## Number Theory

In 2020 the seminar is organized by Jakub Byszewski, Bartosz Naskręcki, Masha Vlasenko and Alex Youcis. We ask our speakers to address the major part of their talk to general audience with interest in arithmetic.

The seminar meets **on Monday**. It is is streamed live on Zoom. We start gathering at 13:00, the **talk begins at 13:15 (Warsaw time)**. Talks are usually 50 minutes and then there is time for discussion. If you are not on the regular mailing list, pelase contact the organizers to get the link to a particular session.

**December 21 Bidisha Roy (IMPAN)**

**December 14 ** **Tim Dokchitser (University of Bristol)**

**December 7 **ONLINE SEMINAR

**Gunther Cornelissen (Utrecht University)**

*An analogy between number theory and spectral geometry*

Sunada’s construction of non-isometric, isospectral manifolds proceeds in the same way as Gassmann’s construction of non-isomorphic number fields with the same zeta function, using a group G with two non-conjugate subgroups H and K such that the permutation representations given by G acting on their cosets are isomorphic. In Gassmann’s example, G was the permutation group on 6 letters and H and K the groups generated by (12)(34) and (13)(24), and (12)(34) and (12)(56), respectively. These can be realized as covering groups of a compact Riemann surface of genus 2. Recently, the speaker and collaborators showed that isomophism of number fields can be detected by equality of suitable L-series. This talk is about the finding the analogous result for manifolds. The result says that if two manifolds are finite Riemannian covers of a developable orbifold, and such that a certain homological condition is satisfied, then the manifolds are isometric if and only if the spectra of finitely many Laplacians twisted by suitable unitary representations of the fundamental group are equal. The result is explicit: in the above example, one needs 56 spectral equalities corresponding to 180-dimensional representations. (Joint work with Norbert Peyerimhoff.)

**November 30 **ONLINE SEMINAR

**Robert Osburn (University College Dublin)**

*Generalized Fishburn numbers, torus knots and quantum modularity*

The Fishburn numbers are a sequence of positive integers with numerous combinatorial interpretations and interesting asymptotic properties. In 2016, Andrews and Sellers initiated the study of arithmetic properties of these numbers. In this talk, we discuss a generalization of this sequence using knot theory and the quantum modularity of the associated Kontsevich-Zagier series.

The first part is joint work with Colin Bijaoui (McMaster), Hans Boden (McMaster), Beckham Myers (Harvard), Will Rushworth (McMaster), Aaron Tronsgard (Toronto) and Shaoyang Zhou (Vanderbilt) while the second part is joint work with Ankush Goswami (RISC).

**November 23 **ONLINE SEMINAR

**Wadim Zudilin (Radboud University, Nijmegen)**

*Dwork-type (*q*-)(super)congruences [video][slides]*

The "microscope" principle is this: If a rational function A(q) of variable q vanishes at every p-th root of unity (for p prime), then A(q) == 0 modulo Φ_p(q), the p-th cyclotomic polynomial; assuming that A(1) is a well-defined rational number with a p-free denominator and specialising the congruence at q=1 we conclude with A(1) == 0 modulo p.

In other words, behaviour of rational functions at p-th roots of unity may be instructive for gaining information about their values at 1 modulo p. With some "creative" extras, we can further consider divisibility by higher powers of primes (and we can even deal with not necessarily primes).

In my talk, partly based on recent joint work with Victor Guo, I plan to highlight some novel outcomes of this "creative microscope" methodology -- examples of Dwork-type supercongruences for truncated hypergeometric sums.

**November 16 **ONLINE SEMINAR** **

**13:15-14:15 Matija Kazalicki (University of Zagreb)**

*Congruences for sporadic sequences, three fold covers of the elliptic modular surfaces and modular forms for non-congruence subgroups [video][slides]*

In 1979, in the course of the proof of the irrationality of zeta(2) Apery introduced numbers b_n defined by binomial sums that are,

surprisingly, integral solutions of recursive relations (n+1)^2 u_{n+1} - (11n^2+11n+3) u_n-n^2 u_{n-1} = 0. Zagier performed a computer search on first 100 million integer triples (A,B,C) and found that the recursive relation generalizing b_n (n+1)u_{n+1} - (An^2+An+B)u_n + C n ^2 u_{n-1}=0, with the initial conditions u_{-1}=0 and u_0=1 has (non-degenerate) integral solution for only six more triples (whose solutions are so called sporadic sequences) .

