The** Number Theory Seminar** usually takes place on Friday 14:15-15:15.

Follow us on www.n-cube.net as well!

**Program 2018-2019:**

**01.03:** Avi Kulkarni (MPI Leipzig)*Room:* Aud 10.*Notice:* at 12:15.*Title:* The arithmetic of uniquely trigonal genus 4 curves.*Abstract:* A uniquely trigonal curve is a smooth algebraic curve that has an essentially unique morphism to $\mathbb{P}^1$ of degree 3. The uniquely trigonal curves of genus 4 are closely related to del Pezzo surfaces of degree one. In this talk, we give two examples of how this connection can be used to generate interesting results in number theory. The first result pertains to class groups of cubic fields; we construct an infinite family of cubic number fields whose class groups have many 2-torsion elements. For the second application, we consider the uniquely trigonal genus 4 curves from the perspective of arithmetic invariant theory.

**Sho Tanimoto (Kumamoto)**

**08.02:***Room:*Store Aud. NEXS

*Notice:*at 14:30.

*Title:*Thin exceptional sets in Manin’s Conjecture

*Abstract:*Manin’s Conjecture predicts the asymptotic formula for the counting function of rational points on a Fano variety after removing the contribution from an exceptional set. We propose a geometric construction of this exceptional set and prove that this is a thin set in sense of Serre. This is joint work with Brian Lehmann and Akash Sengupta.

**08.02:**Neea Palojärvi (Turku).

*Room:*Store Aud. NEXS

*Notice:*at 13:15.

*Title:*On the Riemann-von Mangoldt formula for Selberg class functions.

*Abstract:*Riemann-von Mangoldt formula describes the number of the non-trivial zeros of the Riemann zeta function. There are generalized versions of the Riemann-von Mangoldt formula for other classes of functions. In this talk, I will discuss an explicit Riemann-von Mangoldt formula for functions in the Selberg class.

**24.01:**Ariyan Javanpeykar (Mainz).

*Room:*Aud 10.

*Notice:*at 13:00.

*Title:*Automorphisms of varieties with only finitely many rational points.

*Abstract:*Lang's conjecture on rational points of hyperbolic varieties predicts that a variety with only finitely many rational points has only finitely many automorphisms. We verify this prediction using the theory of dynamical systems in arithmetic geometry.

**16.01:**Ian Petrow (ETH).

*Room:*Aud 10.

*Notice:*at 15:15.

*Title:*The Weyl law for algebraic tori.

*Abstract:*Families are powerful tools across all of mathematics. This talk will be about families of automorphic representations and their applications in number theory. There is no canonical definition of a family of automorphic forms/representations, but several ad-hoc definitions have been proposed. Here, a basic but still difficult question is: given a reductive algebraic group G, how many irreducible automorphic representations of bounded conductor are there? I will present a complete answer to this question in the case that G is a torus defined over a number field.

**11.01:**Paul Helminck (Bremen).

*Room:*Aud 10.

*Title:*Decompositions of tame fundamental groups of nonarchimedean curves using metrized complexes

*Abstract:*In this talk, I’ll discuss a natural functor from the category of tame étale coverings of a punctured nonarchimedean curve to the category of tame étale coverings of a metrized complex associated to the punctured curve. In its simplest form, this functor takes a covering of algebraic curves and assigns to it a covering of intersection graphs arising from a morphism of semistable models. By considering the enhanced category of coverings of metrized complexes with gluing data, we then show that we obtain an equivalence of categories, yielding a natural notion of a profinite fundamental group for metrized complexes since the categories involved are Galois categories.

**08.01:**Peter Koymans (Leiden).

*Room:*Aud 10.

*Notice:*At 13:00.

*Title:*The spin of prime ideals and applications.

*Abstract:*Let be a cyclic, totally real extension of $\mathbb{Q}$ of degree at least and let $\sigma$ be a generator of Gal(K/$\mathbb{Q}$). We further assume that the totally positive units are exactly the squares of units. In this case, Friedlander, Iwaniec, Mazur and Rubin define the spin of an odd principal ideal to be spin(s,a)=(alpha/s(a))_K

** 14.12: **Eugenia Rosu (MPIM)

*Room:*Aud 8.

*Title:*Special cycles on orthogonal Shimura varieties

*Abstract:*I will be talking about joint work with Yott. Extending on the work of Kudla-Millson and Yuan-Zhang-Zhang, we are constructing special divisors for a specific $GSpin$ Shimura variety. We further construct a generating series that has as coefficients the cohomology classes corresponding to the special divisors classes $Z(x, g)_K$ on the $GSpin$ Shimura variety $M_K$ and show the modularity of the generating series in the cohomology group over $C$.

**07.12:**Fabien Pazuki (KU)

*Room:*Aud 10.

*Notice:*At 13:00.

*Title:*Regulators of elliptic curves.

*Abstract:*In a recent collaboration with Pascal Autissier and Marc Hindry, we prove that up to isomorphisms, there are at most finitely many elliptic curves defined over a fixed number field, with Mordell-Weil rank and regulator bounded from above, when the rank is at least 4. We will explain where the result comes from, and discuss links with the Birch and Swinnerton-Dyer conjecture and with asymptotics on the number of rational points of bounded height on elliptic curves.

**26.10:**Laurent Vuillon (Univ Savoie Mont Blanc)

*Room:*Aud 10.

*Notice:*At 13:15.

*Title:*Combinatorics on words for Markoff numbers.

*Abstract:*Markoff numbers are fascinating integers related to number theory, Diophantine equation, hyperbolic geometry, continued fractions and Christoffel words. Many great mathematicians have worked on these numbers and the 100 years famous uniqueness conjecture by Frobenius is still unsolved. In this talk, we state a new formula to compute the Markoff numbers using iterated palindromic closure and the Thue-Morse substitution. The main theorem shows that for each Markoff number m, there exists a word $v\in\{a, b\}^*$ such that $m − 2$ is equal to the length of the iterated palindromic closure of the iterated antipalindromic closure of the word $av$. This construction gives a new recursive construction of the Markoff numbers by the lengths of the words involved in the palindromic closure.

**11.10:**Charles Doran (University of Alberta)

*Room:*DIKU 1034.

*Title:*Arithmetic Mirror Symmetry for K3 Pencils and Hypergeometric Decomposition.

*Abstract:*We will begin with an introduction to Calabi-Yau geometry and survey various forms of mirror symmetry, with an eye towards possible applications in arithmetic geometry. Specializing to a particular construction of mirror pairs, we analyze the factorization of the zeta function for certain symmetric K3 quartic pencils and the relationship with hypergeometric equations.

**14.09:**Peter Humphries

*Room:*Aud 8.

*Title:*The random wave conjecture and arithmetic quantum chaos.

*Abstract:*Berry's random wave conjecture is a heuristic that the eigenfunctions of a classically ergodic system ought to display Gaussian random behaviour, as though they were random waves, in the large eigenvalue limit. We discuss two aspects of this problems for eigenfunctions of the Laplacian on a particular number-theoretic negatively curved surface: Planck scale mass equidistribution, and an asymptotic for the fourth moment. We will highlight how the resolution of these two problems in this number-theoretic setting involves a delicate understanding of the behaviour of certain families of L-functions.

**03-07.09:**Conference Arakelov 2018

*Room:*Aud 1, Frue Plads 4.

The aim of the meeting is to gather experts studying various questions concerning Arakelov geometry and connected topics. It is a yearly tradition, with main coordinators Burgos Gil, Maillot, Moriwaki, ... The Seminar is organized for the first time in Copenhagen this year! For more information: n-cube.net/Arakelov2018.html.

**27-31.08: **Masterclass: Mahler measure and special values of L-functions*Room:* Aud 10, Aud 8.

This Masterclass is meant to illustrate to young researchers the latest developments in the study of special values of L-functions and its connection to Mahler’s measure, polylogarithms, hypergeometric functions and K-theory.

**14.09:**Nadim Rustom (Koc University)

*Room:*Aud 10.

*Title:*New congruences for eigenforms of level 1.

*Abstract:*In this talk, we will state recently discovered congruences satisfied by Hecke eigenforms of level 1. These congruences extend previous results of Hatada (1981), and prove a conjecture of Coleman-Stein (2003). They can be seen as further evidence for a conjecture, formulated jointly with Kiming and Wiese (2016), on the finiteness of congruence classes of eigenforms of fixed level modulo prime powers. The proof uses Merel's description of the action of Hecke operators on modular symbols. We will give a brief sketch of Merel's modular symbol formalism in level 1 and describe the algorithm used to obtain these new congruences.

**Program 2017-2018:**

**11.06:**Martin Speirs (KU)

*Room:*Aud 4.

*Title:*K-theory of coordinate axes in characteristic p.

