## # Number Theory Group

DateSpeakerTitleAbstract

## FS2017

02.03.2017Igor Shparlinksi (Univ. of New South Wales/MPI-Bonn)Multiplicatively dependent algebraic numbers(joint work with Alina Ostafe, Francesco Pappalardi, Min Sha, Cam Stewart) We discuss various questions related to the distribution of vectors of algebraic numbers (u_1, …., u_n) which are multiplicatively dependent. In particular, we present some counting results for the number of such vectors of degree d and height h (from a fixed number field K and from Q-bar). We also give both sided estimates on their density in R^n. Finally we give an analogue of a result of Bombieri-Masser-Zannier (1999) proving the boundedness of the “house" of the shifts v from the abelian closure of a given number field K, for which the vector (u_1-v, …., u_n-v) becomes multiplicatively dependent, rather than the boundedness of their height as in BMZ’99 (but for shifts v from Q-bar). This result has applications to multiplicative dependence in orbits of polynomial dynamical systems, generalising those on roots of unity.
06.04.2017Yuri Bilu (Univ. de Bordeaux)Erratic behavior of the coefficients of modular formsI will speak on a recent joint work with Jean-Marc Deshouillers, Sanoli Gun and Florian Luca. Here is a sample result. Let $\tau(.)$ be the classical Ramanujan $\tau$-function defined by $$q\prod_{n>0} (1-q^n)^{24} = \sum_{n>0} \tau(n) q^n.$$ The classical work of Rankin implies that both inequalities $|\tau(n)|<|\tau(n+1)|$ and $|\tau(n)|>|\tau(n+1)|$ hold for infinitely many $n$. We generalize this for longer segments of consecutive values of $\tau$. Let k be a positive integer such that $\tau(n)$ is not $0$ for $n\le k/2$. (This is known to be true for all $k < 10^{23}$, and, conjecturally, holds for all $k$.) Let s be a permutation of the set $\{1,...,k\}$. Then there exist infinitely many positive integers $n$ such that $|\tau(n+s(1))|<\tau(n+s(2))|<...<|\tau(n+s(k))|$.
11.04.2017Yuri Bilu (Univ. de Bordeaux)Subgroups of Class Groups and the Absolute Chevalley-Weil TheoremThe following conjecture is widely believed to be true: given a finite abelian group G, a number field K and an integer d>1, there exist infinitely many extensions L/K of degree d such that the class group of L contains G as a subgroup. I will speak on some old and recent results on this conjecture, in particular, on my recent joint work with J. Gillibert. (Special location: Rheinsprung 21, Seminarraum 00.002)
20.04.2017Vincent Bosser (Univ. de Caen)Lehmer's problem on Drinfeld modulesThis talk will be about lower bounds for the canonical height associated to a Drinfeld module. I will explain how one can obtain such a lower bound for the canonical height of an algebraic point which is polynomial in the inverse of the degree of the point. This bound is valid for every Drinfeld module (in particular, of arbitrary rank). This is a joint work with Aurélien Galateau.
11.05.2017Fabrizio Barroero (Univ. of Basel)Unlikely intersections in families of abelian varieties and some polynomial Diophantine equationsI will report on some results about unlikely intersections in families of abelian varieties, obtained mostly in collaboration with L. Capuano, and explain how they fit into the framework of the Zilber-Pink Conjectures. Finally, I will present some applications concerning certain polynomial Diophantine equations.
18.05.2017Gabino Gonzales Diez (Univ. Auto. de Madrid)On the action of the absolute Galois group on curves and surfacesA combination of Hodge theory and Serre's GAGA principle implies that many topological invariants of complex projective varieties, such as Betti numbers and Chern classes, remain invariant under Galois action. Nevertheless, in 1964 Serre proved that conjugate varieties need not be homeomorphic, by showing the existence of Galois elements $\sigma \in Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ and projective varieties $X$ such that the fndamental groups of $X$ and its conjugate variety $X^{\sigma}$ are not isomorphic (while their algebraic fundamental groups are). In this talk I will attempt to show that this occurs for every $\sigma \in Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ different from the identity and the complex conjugation. Our projective varieties $X$ will be Beauville surfaces, a kind of rigid surfaces of general type introduced by Catanese, arising as finite quotients of products of triangle curves. This is joint work with Andrei Jaikin-Zapirain.
01.06.2017Carlo Gasbarri (Univ. de Strasbourg)Transcendental Liouville inequality on projective variety and rational points on analytic disksLet $(X, L)$ be a polarized projective variety defined over a number field. If $D$ is a one dimensional analytic disk in $X(\mathbf{C})$ then, a classical theorem of Bombieri and Pila tells us that there are at most $\exp(\epsilon T)$ rational points of height at most $T$ in $D\cap X(K)$. Following an idea due to Masser, one can see that, if $D$ contains some special transcendental point, then the cardinality of the set of points of height at most $T$ in $D\cap X(K)$ is polynomial in $T$. These special points, called points of type $S$, verify a property which is similar to Liouville inequality. We will explain why points of type $S$ are full $X(\mathbf{C})$ (for the Lebesgue measure).