Apery numbers have many interesting properties. For example, Stienstra and Beukers showed that for prime p=a^2+b^2 (where a is odd) there is a congruence b_{(p-1)/2} = 4a^2-2p mod p.

Recently, Osburn and Straub proved similar congruences for all but one of the six Zagier's sporadic sequences (three cases were already known to be true by the work of Stienstra and Beukers) and we proved the congruence for the sixth sequence. In this talk we will describe congruences for the Apery numbers b_{(p-1)/3} (and also for the other sporadic sequences). For that we study Atkin and Swinnerton-Dyer type of congruences between Fourier coefficients of cusp forms for non-congruence subgroups, L-functions of three covers of elliptic modular surfaces and Galois representations attached to these covers.

**November 9 **ONLINE SEMINAR

**13:15-14:15 Masha Vlasenko (IMPAN) **

*Dwork crystals [video][slides]*

In his work on rationality of zeta functions of algebraic varieties Bernard Dwork discovered a number of remarkable p-adic congruences. In this talk I will demonstrate some of these congruences and overview our recent work with Frits Beukers which explains their underlying mechanism.

**November 2 **ONLINE SEMINAR

**13:15-14:15 Julie Desjardins (University of Toronto)**

*Density of rational points on a family of del Pezzo surface of degree 1 [video]*

Let k be a number field and X an algebraic variety over k. We want to study the set of k-rational points X(k). For example, is X(k) empty? If not, is it dense with respect to the Zariski topology? Del Pezzo surfaces are classified by their degrees d (an integer between 1 and 9). Manin and various authors proved that for all del Pezzo surfaces of degree >1 it is dense provided that the surface has a k-rational point (that lies outside a specific subset of the surface for d=2). For d=1, the del Pezzo surface always has a rational point. However, we don't know if the set of rational points is Zariski-dense. In this talk, I present a result that is joint with Rosa Winter in which we prove the density of rational points for a specific family of del Pezzo surfaces of degree 1 over k.

**October 26 **ONLINE SEMINAR

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

*Universal optimality of the E8 and Leech lattices [slides][video]*

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.

**October 19 **ONLINE SEMINAR

**13:15-14:15 Jakub Byszewski (Jagellonian University, Kraków)**

*Finite order automorphisms of the ring of power series over a finite field *

*[video (the record only shows the first 30 minutes of the talk, we are sorry for the technical failure that occured) [slides]*

The Nottingham group at a prime $p$ is the group of (formal) power series $t+a_2 t^2+ a_3 t^3+ \cdots$ in the variable $t$ with coefficients $a_i$ from the field with $p$ elements with the group operation given by composition of power series. This group is known to contain elements of order being an arbitrary power of $p$. Elements of order $p$ have been classified by Klopsch and have a nice description. For higher orders, however, only a handful of examples have been known explicitly.

In the talk we will show how to describe such series in closed computational form through finite automata. This allows us to construct many explicit examples and formulate a number of questions. The talk is based on joint work with Gunther Cornelissen and Djurre Tijsma.

**October 12 **ONLINE SEMINAR

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

*Solution of the conjecture of Schinzel and Zassenhaus and some applications [video][slides]*

We will detail the full proof of an explicit form of the Schinzel-Zassenhaus conjecture: an algebraic integer of degree $n > 1$ is either a root of unity or else has at least one conjugate of modulus exceeding $2^{1/(4n)}$. We furthermore obtain an extension of the original conjecture over to the setting of holonomic functions, with an application to the smallest critical value for (certain) rational functions.

In another application, we would like to take the occasion to raise the apparently unsolved problem of the essential irreducibility (up-to the cyclotomic factor $X^2-X+1$ in degrees a multiple of 12) of $X^{2g} - X^g(1+X+1/X) + 1$, the characteristic polynomial of the integer reciprocal Perron-Frobenius matrix of the smallest spectral radius in each given dimension. Our explicit Schinzel-Zassenhaus bound allows for at most $10$ factors of each of these polynomials.

**October 5 **ONLINE SEMINAR

**13:15-14:15 Dorota Blinkiewicz (Adam Mickiewicz University, Poznań)**

*Classes of extensions of commutative algebraic groups*

During the lecture I give explicit characterization of n-torsion elements in the group of extensions of commutative, smooth algebraic groups.

SUMMER BREAK

**June 29** ONLINE SEMINAR

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

*The block filtration and motivic multiple zeta values [slides]*

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 [video]

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 [video]*

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 *

**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.