Number Theory Seminar 2023/24

Co-organized by Stevan Gajović and Siu Hang “Gordon” Man.

11. 6. Karim Johannes Becher (Universiteit Antwerpen), On the u-invariant of a function field

The u-invariant of a field is the largest dimension of an anisotropic quadratic torsion form over the field. This field invariant was introduced by I. Kaplansky in 1953 and adapted to the study of real fields by R. Elman and T. Y. Lam in 1973. In 2009, D. Harbater, J. Hartmann, and D. Krashen obtained a bound on the u-invariant for nonreal function fields in one variable over a complete discretely valued field. This was extended by C. Scheiderer to cover the case real function fields. In joint work with N. Daans and V. Mehmeti, we obtain a more general version of this result, namely for function fields of curves over a henselian valued field with arbitrary value group.

22. 5. Pavel Čoupek (Michigan State University), Constructing vector-valued automorphic forms for unitary groups

Automorphic forms on unitary groups are “hermitian” analogue of the well-studied case of Siegel modular forms, which themselves can be thought of as a direct multivariate generalization of classical modular forms. Compared to the Siegel case, less is known in terms of explicit examples. In this talk, I will discuss unitary groups and scalar- and vector-valued holomorphic automorphic forms on them. Then I will describe a construction in terms of certain differential operators that from a given scalar-valued automorphic form produces vector-valued automorphic forms on lower-rank unitary groups. This is joint work with T. Browning, E. Eischen, C. Frechette, S. Hong, S. Y. Lee, D. Marcil and A. Shmakov.

15. 5. Yong Hu (SUSTech, Shenzhen), Strong approximation and integral quadratic forms over affine curves
In the classical arithmetic theory of quadratic forms over global fields, strong approximation and the Hasse principle play a very important role. In this talk, we discuss extensions of some results in this direction to function fields of curves defined over more general fields. In particular, we give examples where strong approximation and the Hasse principle for integral quadratic forms hold, and examples where they do not hold. This is based on a joint work with Jing Liu and Yisheng Tian.
30. 4. Jessica Alessandrì (Università dell’Aquila), A local-global problem for divisibility in algebraic groups

Local-global principles have been widely studied during the last century. In this talk, we will present one of them, the so-called Local-Global Divisibility Problem for commutative algebraic groups. It was first stated in 2001 by Dvornicich and Zannier as a modified version of the Hasse principle for quadratic forms. We will also see some results, in particular for algebraic tori: a generalization of the Grunwald-Wang Theorem to any algebraic tori with bounded dimension and a counterexample for the local-global divisibility by any power of an odd prime.  This is based on a joint work with Rocco Chirivì and Laura Paladino.

24. 4. Angelos Koutsianas (Aristotle University of Thessaloniki), Elliptic Curves, Modular Forms and Diophantine Equations

In this talk we will discuss about elliptic curves, modular forms and their relation in the resolution of Diophantine equations (modular method). We will also discuss new directions and results of the modular method.

10. 4. Martin Widmer (Royal Holloway, University of London), Equidistribution in CM-fields

Let K be a number field, and let O_K be its ring of integers. The group of units (O_K)ˣ is the subgroup of elements of Kˣ with absolute value equal to 1 at all non-archimedean places of K. We consider its archimedean counterpart S_K, the subgroup of Kˣ of elements with absolute value equal to 1 at all archimedean places. Is there a counterpart to Dirichlet’s unit theorem? What does this group have to do with CM-fields? And are the elements of S_K equidistributed  in a natural way? The aim of this talk is to answer these, and possibly some other, questions. This is ongoing joint work with Shabnam Akhtari and Jeffrey Vaaler.

3. 4. Matteo Verzobio (IST Austria), An introduction to elliptic divisibility sequences

In this talk, I will introduce a family of sequences of integers called elliptic divisibility sequences. We will discuss its relation with elliptic curves, prove some arithmetic properties of these sequences, and show some applications of the topic to other fields of mathematics.

27. 3. Om Prakash (Charles University), Class numbers of real cyclotomic fields

In 1974, Yamaguchi published a very strong result regarding the divisibility of class numbers of real cyclotomic fields by class numbers of real quadratic fields. However, in this talk, I will give some explicit counterexamples to Yamaguchi’s result. Additionally, within the framework of genus theory, I will describe why Yamaguchi’s result fails to hold in general.

