## Number Theory

We meet on Monday once in two weeks in room 403 at IMPAN. Usually there are two talks and a tea break in room 409.

In 2020 the seminar is organized by Piotr Achinger, Jakub Byszewski, Bartosz Naskręcki, Masha Vlasenko and Alex Youcis.

**June 29** ONLINE SEMINAR: if you are not on the regular mailing list, pelase contact the organizers to get the link to this meeting

**13:15-14:15** **Adam Keilthy (University of Oxford)**

*The block filtration and motivic multiple zeta values*

Multiple zeta values are a class of transcendental numbers, going back to Euler in the 1700s

and with ties to the Riemann zeta function. They are found arising naturally in many areas of mathematics and physics, from algebraic geometry to Feynman amplitudes. Unlike in the case of single zeta values, we know many algebraic relations satisfied by multiple zeta values: the double shuffle relations, the associator relations, the confluence relations. However it is unknown if any of these sets of relations are complete. Assuming Grothendieck's period conjecture, a complete set of algebraic relations are given by the motivic relations, arising from a connection to P^1 minus three points. However these relations are inexplicit.

In this talk, we introduce a new filtration, called the block filtration, on the space of multiple zeta values. By considering the associated graded, we describe several new families of motivic relations, that provide a complete description of relations in low block degree. A generalisation of these results would thus provide a complete description of relations among multiple zeta values and aid in settling several open problems about the motivic Galois group of mixed Tate motives.

**June 22** ONLINE SEMINAR

**13:15-14:15 Jolanta Marzec (TU Darmstadt)**

*Maass relations for Saito-Kurokawa lifts of higher levels [slides]*

It is known that a Siegel modular form is a (classical) Saito-Kurokawa lift of an elliptic modular form if and only if its Fourier coefficients satisfy the so-called Maass relations. The first construction of such a

lift was given by Maass using correspondences between various modular forms. However, in order to generalize this lift to higher levels it is easier to use a construction coming from representation theory. During the talk we present history of this problem and briefly discuss the aforementioned constructions. We also indicate how one can read off the Maass relations from the latter. This work generalizes an approach of Pitale, Saha and Schmidt from the classical to a higher level case.

**June 8** ONLINE SEMINAR

**13:15-14:15 Anna Szumowicz (Sorbonne Université, IMJ-PRG)**

Equidistribution in number fields

The notion of *p*-ordering comes from the work of Bhargava on integer-valued polynomials. Let* k* be a number field and let *O _{k}* be its ring of integers. A sequence of elements in

*O*is a simultaneous

_{k}*p*-ordering if it is equidistributed modulo every prime ideal in

*O*as well as possible. We prove that Q the only number field

_{k}*k*such that

*O*admits a simultaneous

_{k}*p*-ordering. It is a joint work with Mikołaj Frączyk.

**June 1** ONLINE SEMINAR

** 13:15-14:15 Alex Youcis (IMPAN)**

*An approach to the characterization of the p-adic local Langlands correspondence*

In 2013 P. Scholze provided an alternative proof of the local Langlands correspondence (LLC) for GLn and, in doing so, Scholze gave a new characterization of the LLC via a certain trace identity. In this talk the speaker will discuss joint work with A. Bertoloni Meli showing that a generalization of this trace identity characterizes the LLC for much more general groups if one assumes standard expected properties of such a correspondence.

**May 25** ONLINE SEMINAR

**13:15-14:15 Jakub Konieczny (Hebrew University of Jerusalem) **

*Automatic multiplicative sequences [slides]*

Automatic sequences - that is, sequences computable by finite automata - give rise to one of the most basic models of computation. As such, for any class of sequences it is natural to ask which sequences in it are automatic. In particular, the question of classifying automatic multiplicative sequences has attracted considerable attention in the recent years. In the completely multiplicative case, such classification was obtained independently by S. Li and O. Klurman and P. Kurlberg. The main topic of my talk will be the resolution of the general case, obtained in a recent preprint with M. Lemańczyk and C. Müllner.

**May 4** SEMINAR CANCELLED

**13:15-14:15 Vesselin Dimitrov (University of Toronto)**

*A height gap theorem for holonomic functions and a proof of the conjecture of Schinzel and Zassenhaus*

**April 6 ** SEMINAR CANCELLED

**13:15-14:15 Jolanta Marzec (TU Darmstadt)**

*Maass relations for Saito-Kurokawa lifts of higher levels*

**March 23** SEMINAR CANCELLED

**13:15-14:15 Danylo Radchenko (ETH Zürich)**

*Universal optimality of the E8 and Leech lattices *

We look at the problem of arranging points in Euclidean space in order to minimize the potential energy of pairwise interactions. We show that the E8 lattice and the Leech lattice are universally optimal in the sense that they have the lowest energy for all potentials that are given by completely monotone potentials of squared distance.

The proof uses a new kind of interpolation formula for Fourier eigenfunctions, which is intimately related to the theory of modular forms.

The talk is based on a joint work with Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Maryna Viazovska.

**March 9**

**13:15-14:15 Johannes Sprang (University of Regensburg)**

*(Ir)rationality of L-values*

Euler’s beautiful formula ζ(2n) = −(2πi)2n/2(2n)! B2n can be seen as the starting point of the investigation of special values of L-functions. In particular, Euler’s result shows that all critical zeta values are rational up to multiplication with a particular period, here the period is a power of (2πi). Conjecturally this is expected to hold for all critical L-values of motives. In this talk, we will focus on L-functions of number fields. In the first part of the talk, we will discuss the ’critical’ and ’non-critical’ L-values exemplary for the Riemann zeta function. Afterwards, we will head on to more general number fields and explain our recent joint result with Guido Kings on the algebraicity of critical Hecke L-values for totally imaginary fields up to explicit periods.

**14:45-15:45 Shizhang Li (University of Michigan)**

*On integral p-adic Hodge filtrations*

In this talk, we will try to understand the behavior of integral p-adic Hodge filtrations of smooth proper schemes over a p-adic ring of integers, using recent developments in integral p-adic Hodge theory.

**February 24**

**13:15-14:15** **Bartosz Naskręcki (Adam Mickiewicz University, Poznań)**

*Elliptic divisibility sequences over function fields*

In this talk I will describe a recent progress in the estimation of the number of non-primitive elements in the elliptic divisibility sequences over function fields. Compared to the number field case the bounds can be made independent of the given curve and point. I will discuss the necessary height theory and valuation theory used in the results. I will also describe how to effectively compute the non-primitivity bounds for large classes of elliptic curves. This is a joint project with Marco Strong.

**14:45-15:45 Jakub Byszewski (Jagiellonian University, Kraków)**

*Periodic points of endomorphisms of algebraic groups in positive characteristic*

We study periodic points of endomorphisms of algebraic groups and so-called dynamically affine maps over fields of positive characteristic. This includes in particular tori endomorphisms, Chebyshev and Lattès maps, and endomorphisms of semisimple algebraic groups. Periodic points are described via the Artin--Mazur zeta function, which in the case of the Frobenius map on an algebraic variety over a finite field coincides with the classical Weil zeta function. We study the growth rate of the number of periodic points, and relate rationality of the zeta function to arithmetic properties of the map. The talk is based on joint work with Gunther Cornelissen, Marc Houben and Lois van der Meijden.