## HS2016

06.10.2016Martin Widmer (Royal Holloway)Weakly admissible lattices, o-minimality, and Diophantine approximationWe present new estimates for the number of lattice points in sets such as aligned boxes which, in certain cases, improve on Skriganov's celebrated counting results. We establish a criterion under which our error term is sharp, and we provide examples in dimensions $2$ and $3$ using continued fractions. Moreover, we use o-minimality to describe large classes of sets to which our counting results apply. If time permits we also present a similar counting result for primitive lattice points, and apply the latter to the classical problem of Diophantine approximation with primitive points as studied by Chalk, Erd\H{o}s and others.
27.10.2016Lucia Mocz (Princeton / IHES)A New Northcott Property for Faltings' HeightsWe develop explicit techniques using tools from integral $p$-adic Hodge theory to study the change in Faltings' height within an isogeny class of CM abelian varieties. Assuming the Colmez conjecture, this results in a new Northcott property for Faltings' heights for CM points. On the Hilbert modular variety we are moreover able to develop a Colmez-type formula for the Faltings' height of all CM points.
10.11.2016Francesco Veneziano (Basel)The height of subvarietiesHeights are a fundamental tool in diophantine geometry. I will show how to extend the definition of the Weil height for points in the projective space to subvarieties of (multi)projective spaces and what properties can be proved about it.
17.11.2016Adrian Denz (Basel)Bounding the height of certain algebraic numbersEspecially since a paper of Bombieri-Masser-Zannier in 1999, bounding the height became the standard first step in getting finiteness for certain problems. Therefore, bounding the height of certain families of algebraic numbers or points from above is an active area of current research. In my master thesis I showed that the height of an algebraic number $\alpha$ satisfying $\alpha^n + (1-\alpha)^n + (1+\alpha) =1$ for some integer $n\ge2$ or satisfying $\alpha^r + (1-\alpha)^s =1$ for some integers $r\ge1$ and $s\ge1$, not both $1$, is bounded. The proofs are fully effective and I gave concrete values for bounds in these two situations. Both cases are special ones of a much more general result by Amoroso-Masser-Zannier (to appear). In this talk I will present my approach to prove these two results using a version of Siegel's Lemma and I will focus on how I dealt with the non-vanishing problem usually arising in this context. The latter is done differently and much simpler than by Amoroso-Masser-Zannier due to the special cases here, while the general method is the same.
24.11.2016Davide Lombardo (Hannover)On division fields of CM abelian varieties Let K be a number field and A/K be an abelian variety with complex multiplication. We consider the extensions of K generated by torsion points of A and give uniform bounds for their degrees in terms of K and of the dimension of A. This refines a result of Ribet and has applications to some cases of the (uniform) Rasmussen-Tamagawa conjecture.
08.12.2016Michael Stoll (Bayreuth)Simultaneous torsion in the Legendre family of elliptic curves
22.12.2016Lars Kühne (MPIM Bonn)The Bounded Height Conjecture for Semiabelian VarietiesThe Bounded Height Conjecture of Bombieri, Masser and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian variety G there is an upper bound on the Weil height of the points contained in its intersection with the union of all algebraic subgroups having (at most) complementary dimension in G. After partial work of many authors, Habegger proved the conjecture completely for both tori and abelian varieties in 2009. In my talk, I will discuss how to prove the conjecture for general semiabelian varieties.