*Abstract:*I will talk about work in progress on computing the algebraic K-theory of the singular affine curve given by the coordinate axes in affine space over F_p-algebras. In the analogous case over the rational

**14.05:**

**Ganda Day in Copenhagen**

*Room:*Aud 2.

Three recent results in Number Theory will be presented by members of the IRN GANDA, a partnership between France, Denmark, South Africa and Madagascar.

- Jean-Marc Deshouillers (Bordeaux): Arithmetic and automatons.
- Ian Kiming (Copenhagen): Modular forms modulo prime powers.
- Florian Luca (Johannesburg): Prime factors of interesting integers.

More information on http://www.n-cube.net/GandADays2018.html.

**11.05:**Somnath Jha (IIT Kanpur)

*Room:*Aud 8.

*Title:*A duality result for Selmer groups.

*We will discuss a duality result for Selmer groups over $p$-adic Lie extensions. An important ingredient in this is a twisting lemma, which will also be discussed. This talk is based on joint work with T. Ochiai and G. Zabradi.*

*Abstract:***07.05:**Noriko Yui (Queen's University)

*Room:* Aud 9.*Title:* Supercongruences for rigid hypergeometric Calabi-Yau threefolds*Abstract: *We give two proofs of the supercongruences for the fourteen rigid hypergeometric Calabi-Yau threefolds defined over Q. The existence of such supercongruences was conjectured (based on numerical evidence) by F. Rodriguez-Villegas in 2003. This is a joint work with Ling Long, Fang-Ting Tu and Wadim Zudilin.

**30.03:** Linda Frey (Basel)*Room:* Aud 10.*Title:* Explicit Small Height Bound for $\mathbb{Q}(E_{tor})$.*Abstract: *Let $E$ be an elliptic curve defined over $\mathbb{Q}$. We will show that there exists an explicit constant $C>0$ which is only dependent on the conductor and the $j$-invariant of $E$ such that the absolute logarithmic Weil height of a non-zero $a\in{\mathbb{Q}(E_{tor})-\mu}$ is always greater than $C$ where $E_{tor}$ denotes all the torsion points of $E$ and $\mu$ are the roots of unity.

**11.01:** Cody Lee Gunton (Arizona)*Room:* Aud 5.*Notice: *On Thursday at 13:00.*Title:* Néron component groups and crystalline representations*Abstract: *Let $A_K$ be an abelian variety over a finite extension $K$ of the $p$-adic numbers. Attached to $A_K$ are its Néron model $A$, a smooth group defined over the integers $O$ of $K$, and its $p$-adic Tate module $T$, a representation of the absolute Galois group of $K$. The reduction of $A$ modulo the prime ideal of $O$ may not be connected; let $\Phi$ be its component group. There is a subtle and important relationship between the geometry of $A$ and the Galois representation $T$; for instance, it is a result of Coleman and Iovita that one can detect from $T$ whether or not $A$ is proper. In this talk I will present my work, which builds on work of Kim-Marshall in the case of $K$ unramified, showing that the $p$-power torsion in $\Phi$ can be calculated from the cohomology of $T$ if $A$ is semiabelian. Along the way, I will give examples to highlight issues relating to ramification in $K$, and will explain how these issues can be addressed using $p$-adic uniformization and recent work in torsion $p$-adic Hodge theory.

** 9.01: **Alexander Walker (Brown)

*Room:*aud 7.

*Notice:*On Tuesday at 15:15.

*Title:*Second Moment Results in the Gauss Circle Problem.

*Abstract:*The Gauss circle problem is a classic problem in analytic number theory which concerns estimates for the number of lattice points enclosed by a circle of large radius. Improved error bounds in this problem have traditionally come from refinements to the Hardy-Littlewood circle method. In this talk, I will describe a new attack on the Gauss circle problem (and generalizations) based on the meromorphic properties of a type of Dirichlet series called a shifted convolution sum. This talk incorporates joint work with Tom Hulse, Chan Ieong Kuan, and David Lowry-Duda.

**3.01:** Jitendra Bajpai (Göttingen)*Room:* aud 8.*Notice: *On Wednesday at 13:15.*Title:* Arithmeticity and Thinness of hypergeometric groups.*Abstract: *The monodromy groups of hypergeometric differential equations of type nFn-1 are often called hypergeometric groups. These are subgroups of GL_n. Recently, Arithmeticity and Thinness of these groups have caught a lot of attention. In the talk, a gentle introduction and recent progress to the theory of hypergeometric groups will be presented.

**15.12:** Richard Griffon (Leiden).*Room: *Aud 6.*Title:* Bounds on special values of L-functions of elliptic curves in an Artin-Schreier family.*Abstract: *L-functions of elliptic curves over global fields conjecturally encode a lot of information about their arithmetic. In general though, even for elliptic curves $E$ over $F_q(t)$, little is known about their L-functions $L(E, s)$. For example, consider the first non-zero coefficient $L*(E,1)$ in the Taylor expansion of $L(E,s)$ around the point $s=1$ (the "special value"): the size of $L*(E,1)$ remains elusive. One expects that $L*(E, 1)$ is “generically” as big as it possibly can when compared to the conductor of $E$, but this has only been proved in a very limited number of cases. In this talk, I will report on a work in progress about an infinite family of elliptic curves $E$ in an Artin-Schreier family over $K=F_q(t)$. I computed their L-functions explicitly, and I am able to deduce a very precise asymptotic bound on $L*(E,1)$ in terms of the conductor of $E$. Via the Birch and Swinnerton-Dyer conjecture (which is a theorem in this case), one can translate this bound into an asymptotic estimate of certain arithmetic invariants of these elliptic curves $E$.

** 8.12: **DADA Day in Aarhus: http://www.n-cube.net/DADADay2017.html.

** 6.12: **Tiago Jardim da Fonsecca (Paris-Orsay).

*Room:*Aud 8.

*Title:*Higher Ramanujan equations and periods of abelian varieties.

*Abstract:*

*The Ramanujan equations are certain algebraic differential equations satisfied by the classical Eisenstein series $E_2$, $E_4$, $E_6$. These equations play a pivotal role in the proof of Nesterenko's celebrated theorem on the algebraic independence of values of Eisenstein series, which gives in particular a lower bound on the transcendence degree of fields of periods of elliptic curves. Motivated by the problem of extending Nesterenko's transcendence methods to other settings, we shall explain how to generalize Ramanujan's equations to higher dimensions via a geometric approach, and how the values of a particular solution of these equations relate with periods of abelian varieties.*

**24.11: **Alex Kemarski (Copenhagen)*Room:* Aud 10.*Title:* On Galois cohomology.*Abstract:* Following Serre, "Galois cohomology" book, I will explain the notion of a field of type (F), prove that every p-adic field is of type (F) and state the Borel-Serre theorem about finiteness of orbits (theorem 4.4.5 from the book). See also the reference https://www.math.u-psud.fr/~harari/enseignement/cogal/poly.pdf.

**15-17.11:** Conference in Copenhagen: Rational Points and Zariski Density.

**10.11: **Riccardo Pengo (Copenhagen)

*Room:*Aud 8.

*Title:*An adelic description of modular curves.

*Abstract:*The aim of this talk will be to survey the relationships between modular curves and the ring of adèles of the rational numbers. The first ones are moduli spaces of elliptic curves with a prescribed torsion structure, which play a key role in the study of modular forms and in the proof of the modularity theorem. On the other hand the ring of adèles is a really algebraic object aiming to put together all the possible completions of a global field, which is used in many different contexts: class field theory, local-global principles, automorphic forms. We will see how affine modular curves can be described using adèles and we will explain our attempt to generalize this construction to different compactified versions of modular curves.

**03.11:** Simon Kristensen (Aarhus)

*Room:*Aud 8.

*Title:*Diophantine approximation and applications - some ways in which Hardy was wrong

*Abstract:*Hardy famously said: " The Theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics". This has of course been proven wrong several times, especially with the advent of cryptography. In this talk, we will see examples of applications of recent and classical results in Diophantine approximation. We will put particular emphasis on a surprising (at least to the speaker) application of recent results obtained in collaboration with Mumtaz Hussain to wireless communication and the so-called MIMO (Multiple Input and Multiple Output) model, in which one communicates simultaneously on many transmitting as well as receiving channels.

**27.10: **Marc Hindry (Paris).

*Room:*Aud 6.

*Title:*Analogue of Brauer-Siegel theorem for abelian varieties: similarities and differences.

*Abstract:*The classical Brauer-Siegel theorem states that for a sequence of number fields with, say, bounded degree, the product of the class number by the regulator of units behaves asymptotically like the square root of the discriminant. The analogue for abelian varieties of a given dimension defined over a global field (a number field or a function field over a finite field) replaces the three quantities by respectively the cardinality of the Shafarevich-Tate group, the Néron-Tate height regulator and the exponential height. I will describe all the objects involved, discuss the possible analogues and explain why a similar upper bound is "almost certain" and a similar lower bound unlikely.