20. 3. Dušan Dragutinović (Universiteit Utrecht), Supersingular curves of genus four

We consider abelian varieties and curves in positive characteristic and the property of being supersingular, which is related to their number of points over finite fields. In contrast to supersingular abelian varieties, we know little about supersingular curves in general. Our aim is to show that the locus of supersingular genus-4 curves in characteristic p = 2 is 3-dimensional, and we prove this result by considering certain L-polynomials, using an analysis of the data of genus-4 curves and abelian fourfolds over F2. We comment on analogous results in other characteristic p>0.

13. 3. Jakub Krásenský (Czech Technical University), Criterion sets for quadratic forms over number fields

The celebrated 15 theorem of Conway and Schneeberger says that a classical positive definite quadratic form over Z is universal if it represents each element of the set {1,2,3,5,6,7,10,14,15}. Moreover, this is the minimal set with this property. In 2005, B.M. Kim, M.-H. Kim, B.-K. Oh showed that such a finite criterion set exists in a much general setting, but the uniqueness of the criterion set is lost – and the question of uniqueness for some particular situation has been studied by several authors since.

We will discuss the analogous questions for totally positive definite quadratic forms over totally real number fields. There, the existence of criterion sets for universality has been known, and Lee determined the set for Q(√5). We will show the uniqueness and a strong connection with indecomposable integers. A part of our uniqueness result is (to our best knowledge) new even over Z. This is joint work with G. Romeo and V. Kala.

6. 3. Matěj Doležálek (Charles University), Universal quadratic forms through quaternion orders

I present a way of examining some quaternary quadratic forms and their universality using quaternion orders, a topic of my upcoming Master thesis. As a blueprint, I will go over a quaternionic proof of Lagrange’s four-square theorem and how counting factorizations may extend this to a proof of Jacobi’s four-square theorem. Then I will try to show a similar argument in action over the number field Q(√5), examining the critical ingredients of the approach and the complications that occur in them.

28. 2. Stevan Gajović (Charles University), Random p-adic arithmetic statistics

In this talk, we will survey the p-adic arithmetic statistics, starting from some easy problems and generalising them to more interesting questions. The main focus will be one generalisation of the problem of computing the probability that a random p-adic polynomial has a zero in Qp – computing the probability that a random p-adic conic intersects another random p-adic projective curve. We will present the approach in theory, and do some explicit computations for the intersection of two conics. The talk is based on the ongoing joint work with Lazar Radičević.

6. 12. Philip Dittmann (TU Dresden), Characterising fields by Galois-theoretic data

I will discuss various questions around characterisations of fields by their absolute Galois group, i.e. the Galois group of their separable closure. The first non-trivial result here is the 1927 result of Artin and Schreier characterising those fields whose absolute Galois group is Z/2Z. I will also discuss results on p-adic fields and their role in anabelian geometry, as well as recent work by myself on how to obtain a satisfactory result for local fields of positive characteristic.

22. 11. Jędrzej Garnek (IMPAN Warszawa), Cohomologies of p-group covers

Studying cohomology of a variety with an action of a finite group is a classical and well-researched topic. However, most of the previous results focus either on the tame ramification case, on some special groups, or on specific curves. In the talk, we will consider the case of a curve over a field of characteristic p with an action of a finite p-group. Our research suggests that the Hodge and de Rham cohomologies decompose as sums of certain ‘local’ and ‘global’ parts. The global part should be determined by the ‘topology’ of the cover, while the local parts should depend only on an analytical neighborhood of the fixed points of the action. In fact, the local parts should come from cohomologies of Harbater-Katz-Gabber curves, i.e. covers of the projective line ramified only over ∞. During the talk, we present our results related to this conjecture. As an application, we compute the de Rham cohomologies of Z/p-covers and Klein four covers.

15. 11. Mikuláš Zindulka (Charles University), Partitions: Beyond the Integers

Integer partitions are at the intersection of additive number theory and combinatorics. They attracted the attention of great mathematicians such as Euler, Hardy, and Ramanujan, who discovered many beautiful partition identities. It is possible to generalize the notion of partition by replacing positive integers with totally positive integral elements. Can some of the beautiful identities be recovered in this setting?

The first part of the talk will be about partitions in real quadratic fields. I will describe an efficient algorithm for computing the number of partitions of a given element. In which quadratic fields does there exist an element with exactly m partitions? For some particular values of m, we found a complete characterization.

In the second part, I will talk about another class of partitions, a generalization of the so-called m-ary partitions. What happens when we replace m by an algebraic number? This provides a link with the theory of number systems (numeration). The talk is based on joint work with Víťa Kala and David Stern.