## FS2016

25.02.2016Antonella Perucca (Regensburg)Reductions of abelian varietiesLet A and A’ be two abelian varieties defined over a number field. Our goal is comparing A and A', for example we would like to know if they are isogenous. We consider all reductions, and the information at our disposal are the number of points over the residue fields or the size of the reductions of the Mordell-Weil group. This is joint work with Chris Hall (University of Wyoming).
03.03.2016Aurelian Galateau (Besancon)Around the Property (B)The Property (B) was introduced 15 years ago by Bombieri and Zannier to study fields of algebraic numbers on which the Weil height can be bounded from below (outside of the roots of unity). This property is linked to a relative version of the Lehmer problem, which plays an important role in diophantine geometry, for instance towards the Zilber-Pink conjecture. We'll discuss examples that have been found recently of fields with the Property (B).
10.03.2016Margaret Thomas (Konstanz)Effective Pila--Wilkie bounds for restricted Pfaffian surfacesThe counting theorem of Pila and Wilkie has become celebrated for opening up one of the most important developments in applied model theory in recent years. It provides a bound on the density of rational points of bounded height lying on the `transcendental parts' of sets definable in o-minimal expansions of the real field, a result which has had several stunning number theoretic applications (e.g. to the Manin-Mumford and André-Oort Conjectures). However, the proof of the theorem is not effective: it does not give a procedure which, given a definable set, will compute the Pila--Wilkie bound for that set. This of course constrains the effectivity of its applications. I will discuss some recent progress made towards finding an effective version of the Pila--Wilkie Theorem in certain cases. (Joint work with G. O. Jones.)
17.03.2016Lukas Pottmeyer (Basel)Around the Northcott propertyMotivated by the question of counting periodic points of endomorphisms of an abelian variety, Northcott proved in 1949 that every number field contains just finitely many points of bounded height. There are also fields of infinite degree over $\mathbb{Q}$ satisfying this finiteness property. We say that such fields satisfy the Northcott property (N). In this talk we will discuss relations between (N) and other arithmetic properties a field $F\subseteq\overline{\mathbb{Q}}$ may have.
31.03.2016Michael Neururer (TU Darmstadt)Spaces generated by products of two Eisenstein series.Which modular forms are linear combinations of products of two Eisenstein series? Writing a modular form as such a linear combination can be used in many applications, e.g. for calculating Fourier expansions at any cusp. In joint work with Martin Dickson we show that for many congruence subgroups all modular forms are linear combinations of products of Eisenstein series. I will present a proof of this result using the Rankin-Selberg method and modular symbols. In the end I will talk about recent work by Rogers-Zudilin and Brunault on some of Boyd's conjectures concerning relations between Mahler measures and L-values of elliptic curves. They write a newform of weight 2 as a linear combination of products of Eisenstein series in order to study the L-value L(f,2). I will indicate how this approach can be generalised to study the L-value of a K3 surface and connect it to Mahler measures.
08.04.2016Umberto Zannier (SNS Pisa)On the Hilbert Property and the fundamental group of algebraic varietiesThis would concern recent work with P. Corvaja in which we relate the Hilbert Property (a kind of axiom linked with Hilbert Irreducibility) for an algebraic variety with its fundamental group. This leads to new examples (of surfaces) both of validity and failure of the Hilbert Property.