** 20-21.10: **Conference in Stockholm: N-cube Days VII.

**13.10: **Dustin Clausen (Copenhagen)

*Room:*Aud 8.

*Title:*The algebraic topology of p-adic Lie groups.

*Abstract:*Lazard(-Serre) proved that the cohomology of p-adic Lie groups satisfies a version of Poincare duality. This indicates an analogy between p-adic Lie groups and real manifolds. I will explain some results which develop this analogy further.

**06.10: **Daniel Bergh (Copenhagen)

*Room:*Aud 8.

*Title:*Conservative descent for semi-orthogonal decompositions.

*Abstract:*Semi-orthogonal decompositions arise naturally in many situations in the study of derived categories in algebraic geometry. For instance, there are naturally associated semi-orthogonal decompositions of the derived categories of blow-ups, projective bundles, Brauer-Severi schemes, gerbes and root stacks.

I will give a brief introduction to the subject. I will also present some ongoing work (joint with Olaf Schnürer) on how to construct semi-orthogonal decompositions locally, using a technique we call conservative descent. In particular, this allows us to prove existence of the semi-orthogonal decompositions for the geometric constructions mentioned above to the more general context of algebraic stacks, as well as remove some regularity and finiteness assumptions.

** 29.09: **Lars Halle (Copenhagen)

*Room:*Aud 8.

*Title:*Degenerations of Hilbert schemes of points.

*Abstract:*I will present a 'good' compactification of the relative Hilbert scheme of points associated to the smooth locus of a (simple) degeneration of varieties. This compactification is obtained by GIT-methods, and yields a new, refined, approach to earlier constructions of J. Li and B. Wu.

I will also discuss a few applications to degenerations of irreducible holomorphic symplectic varieties.

This is joint work with M. Gulbrandsen and K. Hulek (and partly with Z. Zhang).

**22.09:** Sho Tanimoto (Copenhagen)*Room:* aud 7.*Notice: *At 13:00.*Title:* Rational curves on Fano 3-folds.*Abstract:* Brian Lehmann and I have been studying the geometry of exceptional sets in Manin’s conjecture using birational geometry. Recently we found applications of this study to the study of the space of rational curves. In the upcoming event ‘'Rational points and Zariski density’’, I will talk about these developments focusing on the motivation and the theoretical framework. In this talk, I would like to explain how the geometry of exceptional sets helps to understand dimension and number of components of moduli spaces of rational curves by working out some examples of Fano 3-folds. This is joint work with Brian Lehmann.

**15.09: **Fabien Pazuki (Copenhagen)*Room:* aud 9.*Title:* Elliptic curves and isogenies.*Abstract:* Two elliptic curves $E$ and $E'$ defined over a number field $K$

are isomorphic over the algebraic closure of $K$ if and only if they

have the same j-invariant. A natural question is: how is this

invariant transformed by general isogenies? We prove a new height

bound on the difference of heights of the j-invariants of isogenous

elliptic curves, and derive several consequences, for instance bounds

for the height of modular polynomials and for Vélu's formulas. If time

permits, we will add a remark on Mordell-Weil ranks of elliptic curves.

**21-25.08:** International PhD course: Representation theory - Number theory (Dipendra Prasad, Gautam Chinta, Alexei Entin).

**Program 2016-2017:**

**26-30.06: **Masterclass with Matthew Morrow (Jussieu-Paris) and Thomas Nikolaus (MPI Bonn): Stable Homotopy Theory and p-adic Hodge Theory.

** 19.06: **Hartmut Monien (Bonn)

*Room:*aud 9.

*Notice:*On Monday at 14:15.

*Title:*Belyi maps of non-congruence subgroups of the modular group associated to sporadic groups.

*Abstract:*Determining Fourier coefficients of modular forms of a finite index noncongruence subgroups of $PSL_2(\mathbb{Z})$ is still a non-trivial task. In my talk I will describe a new algorithm to reliably calculate an approximation for a modular form of given weight. As an application we have calculated Belyi maps which realize sporadic groups as the Galois group of three-point ramified cover over $\mathbb{P}^1(\mathbb{C})$ for all genus zero sporadic groups.

**15.06: **Ariyan Javanpeykar (Mainz)*Room: *aud 5.*Notice: *On Thursday at 13:15.*Title: *Arithmetic, algebraic, and analytic hyperbolicity.*Abstract: *The Lang-Vojta conjecture predicts that a complex algebraic variety is algebraically hyperbolic if and only if it is analytically hyperbolic if and only if it is arithmetically hyperbolic. This conjecture is known for the moduli space of polarized abelian varieties by work of Borel, Faltings, and Zuo. In this talk, I will explain this conjecture, discuss its consequences for smooth hypersurfaces and Calabi-Yau varieties, and present a p-adic extension of the Lang-Vojta conjecture for the moduli space of polarized abelian varieties, using Scholze's theory of perfectoid spaces. Finally, I will discuss the conjecture for the moduli space of canonically polarized varieties.

** 13.06: **Peter Sarnak (Princeton)

*Room:*aud 4.

*Notice:*On Tuesday at 15:15, joint with Harald Bohr Lecture.

*Title:*Navigating $U(2)$ with Golden Gates.

*Abstract:*The problem of devising optimally efficient universal gates for quantum computing is one of finding the best generators for rotation groups. We will discuss recent developments concerning ‘Golden Gates,’ which are number theoretic generators of $U(2)$. The tools range from groups associated with the platonic solids to modern diophantine problems of sums of squares.

** 19.05: **Yohan Brunebarbe (Univ Zürich)

*Room:*aud 8.

*Notice:*On Friday at 13:15.

*Title:*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 level-n structure (it is a smooth quasi-projective variety for n>2). Building on works 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.

**15.05: **Yiannis N. Petridis (UCL/UCPH)*Room: *aud 10.*Notice: *On Thursday at 12:45, joint with Geometry and Analysis Seminar.*Title: *Arithmetic Statistics of modular symbols.*Abstract: *Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. In joint work with Morten S. Risager we prove these on average using analytic properties of Eisenstein series twisted by modular symbols. We also prove another conjecture predicting the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps.

**11.05: **Sho Tanimoto (KU)*Room: *aud 8.*Notice: *On Thursday at 15:15, joint with Geometry and Analysis Seminar.*Title: *Zero-loci of Brauer elements on semisimple groups.*Abstract: *Suppose that we have a family of smooth projective varieties defined over a number field k whose base is a projective space. One can consider the number of varieties in the family containing a k-rational points and its asymptotic formula. Such a problem was first considered by Serre for a family of conics. In this talk I would like to explain a similar problem for certain families of varieties over semisimple groups and a solution to that problem using spectral theory of automorphic forms. This is joint work with Daniel Loughran and Ramin Takloo-Bighash.

**8.05-11.05: **Master Class: Cohomology of Arithmetic Groups.

**8.05: **Herbert Gangl (Durham University)*Room: *aud 8.*Notice: *On Monday at 15:15, joint with Algebra and Topology Seminar.*Title: *Zagier's polylogarithm conjecture revisited.*Abstract: *In the early nineties, Goncharov proved the weight 3 case of Zagier's Conjecture stating that the special value $\zeta_F(3)$ of a number field $F$ is essentially expressed as a determinant of trilogarithm values taken in that field. He also envisioned a vast--partly conjectural--programme of how to approach the conjecture for higher weight. We can remove one important obstacle in weight 4 by solving one of Goncharov's conjectures. It further allows us to deduce a functional equation for $Li_4$ in four variables as one expects to enter in a more explicit definition of $K_7(F)$.

** 5.05: **Henrik Schlichtkrull (KU)

*Room:*aud 8.

*Notice:*On Friday at 12:30, joint with Geometry and Analysis Seminar.

*Title:*Asymptotic density of integer points on wavefront spherical spaces.

*Abstract:*Given a homogeneous space Z of an algebraic real reductive group G and an orbit D in Z of a discrete subgroup of G with cofinite volume, and given an increasing and exhaustive family of compact subsets B_R (called balls) of Z, one wants to determine the expected number of points from D in B_R for large R. For the Euclidean plane Z this is the famous circle problem of Gauss. The talk concerns the solution of this problem, including an error estimate, for a particular family of homogeneous spaces and a geometrically defined family of balls.

** 3.03: **David Holmes (Leiden)

*Room:*aud 10.

*Notice:*On Friday at 13:15.

*Title:*Rational torsion points and String theory.

