Teoria liczb

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.


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.

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.


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.


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

Maass relations for Saito--Kurokawa lifts of higher levels

April 20

May 4

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

May 18


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