26.04.2016Pierre Le Boudec (EPLF)Height of rational points on algebraic varietiesGiven a finite set S of irreducible homogeneous polynomials in several variables and with integral coefficients, a classical problem in number theory is to describe the set V(Q) of rational points on the variety V defined by S, that is the set of common zeros with rational coordinates of the polynomials in S. Typical questions are: when is V(Q) non-empty or infinite? If V(Q) is infinite, can we measure its size? If V(Q) is non-empty, what is the size of its smallest element? We will be concerned in particular with the latter problem. While all these questions are too hard in such generality, we will try to explain how they become easier when investigated on average over families of varieties.
19.05.2016Ezra Waxman (ETH / Tel Aviv)The Distribution of Slopes of Gaussian Primes for the Rational Function FieldHecke proved that the Gaussian primes are equidistributed across sectors of the complex plane, by making use of (infinite) Hecke characters and their associated L-functions. In this talk I will present a function field analogue to this result, by making use of "super even" characters and their associated L-functions. By applying a recent result of N. Katz concerning the equidistribution of super even characters, I will also provide a result for the variance of function field Gaussian primes across sectors; a computation whose analogue in number fields is unknown beyond a trivial regime.
26.05.2016Yuri Zarhin (Penn State)Compatible systems of $\ell$-adic representations arising from abelian varieties.Famous (and still unproven in full generality) conjectures of Serre-Grothendieck, Tate and Fontaine-Mazur describe $\ell$-adic representations that arise from the action of the absolute Galois group of a number field $K$ on the (twisted) $\ell$-adic cohomology groups of projective algebraic varieties that are defined over $K$. Assuming all these conjectures (and the Hodge conjecture), we discuss the following question: which $\ell$-adic representations correspond to the $\ell$-adic Tate modules of an abelian variety? We give an answer for abelian varieties without nontrivial endomorphisms. This is a report on a joint work with Stefan Patrikis and Felipe Voloch.
02.06.2016Emmanuel Breuillard (Univ. Münster)Two reformulations of the Lehmer conjectureSalem numbers are those numbers (real >1) with at most one conjugate outside the unit disc and at least one on the unit circle. Salem's conjecture asserts that they cannot be too close to 1. More generally, Lehmer's conjecture asserts that given any algebraic unit, which is not a root of unity, the modulus of the product of the conjugates lying outside the unit disc (the Mahler measure) is bounded away from 1. I will present two joint works, one with B. Deroin and the other with P. Varju, in which we give two unrelated ways to reformulate Salem's and Lehmer's conjecture. The first as a uniform spectral gap for a certain family of hyperbolic surfaces, the second as an elementary counting problem in finite fields.
09.06.2016Ziyang Gao (CNRS Paris 6)Polarized isogeny orbits in families of abelian varietiesAbstract: Let $\mathcal{A}$ be an abelian scheme over $B$. Fix a point $s$ in $\mathcal{A}$ and let $\Sigma$ be the polarized isogeny orbit generated by $s$. We characterize curves in $\mathcal{A}$ whose intersections with $\Sigma$ is Zariski dense and give the conjecture for higher dimensional subvarieties.