*Abstract:*I will try to explain how I started working to understand rational torsion points on abelian varieties, and ended up studying problems related to string theory. The key player in this story is the double ramification cycle (DRC). For a number theorist, the DRC can behave like a generalisation of a modular curve to abelian varieties of higher dimension, and understanding its rational points would give a lot of information about rational torsion points. The DRC is also of interest in enumerative geometry, for example as a prototype for defining Gromov-Witten invariants of Artin stacks. After explaining some of this story, I will describe new results on compactifications and integral models of the DRC.

**27.02: **Joseph Ayoub (Zurich)*Room: *aud 8.*Notice: *On Monday at 15:15, joint with Algebra/Topology seminar.*Title: *Motivic Galois groups and periods.*Abstract: *I'll explain the construction of a motivic Galois group $G_{mot}(k,\sigma)$ associated to a field $k$ endowed with a complex embedding $\sigma$. I'll discuss some known results and some conjectures. In particular, I'll describe the conjectural relation to periods of algebraic varieties and the non conjectural relation to solutions of Gauss-Manin connections.

**20.02:** Dustin Clausen (Copenhagen)*Room: *aud 8.*Notice: *On Monday at 15:15, joint with Algebra/Topology seminar.*Title: *A general approach to Artin maps.*Abstract: *If $F$ is a global field, then the Artin map for $F$ is a certain homomorphism from $(A_F)^*/F^*$ to the abelianized absolute Galois group of $F$. Its very existence implies the Artin reciprocity law, a generalization of the quadratic reciprocity law. But there are other fields of arithmetic significance than the global fields, and many of them also have Artin maps. For example, if $F$ is a local field, then the Artin map has source $F^*$ instead. We will describe a way to produce these Artin maps which is uniform in the field, and works even in much more general situations, e.g. for an arbitrary non-commutative ring. A consequence is a new proof of the Artin reciprocity law. The methods are homotopy theoretic, based on a simple topological construction with tori and two new kinds of K-theory.

**20.01: **Giovanni Rosso (Cambridge)*Room: *aud 7.*Notice: *On Friday at 13:15.*Title: *Eigenvarieties for non-cuspidal Siegel modular forms.*Abstract: *Since the seminal work of Serre and Swinnerton-Dyer, people have been interested in congruences between eigenforms. After recalling some foundational results of Hida and Coleman on p-adic families of modular forms, we shall explain how Andreatta, Iovita and Pilloni constructed families of cuspidal Siegel forms and how their work can be generalised to non-cuspidal forms. This is joint work with Riccardo Brasca.

**21.12: **Farbod Shokrieh (Cornell Univ.)*Room: *aud 5.*Notice: *On wednesday at 14:00.*Title: *Metric graphs, potential theory, and applications.*Abstract: *A metric graph can be viewed, in many respects, as a non-Archimedean analogue of an algebraic curve. For example, there is a notion of Jacobian for graphs. More classically, metric graphs can be viewed as electrical networks. I will discuss the interplay between these two points of view, and discuss some applications.

** 21.12: **Johannes Nicaise (Imperial College)

*Room:*aud 5.

*Notice:*On wednesday at 13:00.

*Title*

*:*

*From elementary number theory to string theory and back again.*

*Abstract:*I will describe some surprising interactions between number theory, algebraic geometry and mirror symmetry that have appeared in my recent work with Mircea Mustata and Chenyang Xu and that have led to a solution of Veys' 1999 conjecture on poles of maximal order of Igusa zeta functions. The talk will be aimed at a general audience and will emphasize some key ideas from each of the fields involved rather than the technical aspects of the proof.

**16.12**: Ziyu Zhang (Hannover) *Room:* aud 8.*Notice:* On Friday at 11:00.*Title:* Holomorphic symplectic manifolds among Bridgeland moduli spaces*Abstract:* We consider moduli spaces of semistable complexes on a projective K3 surface with respect to generic Bridgeland stability conditions. Similar to the sheaf case, the smooth ones among them are holomorphic symplectic manifolds, and the 10-dimensional singular ones admit symplectic resolutions. I will explain why these examples of holomorphic symplectic manifolds are all deformation equivalent to the known ones. By generalizing the prominent work of Bayer and Macri, we can also study the birational geometry of the 10-dimensional singular moduli spaces via wall-crossing on the stability manifold. This is a joint work with C.Meachan.

**9.12: **Marius Leonhardt (Univ. of Cambridge)*Room: *aud 8.*Notice: *On Friday at 13:15.*Title: *Galois characteristics of local fields.*Abstract: *What characteristics of a field can be deduced from its absolute Galois group?Does the Galois group uniquely determine the field? It turns out that the answer to this question depends on the "type" of field. For example, any two finite fields have isomorphic absolute Galois groups, whereas two number fields are isomorphic if and only if their Galois groups are.

In the case of finite extensions of Q_p, there are non-isomorphic fields with isomorphic Galois groups. However, if one requires the group isomorphism to respect the filtration given by the ramification subgroups, then S. Mochizuki has shown that one can fully reconstruct the field.

In this talk I will give an overview of the methods involved in Mochizuki's proof, focusing on the use of Hodge-Tate representations in the construction of an isomorphism between two given fields.

**28.11: **Hui Gao (Helsinki)*Room: *4.4.20*.**Notice: *On monday at 13:15.*Title: *Overconvergence of \'etale $(\varphi, \tau)$-modules*Abstract: *The category of \'etale $(\varphi, \tau)$-modules, similar as the category of \'etale $(\varphi, \Gamma)$-modules, is equivalent to the category of $p$-adic Galois representations. A classical theorem of Cherbonnier-Colmez says that all \'etale $(\varphi, \Gamma)$-modules are overconvergent. In this talk, we show that all \'etale $(\varphi, \tau)$-modules are also overconvergent. Our method is completely different from that of Cherbonnier-Colmez. The key idea is a certain crystalline approximation technique. This is joint work with Tong Liu.

**8.11: **Robin de Jong (Leiden)*Room: *Aud 10*.**Notice: *On Tuesday at 16:15.*Title: *Neron-Tate heights of cycles on jacobians.*Abstract: *We discuss a method to calculate explicitly the Neron-Tate height of

certain integral cycles on jacobians of curves defined over global fields.

For example, we obtain closed expressions for the Neron-Tate height of the

difference surface, the Abel-Jacobi image of (the square of) the curve,

and of any symmetric theta divisor on the jacobian. We discuss

applications to the effective Bogomolov conjecture. The results are

phrased in terms of arithmetic intersection theory on the curve.

**28.10: **Brian Lehmann (Boston College)*Room:* Aud 10.*Title:* Positivity for curves.*Abstract: *The best way to capture the geometry of a divisor is to study the asymptotic behavior of sections of multiples of the divisor. This leads to a rich theory of "positivity" relating asymptotic and intersection-theoretic invariants. I will discuss recent progress in understanding the analogous picture for curves. Much of this work was done jointly with Mihai Fulger or Jian Xiao.

**14.10:** Jesse Jääsaari (University of Helsinki)*Room:* Aud 10.*Notice:* from 13:15 to 14:15.*Title:* On exponential sums involving Fourier coefficients of automorphic forms.*Abstract:* In this talk I will discuss moments and pointwise bounds for exponential sums involving Fourier coefficients of automorphic forms. This is joint work with Esa V. Vesalainen.

**Program 2015-2016:**

**15.06: **Jasper Van Hirtum (KU Leuven and Univ. Luxembourg)*Room:* Aud 8.*Notice:* on Wednesday from 10:00 to 11:00.*Title:* On the distribution of Frobenius of weight 2 eigenforms with quadratic coefficient field.*Abstract: *The coefficients of a modular form without so called inner twists are elements of a totally real number field. If this number field is different from Q then one can study the set of primes p such that the p-th coefficient is a rational number. This set is known to be of density zero. However only conjectural statements exists on its size. Using the latest results on the Sato-Tate conjecture for Abelian varieties we obtain a heuristic model for the asymptotic size of this set under reasonable assumptions. More precisely we treat the case of weight 2 eigenforms with quadratic coefficient field without inner twist. Moreover we present numerical data which agrees with our model and the assumptions we made to obtain it*.*

** 9.06: **Olivier Taïbi (Imperial)

*Room:*Aud 5.

*Notice:*on Thursday from 13:15 to 14:15.

*Title:*Arthur's multiplicity formula for certain inner forms of special orthogonal and symplectic groups.

*Abstract:*I will explain the formulation and proof of Arthur's multiplicity formula for automorphic representations of special orthogonal groups and certain inner forms of symplectic groups G over a number field F. I work under an assumption that substantially simplifies the use of the stabilisation of the trace formula, namely that there exists a non-empty set S of real places of F such that G has discrete series at places in S and is quasi-split at places outside S, and restricting to automorphic representations of G(A_F) which have algebraic regular infinitesimal character at the places in S. In particular, this proves the general multiplicity formula for groups G such that F is totally real, G is compact at all real places of F and quasi-split at all finite places of F. Crucially, the formulation of Arthur's multiplicity formula is made possible by Kaletha's recent work on local and global Galois gerbes and their application to the normalisation of Kottwitz-Langlands-Shelstad transfer factors. I will also explain why these particular hypotheses are often enough to handle arithmetic applications.