8. 11. Andrew Granville (Université de Montréal), A new proof of Hoheisel’s Theorem

Hoheisel’s theorem states that there exists ε>0 such that all intervals [x,x+x^(1-ε)] contain primes, once x is sufficiently large. Traditional proofs involve difficult zero density estimates for the Riemann-zeta function. In 2016 Harper, Soundararajan and I constructed a much simpler “pretentious proof” not involving zeros. Now Matomäki, Merikoski and Teräväinen have given an even easier proof using fairly simple ideas from the theory of the linear sieve and some additive combinatorics. We will present our version of their proof.

1. 11. Connor Cassady (Ohio State University), Universal quadratic forms over semi-global fields

Given a quadratic form (homogeneous degree two polynomial) q over a field k, some basic questions one can ask are:

Over a global field F, the Hasse-Minkowski Theorem, which is one of the first examples of a local-global principle, allows us to use answers to these questions over the completions of F to form answers to these questions over F itself. In this talk, we will focus primarily on quadratic forms over semi-global fields (function fields of curves over complete discretely valued fields) and see how a local-global principle of Harbater, Hartmann, and Krashen can be used to study universal quadratic forms over semi-global fields.

26. 10. Vítězslav Kala (Charles University), Universal quadratic forms and Northcott property of infinite number fields

Universal quadratic forms generalize the sum of four squares about which it is well known that it represents all positive rational integers. In the talk, I’ll start by discussing some results on universal quadratic forms over totally real number fields. Then I’ll move on to the – markedly different! – situation over infinite degree extensions K of Q. In particular, I’ll show that if K doesn’t have many small elements (i.e., “K has the Northcott property”), then it admits no universal form. The talk should be broadly accessible, and is based on a very recent joint work with Nicolas Daans and Siu Hang Man.

Seminar website

25. 10. Giuliano Romeo (Politecnico di Torino), On the properties of p-adic continued fractions

Continued fractions over the field of real numbers are a classical and powerful tool in Diophantine approximation. Therefore, it has been natural to introduce them also in the field of p-adic numbers, where they behave quite differently from the classical case. In fact, unlike continued fractions over the real numbers, there is not a unique standard algorithm, due to the fact that there is not a unique canonical way to define the integral part of a p-adic number. One of the main open problems in this framework is the research of an algorithm that produces an ultimately periodic continued fraction for every p-adic quadratic irrational, i.e. the analogue of the famous Lagrange’s Theorem. In this talk we provide an overview on the theory of p-adic continued fractions and we discuss the most important algorithms that have been defined throughout the years. We focus in particular on the properties of convergence, finiteness and periodicity, together with the most recent developments towards the proof a p-adic version of Lagrange’s Theorem.

18. 10. Didier Lesesvre (Université de Lille), The Weyl law with uniform power savings

For a compact Riemannian manifold, the Weyl law describes the asymptotic behavior of the number of eigenvalues of the underlying Laplace operator. Understanding lower order or error terms remains particularly challenging. In the more general context of locally symmetric spaces, the spectral theory of the Laplacian is intimately related to the theory of automorphic forms (among which are elliptic curves, modular or Maass forms, Galois representations…) and similar questions arise.

It is therefore natural to ask for such a Weyl law to hold for families of all automorphic forms of a given reductive group. Until recently, however, all the known asymptotics were for automorphic forms with fixed aspects. In some sense, this amounts to picking a “slice” of the space of automorphic forms only. Unfortunately, making explicit the hidden dependencies in the featured error term does not allow to sum over these aspects to obtain a uniform counting law: existing results did not allow to patch back together the slices.

In their recent achievement, Brumley and Milićević obtained a uniform Weyl law for GL(2), using the trace formula of Arthur, but with an error term saving only by a power of log. Simplifying the very general setting of this work, and going back to ideas used a long time ago by Drinfeld in the setting of function fields, we obtained a power savings in the smooth Weyl law for the universal family of all automorphic forms of GL(2). The idea is to study a suitable “conductor zeta function”, and to deduce a counting law by Tauberian arguments, mimicking a standard strategy in the realm of counting rational points on varieties.

11. 10. Daejun Kim (Korea Institute for Advanced Study), Lifting problem for universal quadratic forms over totally real cubic number fields

Lifting problem for universal quadratic forms asks for totally real number fields K which admits a positive definite quadratic form with rational integer coefficients that is universal over the ring of integers of K. In this talk, we overview related history on this lifting problem including recent results given by Kala-Yatsyna. Also we discuss a recent work which shows that there is only one such totally real cubic field, and that there is no such biquadratic field. This is a joint work with Seok Hyeong Lee.

Korzok village