## HS2015

17.09.15Ariyan Javanpeykar (Mainz)The Lang-Vojta conjecture and arithmetic finiteness results for smooth hypersurfacesIn 1983, Faltings proved the Shafarevich conjecture: for a finite set of finite places of a number field K and an integer g>1, the set of isomorphism classes of curves of genus g over K with good reduction outside S is finite. In this talk we shall consider analogues of the Shafarevich conjecture for hypersurfaces. We will prove, assuming the conjecture of Lang-Vojta, the analogous finiteness statement for hypersurfaces of fixed degree and fixed dimension. Unconditionally, we prove the Shafarevich conjecture for hypersurfaces of Hodge level at most one, and some hypersurfaces of Hodge level 2. This is joint work with Daniel Loughran.
09.10.15Daniel Bertrand (Paris)Galois groups of logarithmic equationsI will describe a recent joint work with A. Pillay, where we extend to semi-abelian schemes the classical theorem of Ax on the exponential of algebraic functions. The proof is based on my co-author's theory of logarithmic differential equations, combined with an argument of Galois descent reminiscent of Kummer theory
15.10.15Jonathan Pila (Oxford)Multiplicative and modular diophantine problemsI will describe some diophantine results and conjectures, from the Mordell conjecture of 1922 (theorem of Faltings) to the open and very general Zilber-Pink conjecture. I will describe a recent result and conjecture of similar flavour which are not formally consequences of the Zilber-Pink conjecture.
22.10.15Christian Elsholtz (TU Graz/FIM)A solution of a problem of Ramanujan of 1915It is well known that the number of divisors $d(n)$ is large, when $n$ is a product of many primes. Wigert determined the maximal order of magnitude of the divisor function: $d(n) \leq \exp ( (\log 2 +o(1)) \frac{\log n}{\log \log n},$ or in other words $\max_{n\le x} \log d(n) \sim (\log 2){\frac{\log x}{\log \log x}}.$ Ramanujan (1915) was the first to investigate the maximal order of magnitude of the iterated divisor function $d(d(n))$, giving an example that $d(d(n) \ge (\sqrt{2}\log 4 - o(1)) \frac{\sqrt{\log n}}{\log \log n}$ for infinitely many $n$. Erd\H{o}s, K\'{a}tai, Ivi\'{c} and Smati gave upper bounds, but determining an analogue of Wigert's result, i.e. a best possible upper bound, was an open problem. In this talk we give a solution to this problem: $\max_{n\le x} \log d(d(n))= \frac{\sqrt{\log x}}{\log_2 x} \left( c + O(\frac{\log_3 x}{\log_2 x} )\right),$ where $c =\Bigg( 8 \sum_{j=1}^\infty \log^2 (1+1/j) \Bigg)^{1/2} = 2.7959802335\ldots.$ (Joint work with Yvonne Buttkewitz, Christian Elsholtz, Kevin Ford, Jan-Christoph Schlage-Puchta.)
29.10.15Roland Paulin (Salzburg)An explicit Andr\'e-Oort type result for P^1(C) x G_m(C).We will discuss a problem of Andr\'e-Oort type for $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$. In this variation the special points of $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$ are of the form $(\alpha, \lambda)$, with $\alpha$ a singular modulus and $\lambda$ a root of unity. The qualitative version of our result states that if $\mathcal{C}$ is a closed algebraic curve in $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$, defined over a number field, not containing a horizontal or vertical line, then $\mathcal{C}$ contains only finitely many special points. We discuss two approaches, one using logarithmic forms, and another using class field theory. Both approaches give explicit results.
12.11.15Thanasis Bouganis (Durham)p-adic measures of Hermitian modular forms and the Rankin-Selberg methodp-adic measures are playing an important role in the various Main Conjectures of Iwasawa Theory. In this talk I will start by presenting some basic properties of the classical L functions associated to a Dirichlet character, such as the Kummer congruences, and then explain how these properties can be understood in a broader context, namely that of the existence of a p-adic measure. Then after discussing some basics of Hermitian modular forms, (automorphic forms associated to unitary groups) I will present the construction of various p-adic measures, which can be associated to a Hermitian modular form by employing the so-called Rankin-Selberg method.
03.12.15Javier Fresan (ETHZ)Gamma values: regular and irregularThe values of the gamma function at rational numbers remain quite mysterious, one of the reasons being that (conjecturally) they are not periods in the usual sense of algebraic geometry. However, the theory of regular singular connections allows to show that suitable products of them are periods of Hodge structures with complex multiplication, as predicted by Gross and Deligne. To deal with single gamma values, one needs to consider irregular singular connections instead. After an exposition of my results on the Gross-Deligne conjecture, I will explain how irregular singular connections may shed some light on the arithmetic nature of these numbers.
10.12.15Martin Bays (Munster)Algebraic structures of exponential maps I will discuss the problem of giving a complete algebraic description of the interaction between algebraic geometry and the exponential map of a complex algebraic group, focusing on the case of a simple abelian variety. The problem and many of the tools originate in model theory, but I will concentrate more on the role played by transcendence theory and the Faltings-Ribet Kummer theory for abelian varieties. The talk will be based on recent work with Jonathan Kirby, extending work of Zilber on the case of the multiplicative group.