**3-4.06: **N-cube Days IV**.**

**31.05: **Lance Gurney (Univ. Amsterdam)*Room:* Aud 10.*Notice:* on Tuesday from 16:15 to 17:15.*Title:* Arithmetic of the moduli space of CM elliptic curves.*Abstract:* In this talk I will describe certain arithmetic structures possessed by the moduli space M of elliptic curves with CM by the ring of integers O_K of a fixed imaginary quadratic field K. In the first part of the talk I will explain how M admits a commuting family of endomorphisms \psi_P: M --> M lifting the various Frobenius endomorphisms modulo each prime P of O_K. Exploiting this structure leads to generalisations of results of Gross on the existence of global minimal models and of Taylor-Cassou-Nogues on the monogenicity of the rings of integers of the ray class fields of K. In the second part of the talk I will explain how the existence of the endomorphisms \psi_P is a consequence of the fact that M admits a Lambda-O_K-structure (a notion due to Borger). This implies the existence of a global, multi-prime theory of canonical lifts for CM elliptic curves analogous to the usual theory of Serre-Tate for ordinary elliptic curves. This allows for two curious constructions. The first is an infinite dimensional, smooth and affine cover of M and the second is an exact sequence related to both p-adic and complex periods.

** 20.05: **Daniel Loughran (Hannover University)

*Room:*Aud 5.

*Title:*Fibrations with few rational points.

*Abstract:*In this talk, we study the problem of counting the number of varieties in fibrations over projective spaces which contain a rational point. We obtain geometric conditions that force very few of the varieties in the family to contain a rational point, in a precise quantitative sense. This generalises the special case of conic bundles treated by Serre.

**20.05:** Taylor Dupuy (Hebrew University) *Room:* Aud 5.*Notice:* on Friday from 13:00 to 14:00.*Title:* Some constructions used in Mochizuki's IUT papers. *Abstract: *I will touch on various constructions that appear in Mochizuki's IUT papers. Among other things, I will explain what an Etale Theta class is and the relevance of this construction in his approach to the ABC conjecture..

** 19.05: **Jan Steffen Müller (University Oldenburg)

*Room:*Aud 3.

*Notice:*on Thursday from 14:15 to 15:15.

*Title:*Canonical heights on Jacobians of curves of genus two.

*Abstract:*To find explicit generators for the Mordell-Weil group of an abelian variety over a global field, one needs algorithms to compute canonical (i.e. N\'eron-Tate) heights of rational points and to enumerate all rational points of bounded canonical height. In my talk, I will discuss how this can be done efficiently for Jacobians of curves of genus 2. The idea is to decompose the difference between the canonical height and the corresponding naive height into a sum of local error functions. The non-archimedean local error functions can be analyzed using Picard functors and Zhang's theory of harmonic analysis on reduction graphs. This is joint work with Michael Stoll.

** 19.05: **Christelle Vincent

*Aud 4.*

*Room:**on Thursday from 13:00 to 14:00.*

*Notice:**Computing equations of hyperelliptic curves whose Jacobian has CM*

*Title:**.*

It is known that given a CM sextic field, there exists a non-empty finite set of abelian varieties of dimension 3 that have complex multiplication by this field. Under certain conditions on the field and the CM type, this abelian variety can be guaranteed to be principally polarizable and simple. This ensures that the abelian variety is the Jacobian of a hyperelliptic curve or a plane quartic curve.

*Abstract:*In this talk, we begin by showing how to generate a full set of period matrices for each isomorphism class of simple, principally polarized abelian variety with CM by a sextic field K. We then show how to determine whether the abelian variety is a hyperelliptic or plane quartic curve. Finally, in the hyperelliptic case, we show how to compute Rosenhain invariants for the curve.

This is joint work with J. Balakrishnan, S. Ionica and K. Lauter.

**13.05:** Samuel Edwards (Uppsala University) *Room:* Aud 5.*Notice:* on Friday from 13:15 to 14:15.*Title: *Effective equidistribution of horospherical orbits.*Abstract: *We will review equidistribution properties of expanding translates of horospherical orbits on homogeneous spaces, and connections to a few problems of a number-theoretic nature. In the course, we will discuss various techniques (both spectral and ergodic) of obtaining rates of effective equidistribution, and report on recent progress in generalizing a method of Burger*.*

**2.05:** Jim Stankewicz (University of Bristol) *Room:* Aud 6.*Notice:* on Monday from 16:15 to 17:15.*Title:* Coleman's conjecture and abelian surfaces.*Abstract: *In his Disquisitiones over 200 years ago, Gauss made his famous class number conjecture. In modern language this conjecture implies that there are only finitely many endomorphism rings of elliptic curves over the rational numbers. This conjecture was finally proven in the mid 1900s by Heegner, Baker, and Stark. In this talk we will review a higher dimensional analogue of this conjecture, due to Coleman. We will also give some new results and present some possible next steps in the study of Coleman's conjecture.

**29.04: **Rachel Newton (Univ. Reading)*Room:* TBA.*Title:* The Hasse norm principle for abelian extensions.*Abstract:* Let L/K be an extension of number fields and let J_L and J_K be the associated groups of ideles. Using he diagonal embedding, we view L* and K* as subgroups of J_L and J_K respectively. The norm map N: J_L--> J_K restricts to the usual field norm N: L*--> K* on L*. Thus, if an element of K* is a norm from L*, then it is a norm from J_L. We say that the Hasse norm principle holds for L/K if the converse holds, i.e. if every element of K* which is a norm from J_L is in fact a norm from L*.

The original Hasse norm theorem states that the Hasse norm principle holds for cyclic extensions. Biquadratic extensions give the smallest examples for which the Hasse norm principle can fail. One might ask, what proportion of biquadratic extensions of K fail the Hasse norm principle? More generally, for an abelian group G, what proportion of extensions of K with Galois group G fail the Hasse norm principle? I will describe the finite abelian groups for which this proportion is positive. This involves counting abelian extensions of bounded discriminant with infinitely many local conditions imposed, which is achieved using tools from harmonic analysis.

This is joint work with Christopher Frei and Daniel Loughran.

**21.04: **Christian Maire (Univ Franche-Comté)*Room:* Aud 10.*Notice:* on Thursday from 14:15 to 15:15.*Title:* Decomposition in infinite extensions of number fields.*Abstract: *The aim of the talk is to show in two contexts the importance of the knowledge of the set of decomposed places in an infinite extension : (i) for the mu-invariant ; (ii) for the exponent of the class group along a p-tower.

**08.04: **Kazim Buyukboduk (Koc Univ)*Room:* Aud 9.*Title:* Perrin-Riou's conjecture and rational points on supersingular elliptic curves.*Abstract:* I will report on some work in progress towards Perrin-Riou's conjecture (that predicts the non-vanishing of the $p$-adic Beilinson-Kato elements when the analytic rank of an elliptic curve $E$ equals one). We will simultaneously treat all $p$-semistable and $p$-supersingular elliptic curves, and in the case of the latter, we will also show how one can then construct a $\mathbb{Q}$-rational point on $E$ of infinite order, starting off with the special values of $p$-adic L-functions. Our approach is altogether different from those of Bertolini and Darmon (who handle elliptic curves with good ordinary reduction) and it is based on the general theory of $\Lambda$-adic Kolyvagin systems.

See notes.

**26.02**: Helge Ruddat (Univ. of Mainz)*Room:* Aud 8.*Title:* Tropical descendent log Gromov-Witten invariants.*Abstract:* Descendent Gromov-Witten invariants play a central role in canonical deformations of Landau-Ginzburg models as well as the multiplication rules of generalized theta functions, both relevant for (homological) mirror symmetry. In a joint work with Travis Mandel, I prove that tropical Gromov-Witten invariants with psi class conditions coincide with descendent log Gromov Witten invariants for smooth toric varieties whenever non-superabundance is given. We use toric degenerations a la facon de Siebert-Nishinou and we expect that our approach will be generalizable to Mumford or Gross-Siebert type degenerations.

**19.02: **Julia Brandes (Chalmers Univ. Göteborg)*Room:* Aud 8.*Title:* On the number of linear spaces on hypersurfaces with a prescribed discriminant.*Abstract: *For a given form $F \in \Z[x_1,\dots,x_s]$ we apply the circle method in order to give an asymptotic estimate of the number of $m$-tuples $\mathbf x_1, \dots, \mathbf x_m$ on the hypersurface $F(X) = 0$ having $\det ( \mathbf x_1, \dots, \mathbf x_m)^t \, (\mathbf x_1, \dots, \mathbf x_m)) = b$. As a corollary, we obtain a count of rational linear spaces contained in the hypersurface $F(\mathbf x) = 0$ having dimension exactly $m$, thus addressing a weakness of previous results.

