theory SeminarMay 19, UvA and VU Amsterdam. All lectures are in room A1.10, at Science Park 904.11:0012:00  Ana Caraiani (Bonn), Galois representations and torsion classes Abstract. 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 CalegariGeraghty 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.  13:0014:00  Arne Smeets (Njmegen), Pseudosplit fibres and the AxKochen theorem Abstract. In 1965 Ax and Kochen proved a famous theorem concerning the existence of padic 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 ColliotThélčne, 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 AxKochen theorem holds. This work is joint with Dan Loughran and Alexei Skorobogatov.  14:3015:30  Maxim Mornev (Leiden & Amsterdam), Shtuka cohomology and special values of Drinfeld modules Abstract. The motivic Tamagawa number conjecture gives a formula for special
values of Lfunctions 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
Lfunctions are replaced by Goss Lfunctions which take values in a
positive characteristic field, and one is interested in Goss Lfunctions
attached to Drinfeld modules. In the talk I will explain how special
values of these Lfunctions can be expressed in terms of shtuka
cohomology which plays the role of motivic cohomology in this story.  16:0017:00  Jeroen Sijsling (Ulm), Computing endomorphisms of Jacobians Abstract. 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 Z, 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 (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. 

DIAMANT SymposiumJune 2, Breukelen. This is part of a twoday event: June 12. 
Intercity Number Theory SeminarNovember 3, UvA and VU Amsterdam. All talks will be in the main building (Hoofdgebouw) at the VU Campus. The first talk in room HG05A24 followed by two talks in room HG15A16. Lunch will be provided in the latter room before the start of the second talk.12:0512:50  Julie Desjardins (Bonn), Density of rational points on elliptic surfaces Abstract. Let X be an algebraic surface. We are interested in the set of rational points X(â„š). Is it nonempty ? Is it infinite ? Is it Zariskidense ? We are concerned with elliptic surfaces, i.e. 1parameter 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 nonisotrivial 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.  13:1514:00  Jan Tuitman (Leuven), An update on effective Chabauty Abstract. The method of ChabautyColeman 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 MordellWeil 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.  14:0514:50  Giulio Orecchia (Leiden), A monodromy criterion for existence of Neron models Abstract. 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
toricadditivity) on the action of monodromy on the ladic Tate module
of the generic fibre, for a prime l invertible on the base. 

Intercity Number Theory SeminarNovember 17, Leiden. Snellius, room 312. (The Snellius restaurant will be closed; the restaurants in the Huygens and Gorlaeus buildings are open.)13:0014:00  Reinier Bröker (Brown University), Lower bounds for Hilbert class polynomials Abstract. Hilbert class polynomials (minimal polynomials of jinvariants 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.  14:3015:30  Christopher Lazda (Universiteit van Amsterdam), A NĂ©ronOggShafarevich criterion for K3 surfaces Abstract. The classical NeronOgg Shafarevich criterion gives a criterion for good reduction of abelian varieties over a padic 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} (p ≥ 5) which only admit good reduction over the unique unramified quadratic extension of Q_{p}. Assuming potential semistable 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 semistable 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 â€ścanonical reductionâ€ť of X. This is joint work with B. Chiarellotto and C. Liedtke.  15:4516:45  Nils Bruin (Simon Fraser University), Genus 2 Jacobians with full level 3 structure Abstract. The family of genus 2 curves with fully marked 3torsion 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 3torsion.
This is joint work with Brett Nasserden. 

DIAMANT SymposiumDecember 1, Breukelen. This is part of a twoday event: November 30December 1. 
DutchBelgian Algebraic Geometry DayDecember 15, Nijmegen. See the website. 