## FS2015

21.04.15Gael Rémond (Bordeaux, CNRS)Abelian varieties and maximal ordersI study abelian varieties whose endomorphism ring is a maximal order. The algebraic properties of these orders, which can be seen as non-commutative analogs of Dedekind domains, allow to prove structure theorems for the aforesaid abelian varieties. For example, they are always products of simple varieties. Furthermore, this kind of results also give information for an arbitrary abelian variety, since it is isogenous to another one as above. I give also an application to the size of the torsion part of the group of rational points over a number field.
28.04.15Stefan Schmid / Harry Schmidt (Basel)Singular modular that are S-units / Generalized Jacobians and additive extensions of elliptic curves
07.05.15Fabrizio Barroero (SNS Pisa)Unlikely Intersections in certain families of abelian varieties and the polynomial Pell equation.Let E_t be the Legendre elliptic curve of equation Y^2=X(X-1)(X-t). In 2010 Masser and Zannier proved that, given two points on E_t with coordinates algebraic over Q(t), there are at most finitely many specializations of t such that the two points become simultaneously torsion on the specialized elliptic curve, unless they were already generically linearly dependent. This fits inside the framework of the so-called Unlikely Intersections. As a natural higher-dimensional analogue, we considered the case of n generically independent points on E_t with coordinates algebraic over Q(t). Then there are at most finitely many specializations of t such that two independent relations hold between the specialized points. We recently also dealt with the case of points on Jacobians of genus two curves. This has applications in the study of solvability of the polynomial (almost) Pell equation. This is joint work with Laura Capuano.
21.05.15François Charles (Paris)Frobenius distribution for pairs of elliptic curves and exceptional isogeniesWe will discuss a proof of the following result: if E and E' are two elliptic curves over a number field K, then there exist infinitely many primes p of K such that the reductions of E and E' modulo p are geometrically isogenous. The proof relies on the arithmetic dynamics of Hecke correspondences.
28.05.15Fabien Pazuki (Copenhagen)Bad reduction of curves with CM jacobiansAn abelian variety defined over a number field and having complex multiplication (CM) has potentially good reduction everywhere. If a curve of positive genus which is defined over a number field has good reduction at a given finite place, then so does its jacobian variety. However, the converse statement is false already in the genus 2 case, as can be seen in Namikawa and Ueno's classification table of fibres in pencils of curves of genus 2. In this joint work with Philipp Habegger, our main result states that this phenomenon prevails for certain families of curves. We prove the following result: Let F be a real quadratic number field. Up to isomorphisms there are only finitely many curves C of genus 2 defined over the algebraic closure of the rationals with good reduction everywhere and such that the jacobian Jac(C) has CM by the maximal order of a quartic, cyclic, totally imaginary number field containing F. Hence, except for finitely many examples, such a curve will always have stable bad reduction at some prime whereas its jacobian has good reduction everywhere. A remark is that one can exhibit infinite families of genus 2 curves with CM jacobian such that the endomorphism ring is the ring of algebraic integers in a cyclic extension of the rationals of degree 4 that contains F for some specific F
19.06.15Rafael von Känel (Princeton) Solving S-unit and Mordell equations via Shimura-Taniyama conjectureJoint work with Benjamin Matschke. In the first part of this talk, we shall present new practical algorithms which solve S-unit and Mordell equations by combining the method of Faltings (Arakelov, Parshin, Szpiro) with the Shimura-Taniyama conjecture. Our algorithms do not use lower bounds for linear forms in logarithms and they considerably improve the actual best algorithms. In the second part we plan to discuss in detail the construction of a sieve used in the algorithm for Mordell equations. Our sieve settles in particular the open problem of efficiently enumerating integral points of bounded height on elliptic curves.