**20.01: **Daniel Bergh (MPI Bonn)*Room:* Aud 5.*Notice:* on Wednesday from 13:15 to 14:15.*Title:* Applications of destackification.*Abstract: *Stacky blow-ups are particularly well-behaved birational modifications of algebraic stacks. We give an algorithm which can be used to modify certain smooth stacks such that they become smooth schemes by using sequences of stacky blow-ups.Since this removes the "stackiness" from the stack in a controlled way, we call the process destackification. I will briefly describe what destackification is and mention some possible applications. More specifically, I will mention some results from two ongoing projects:

* Geometricity for derived categories of algebraic stacks (with Lunts, Schnürer).

* A comparison theorem for equivariant categorical measures (with Gorshinsky, Larsen, Lunts).

**19.01: **Davide Veniani (Hannover)*Room:* Aud 8.*Notice:* on Tuesday from 15:15 to 16:15.*Title:* Lines on K3 quartic surfaces.*Abstract: *On a smooth complex quartic surface there are at most 64 lines: this theorem was first stated by B. Segre in 1943, then correctly proven by Rams and Schütt about 70 years later. I will talk about a new geometric proof which extends the theorem to quartic surfaces with isolated ADE singularities. This new proof gives a deeper insight on particular configurations of lines. If time allows, I will talk about the same problem over fields of positive characteristic.

** 18.01: **Robert Kucharczyk (Bonn)

*Room:*Aud 7.

*Notice:*on Monday from 10:15 to 11:15.

*Title:*On congruence subgroups of Fuchsian groups

*Abstract:*Subgroups of finite index in SL(2,Z) have been intensively studied since the 19th century. These can be divided into congruence and non-congruence subgroups. The former feature prominently in the theory of elliptic curves and have many remarkable special properties, but are much rarer. A similar distinction can be made for general arithmetic subgroups of SL(2,R).

In my talk I will argue that congruence subgroups can also be defined in a natural way for some non-arithmetic lattices in SL(2,R) appearing in algebraic geometry, and present some results suggesting that this generalised notion again produces a sharp dichotomy for these groups. These include a rigidity theorem for lattices in SL(2,R) that serves as a replacement for Mostow's rigidity theorem (which is false for SL(2,R)) and makes use of this generalised notion of congruence subgroups, hence acquiring an adelic flavour.

**16.12: **Pierre Le Boudec (EPFL Lausanne)*Room:* 4013 in the Bio Center.*Notice:* on Wednesday from 14:15 to 15:15.*Title:* Height of rational points on elliptic curves in families.*Abstract:* Given a family F of elliptic curves defined over Q, we are interested in the set H(Y) of curves E in F, of positive rank, and for which the minimum of the canonical heights of non-torsion rational points on E is bounded by some parameter Y. When one can show that this set is finite, it is natural to investigate statistical properties of arithmetic objects attached to elliptic curves in the set H(Y). We will describe some problems related to this, and will state some results in the case of families of quadratic twists of a fixed elliptic curve..

**11.12:** Valentijn Karemaker (Utrecht)** **

*Room:*Aud 10.

*Notice:*on Friday, 13:15-14:15.

*Title:*Local and adelic Hecke algebra isomorphisms.

*Abstract:*First, let K and L be number fields and let G be a linear algebraic group over Q. Suppose that there is a topological group isomorphism of points on G over the adeles of K and L, respectively. We establish conditions on the group G, related to the structure of its Borel groups, under which K and L have isomorphic adele rings.

As a corollary, we show that when K and L are Galois over Q, an isomorphism of Hecke algebras for GL(n), which is an isometry in the L1- norm implies that K and L are isomorphic as fields.

Secondly, let K and L be non-archimedean local fields of characteristic zero. Analogous results to the number fields case still hold. However, we show that the Hecke algebra for GL(2) for any local field is Morita equivalent to the same complex algebra, determined by the Bernstein decomposition.

**1.12: **Samuele Anni (University of Warwick)*Room:* Aud 9.*Notice:* on Tuesday, 16:15-17:15.*Title:* The inverse Galois problem, abelian varieties and uniform realizations.*Abstract:* Let $\overline{\mathbb{Q}}$ be an algebraic closure of

$\mathbb{Q}$, let $n$ be a positive integer and let $\ell$ a prime

number. Given a curve $C$ over $\mathbb{Q}$ of genus $g$, it is

possible to define a Galois representation $\rho:

\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to

\mathrm{GSp}_{2g}(\mathbb{F}_\ell)$, where $\mathbb{F}_\ell$ is the

finite field of $\ell$ elements and $\mathrm{GSp}_{2g}$ is the general

symplectic group in $\mathrm{GL}_{2g}$, corresponding to the action of

the absolute Galois group

$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on the $\ell$-torsion

points of its Jacobian variety $J(C)$. If $\rho$ is surjective, then

we realize $\mathrm{GSp}_{2g}(\mathbb{F}_\ell)$ as a Galois group over

$\mathbb{Q}$.

In this talk I will describe a joint work with Pedro Lemos and Samir

Siksek, concerning the realization of

$\mathrm{GSp}_6(\mathbb{F}_\ell)$ as a Galois group for infinitely

many odd primes $\ell$. Moreover I will describe uniform realizations

of linear groups.

**27.11: **Ian Petrow (EPFL)*Room:* Aud 10.*Title:* A twisted Motohashi formula.*Abstract:* Some of the strongest currently-known subconvex bounds are for L-functions of cusp forms twisted by quadratic characters and are due to Conrey and Iwaniec. Their estimate is derived from an estimate of the cubic moment of those L-functions. I will present a Motohashi-type formula which describes the dual sums of this cubic moment. This Motohashi formula is crucial in extending Conrey and Iwaniec's results to the particularly challenging case of weight 2 cusp forms.

# 12-13.11 : Arithmetic Geometry Days 2015

Speakers:

Jean-Benoît Bost (Paris-Sud)

Klaus Hulek (Hannover)

Dan Petersen (Copenhagen)

Martin Westerholt-Raum (Gothenburg)

Stefan Schröer (Düsseldorf)

Michael Stoll (Bayreuth)

Sho Tanimoto (Copenhagen)

Gabor Wiese (Luxembourg)

**10.11: **Alan Hertgen (Univ Bordeaux)*Room:* Aud 8.*Notice:* on Tuesday from 16:15 to 17:15.*Title: *Splitting properties of the reduction of semi-abelian varieties.*Abstract:* Let G/K be a semi-abelian variety over a discrete valuation field. The special fiber of the Neron model of G/K is an extension of the connected component of 0 by the group of components. We say that G/K has split reduction if this extension is split. We will recall some results obtained by Liu and Lorenzini and talk about some generalizations of these. For instance, whereas G/K has always split reduction if the residue characteristic is 0, we prove that it is no longer the case if the residue characteristic is> 0 even if G/K is tamely ramified. If J/K is the Jacobian variety of a smooth proper and geometrically connected curve C/K of genus g, we prove that for any tamely ramified extension M/K of degree greater than a constant, depending on g only, J_M/M has split reduction.

**9.11:** Aftab Pande (Universidade Federal do Rio de Janeiro)*Room:* Aud 9.*Notice:* on Monday from 16:15 to 17:15.*Title :* An elementary construction of p-adic families of Hilbert Modular forms.

*Abstract*We use an idea of Buzzard and give an elementary construction of a p-adic family of Hilbert Modular eigenforms. In previous work of the author, results on local constancy of slope alpha spaces of Hilbert modular forms were obtained in the spirit of the Gouvea-Mazur conjectures. Assuming the dimensions of the slope alpha spaces is 1, we are able to obtain a p-adic family of Hilbert Modular Forms.

**:****6.11**: Niko Laaksonen (UCL, visiting KU)*Room*: TBA.*Title*: Lattice Point Counting in Sectors of Hyperbolic Space.*Abstract*: Huber demonstrated how the hyperbolic lattice point problem in conjugacy classes corresponds to counting lattice points in a sector of the hyperbolic plane.

This is equivalent to counting geodesic segments according to length. For this problem, Good and Chatzakos--Petridis proved separately an error term analogous to that of Selberg.

We show how this generalises to three dimensions and prove a similar strong bound on the error term. We will also apply the work of Chamizo on large sieve inequalities in hyperbolic spaces to our problem in the radial and spatial aspects. In particular, we will discuss why these yield diminishing returns in higher dimensions.

**25.09: **Andre Chatzistamatiou (MPI Bonn)*Room:* Aud 8.*Title:* Integrality of p-adic integrals.*Abstract:* The basic idea for integration of functions on real manifolds fails for p-adic spaces due to their totally disconnected nature. Nevertheless, Coleman was able to construct a p-adic theory of iterated integration. We will report on integrality results for such integrals. As an application we will provide lower bounds for the valuations of p-adic multiple zeta values.

**22.09: **Amilcar Pacheco (Rio, UFRJ)*Room:* 4.4.20*Notice:* on Tuesday from 10:15 to 11:15.*Title:* An analogue of the Brauer-Siegel theorem for abelian varieties over function fields. *Abstract:* Consider a family of abelian varieties $A_i$ of fixed dimension defined over the function field of a curve over a finite field. We assume finiteness of the Shafarevic-Tate group of $A_i$. We ask then when does the product of the order of the Shafarevic-Tate group by the regulator of $A_i$ behave asymptotically like the exponential height of the abelian variety. We give examples of families of abelian varieties for which this analogue of the Brauer-Siegel theorem can be proven unconditionally, but also hint at other situations where the behaviour is different. We also prove interesting inequalities between the degree of the conductor, the height and the number of components of the Néron model of an abelian variety.

** 18.09: **Kentaro Mitsui (Kobe University)

*Room:*Aud 10.

*Title:*Closed points on torsors under abelian varieties.

*Abstract:*We show the existence of a separable closed point of small degree on any torsor under an abelian variety over a complete discrete valuation field under mild assumptions on the residue field of the valuation ring and the reduction of the abelian variety. To show the existence, we introduce and study minimal models of torsors under quasi-projective smooth group schemes.

**04.09:** Yuri Bilu (Bordeaux)*Room:* Aud 10.*Title:* Special Points on Straight Lines and Hyperbolas.*Abstract:* I will speak on the recent joint work with Bill Allombert, Florian Luca, David Masser and Amalia Pizarro-Madariaga about the special points on the simplest algebraic curves. Call a point "special" if its coordinates are j-invariants of lattices with complex multiplication. I will state the general theorem of Yves André about finiteness of the number of such point of a "general" algebraic curve. Next, I will show that one can say much more in the case of the "simple" curves, like straight lines and hyperbolas.

**Program 2014-2015:**

** 16.06-17.06: **N³ Days II

**.****12.06: **Chandan Dalawat (Harish-Chandra Institute)

*Room:*Aud 10

*.*

Title:The compositum of all degree-p extensions of a local field of residual characteristic p.

Title:

*Abstract:*We determine the Galois group of this extension, along with its ramification filtration. This provides a parametrisation of degree-p extensions, and a conceptual proof Serre's mass formula in the case of degree p.

**04.06:** Matthieu Gendulphe (Rome)*Room:* Aud 8.*Notice:* on Thursday from 13:00 to 14:00.*Title:* The asymptotic of the number of simple closed geodesics on a hyperbolic surface according to M Mirzakhani.*Abstract: *Last summer, Maryam Mirzakhani received the Fields medal for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces." We will describe one of her most famous results, which illustrates the sentence above. This talk is intended for non-experts.

**01.06**: Panagiotis Tsaknias (Univ. of Luxembourg)*Room:* Aud. 7.*Notice:* at 16:15.*Title:* Generalizations of Maeda's Conjecture.*Abstract: *I will report on joint work with L. Dieulefait, currently in progress, on

generalizations of the Maeda conjecture. I will provide a precise generalized

version of its weak form regarding the number of newform Galois orbits for

arbitrary level and trivial nebentypus. I will also describe further ways to

generalize the original conjecture (e.g. non-trivial nebentypus, Hilbert modular

forms, strong form for arbitrary levels).

**15.05:** James Park (Lethbridge)

*Room:*Aud 10.

*Notice:*at 14:30.

*Title:*Averages of elliptic curves over finite fields.

*Abstract:*Let E be an elliptic curve over the rationals. For any prime p, let F_p denote the finite field with p elements. If p is a prime of good reduction then we define the group of points on the reduced elliptic curve over F_p as E_p(F_p). In this talk we consider two functions defined on the primes. In particular, we define a function that counts the number of primes such that #E_p(F_p)=N, for a fixed integer N and a function that counts pairs of distinct primes (p,q) such that #E_p(F_p)=q and #E_q(F_q)=p. We present results on the average of these functions over a family of elliptic curves.

** 15.05: **Daniel Loughran (Hannover)

*Room:*Aud 10.

*Notice:*at 13:15.

*Title:*Good reduction of complete intersections.

*Abstract:*In 1983, Faltings proved the famous Mordell conjecture on the

finiteness of the set of rational points on curves of higher genus.

Along the way he proved numerous other finiteness statements, including

the Shafarevich conjecture, which states that there are only finitely

many curves of higher genus over a number field which have good

reduction outside any given set of prime ideals. In this talk we shall

consider analogues of the Shafarevich conjecture for certain classes of

Fano varieties given as complete intersections in projective space. This

is joint work with Ariyan Javanpeykar.

** 20.04:** Jean-Pierre Wintenberger (Strasbourg).

*Room:*Aud 8

*.*

*Notice:*at 9:00.

*Title:*Ramification and Iwasawa modules.

*Abstract:*We state a criteria for Leopoldt conjecture and construct a Zp-extension

whose ramification satisfies most of the properties of the criteria ( jw

with Chandrashekhar Khare).

**13.04: **Lior Rosenzweig (KTH Stockholm).

*Room:*Aud 6.

*Notice:*at 14.15.

*Title:*Diophantine approximation in nilpotent Lie groups.

*Abstract:*In classical Diophantine approximation one asks how well a real number x can be

approximated by an irreducible fraction whose numerator and denominator are integers. This formulation can be extended to the non-abelian world of Lie groups: A finitely generated subgroup of a real Lie group G is said to be Diophantine if there is b>0 such that non-trivial elements in the word ball B(n) centered at the identity never approach the identity of G closer than 1/|B(n)|^b, and the Lie group G is said to be Diophantine if for every k>0, a random k-tuple in G generates a Diophantine subgroup. In this talk I will discuss the case of nilpotent subgroups. We will show a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most 5, or derived length at most 2, as well as rational nilpotent Lie groups are Diophantine. Time permitting we will also discuss further progress of this question involving the optimality of the exponent b for a given nilpotent Diophantine group G, related to the works of Kleinbock and Margulis. This is joint work with M. Aka, E. Breuillard and N. de Saxce.

**06-09.04: **Christopher Deninger (Münster)*Room:* Aud 5.*Notice:* Tuesday 13:15-14:15, Wednesday-Friday 14:15:15:15.*Title:* Arithmetic geometry and foliated dynamical systems.*Abstract: *Cohomological interpretation of the Dedekind zeta function.

**24.03: **Pankaj Vishe (The University of York)*Room:* Aud 8.*Notice:* Tuesday, March 24, at 13:15.*Title:* Uniform bounds for Period integrals and sparse equidistribution.*Abstract: *Let M be a compact quotient of SL(2,R) and let f be a smooth function of zero average on M. We use spectral methods to get uniform (i.e. independent of spectral gap) bounds for twisted averages of f along long horocycle orbit segments. We apply this to obtain an equidistribution result for sparse subsets of horocycles on M. This is a joint work with J. Tanis.

**16.03**: Orsola Tommasi (Hannover)*Room: *Aud 8.

*Monday 16 at 15:15, joint with Top.*

Notice:

Notice:

*Cohomological stabilization of complements of discriminants*

Title:

Title:

*.*

Abstract:The discriminant of the space of complex polynomials of degree d in one variable is the locus of polynomials with multiple roots. Arnol'd proved that the cohomology of the complement of this discriminant stabilizes when the degree of the polynomials grows, in the sense that the k-th cohomology group of the space of polynomials without multiple roots is independent of the degree of the polynomials considered.

Abstract:

In this talk, I will present a similar stability result for the space of non-singular complex homogeneous polynomials in a fixed number of variables and its rational cohomology and discuss an extension to the more general situation of the space of sections of a very ample line bundle on a fixed non-singular variety. This is inspired by work of Vakil and Wood on stabilization behaviour in the Grothendieck group of varieties.

**13.03:** Bruno Winckler (Université de Bordeaux)*Room:* Aud 10.*Title:* Lehmer's problem and arithmetic intersection.*Abstract: *I will talk about a work in progress regarding lower bounds for the canonical height of algebraic points on elliptic curves with complex multiplications, aiming for the conjecture of Lehmer. I try to prove a better and more explicit version of a weaker form of this inequality, due to Laurent, by using Arakelov's intersection theory: a theorem of Faltings and Hriljac links the canonical height on a elliptic curve and an intersection number on its regular minimal model.

This computation needs some explicit bounds of sums indexed by well-chosen prime numbers, obtained thanks to an explicit version of Cebotarev theorem. I will give such a version.

**03.03:** Arne Smeets (Université Lille 1)*Room:* Aud 4.*Notice:* Tuesday 3.03, at 15:15.*Title:* Logarithmic good reduction, monodromy and the rational volume.*Abstract: *Let $R$ be a local ring which is complete for a discrete valuation, with fraction field $K$ and algebraically closed residue field $k$. Let $X$ be a smooth, proper variety over $K$. Nicaise conjectured that the rational volume of $X$ is equal to the trace of the tame monodromy operator on the $\ell$-adic cohomology if $X$ is cohomologically tame. We prove his conjecture for a large class of such varieties: those having logarithmic good reduction.

**27.02: **Qizheng Yin (ETH Zürich)*Room:* Aud 10.*Title:* Curve counting on abelian surfaces and threefolds.*Abstract: *In this talk, we will explain a way to link Gromov-Witten and Donaldson-Thomas theories on curve counting on specific smooth projective varieties.

**13.02:** Cormac O'Sullivan (New York)*Room:* Aud 7.*Title:* Hyperbolic Fourier coefficients of modular forms.*Abstract:* The usual Fourier coefficients of modular forms are

well-known and important in number theory and other fields too. These

coefficients are associated to a cusp which is just a parabolic fixed

point. In the 1940s Petersson also introduced Fourier coefficients

associated to hyperbolic fixed points. In this talk I describe what is

known about these coefficients, based on two current projects. The

elliptic case, corresponding to Taylor coefficients, will also be

discussed if time allows.

**19.01:** Simon Rose (Bonn)*Room:* Aud 8.*Notice:* Monday 19, 14:15-15:15.*Title:* Beyond hyperelliptic curves on abelian surfaces.*Abstract:* If one is willing to assume the crepant resolution conjecture, then there is a beautiful formula which provides a generating function for the number of hyperelliptic curves on a polarized abelian surface in terms of certain quasi-modular forms. In this talk I will go over this formula, and talk about a number of other natural other ideas that arise from looking at this problem, including mirror symmetry, Hurwitz-Hodge integrals, and Shioda-Inose K3 surfaces.

**16.01:** Sho Tanimoto (Rice University).*Room:* Aud 8.*Notice:* Friday 16, 14:15-15:15.*Title:* Distribution of rational points on algebraic varieties.*Abstract:* When we count rational points of bounded height on algebraic varieties, it is important to exclude exceptional sets to capture the generic distribution of rational points on underlying varieties. This idea leads to the notion of balanced line bundles, and the study of balanced line bundles can be achieved through the study of birational geometry, *e.g.*, the Minimal Model Program. In this talk, first I introduce height functions and counting problems, and I discuss the notion of balanced line bundles. Then I will talk about how useful birational geometry is to study this notion.

**15.01:** Marta Pieropan (Hannover)*Room:* Aud 6.*Notice:* Thursday 15, 14:15-15:15.*Title:* On the distribution of rational points on Fano varieties over number fields.*Abstract:* After introducing the natural questions around rational points on varieties over number fields, we focus on Manin's conjecture about the distribution of rational points on Fano varieties. This conjecture predicts an asymptotic formula for the number of points of bounded anticanonical height that is intimately related to the geometry of the variety. We see how Cox rings and universal torsors can be employed to produce parameterizations that are used to verify the conjecture in some specific cases.

** 9.12: **Luigi Lombardi (Bonn)

Room:

Room:

*Notice:*Friday 9 Jan, 14:15-15:15.

**Title:**

**Deformations of minimal cohomology classes and regularity.**

*Abstract:*

**In this talk I will present a conjecture of Debarre, Pareschi and Popa on the classification and characterization of subvarieties of abelian varieties representing a minimal cohomology class. I will discuss an approach via deformation theory to give a further evidence to the conjecture. In the second part of the talk I will establish a regularity bound called Theta-regularity for ideal sheaves of curves embedded in abelian varieties. This is a further step towards the conjecture mentioned above as Theta-regularity should characterize minimal cohomology classes as those having the least possible regularity.**

**15.12-16.12:** Number Theory Days (I)

Monday 15.12 | Tuesday 16.12 | |

09:10-10:00 | Talk 5 in Aud 1 | |

10:10-11:00 | Talk 6 in Aud 1 | |

11:10-12:00 | Talk 7 in Aud 1 | |

12:00-13:00 | ||

13:15-14:05 | Talk 1 in S11 | |

14:10-15:00 | Talk 2 in S11 | |

15:15-16:15 | Talk 3 in Aud 10 | |

16:30-17:20 | Talk 4 in Aud 6 | |

18:30 | Dinner |

Program:

T1: Simon Kristensen (Univ Århus): *Multiplicative problems in Diophantine approximation.*T2: Tomas Persson (Univ Lund):

*Hausdorff dimensions of sets in Diophantine*

*approximation.*

T3: Lars Winther Christensen (Texas Tech Univ, joint talk with Alg-Top):

*Tate (co)homology over associative rings.*

T4: Ulf Kühn (Univ Hamburg):

*On the generators of a certain algebra of multiple q-zeta values.*

T5: Dan Petersen (Univ Copenhagen):

*Cohomology of local systems on the moduli of principally polarized abelian surfaces.*

T6:

*Per Salberger (Univ Gothenburg):*

*Counting rational points on projective varieties.*

T7:

*Nadim Rustom (Univ Copenhagen):*

*Finiteness questions in the arithmetic of modular forms mod p^m.*

**See the detailed program**

*(PDF)*The room S11 is in the D part of the HCØ building, nano-science center, ground floor.

**.****5.12:** Anna Arnth-Jensen*Room:* Aud 8

*Notice:*Friday 5 Dec, 13:15-14:15.

*Title:*Exhibiting non-trivial elements of the Tate-Shafarevich group for the Jacobian of a hyperelliptic curve.

*By the Mordell-Weil theorem, the set of K-rational points on the Jacobian of a hyperelliptic curve over a number field K is a finitely generated abelian group. Via descent methods, one may hope to determine the Mordell-Weil group completely. However, the Tate-Shafarevich group constitutes an obstruction to the successful application of this methodology, since there is no known algorithm for computing this group.*

Abstract:

Abstract:

In this talk I will show how non-trivial elements of the Tate-Shafarevich group for the Jacobian of a hyperelliptic curve may be exhibited by exploiting information from descents on isogenous abelian varieties. I will present some explicit examples of infinite families of higher genus curves whose Jacobians have a non-trivial Tate-Shafarevich group.

**28.11:** Alexander Molnar, Queen's University.*Room:* Aud 8.*Notice: *Friday 28 Nov, 12:30-13:30.*Title: *Arithmetic on intermediate Jacobians of some rigid Calabi-Yau

threefolds.*Abstract: *Generalizing the Jacobian variety of a curve, one may

associate to any higher dimensional complex variety X some (complex)

varieties defined in terms of cohomological quotients of X, the

intermediate Jacobians. These receive cycle class maps, so there is

much interest in being able to define them over number fields in order

to study the many open conjectures on cycles and Chow groups of

varieties.

We will discuss some examples of rigid Calabi-Yau threefolds where we

may compute the intermediate Jacobians as complex tori, and show that

each model of the threefolds over the rational numbers leads to a

natural rational model of the intermediate Jacobians. This allows us

to consider (quadratic) twists of the threefolds, see how this affects

the intermediate Jacobian, and compute the L-functions of the twisted

threefolds and the respective twisted intermediate Jacobians, as well

as their special values and look for links between them.

**10.11:** Anders Södergren, Univ Copenhagen.*Room:* Aud 6.*Title:* The generalized circle problem, mean value formulas and Brownian motion.*Abstract:* The generalized circle problem asks for the number of lattice points of an n-dimensional lattice inside a large Euclidean ball centered at the origin. In this talk I will discuss the generalized circle problem for a random lattice of large dimension n. In particular, I will present a result that relates the error term in the generalized circle problem to one-dimensional Brownian motion. The key ingredient in the discussion will be a new mean value formula over the space of lattices generalizing a formula due to C. A. Rogers. This is joint work with Andreas Strömbergsson.

**27.10:** Oscar Marmon, University of Göttingen.*Room:* Aud 1.*Title:* Counting integral points on complete intersections.*Abstract:* We study the density of solutions to a general system of Diophantine

equations for which the underlying variety is a non-singular complete

intersection. If the number of variables is large enough in terms

of the number of equations and their degree, then the Hardy-Littlewood

circle method gives precise information about the density of solutions.

We derive upper bounds that are valid for a considerably smaller number of

variables, using a multidimensional q-analogue of van der Corput

differencing due to Heath-Brown.