## Number Theory

In fall 2023 we have a study group on p-adic modular froms. The group meets at 14:30 on **Thursdays ****at IM PAN Warsaw, room 405**. Usually there is an **one hour prepared talk starting at 14:45**.

**January 18:** **Sebastian Gontarek (University of Wroclaw - MIM UW)*** Contraction property II*

**January 11: Piotr Achinger (IMPAN)** *Katz's geometric definition of p-adic modular forms *

**January 4: Piotr Achinger (IMPAN)** Tate's curve and q-expansion* *

CHRISTMAS BREAK

**December 21: Anna Goncharuk (Vasyl Karazin National University, Kharkiv) ***Ramanujan's congruences* - remote talk

**December 14: Masha Vlasenko (IMPAN)** *Analytic families of p-adic modular forms*

**December 7: Błażej Żmija (IMPAN)** *Lehner's congruences*

**November 30: Veronika Ertl (IMPAN)** *Modular *f*orms on Γ _{0}(p) are p-adically on SL_{2}(Z)*

**November 23: Anna Szumowicz (IMPAN)** *Application of the contraction property to the computation of the constant term of p-adic modular forms*

**November 16: Anna Szumowicz (IMPAN) ***Computation of the constant term of p-adic modular forms*

**November 9:*** ***Sebastian Gontarek (University of Wroclaw - MIM UW)*** The contraction property*

**November 2: **no talk

**October 26: Feliks Rączka (IMPAN)** *Hecke operators on p-adic modular forms*

**October 19: Daniel Vargas-Montoya (IMPAN)** *p-adic modular forms after Serre II*

**October 12: Daniel Vargas-Montoya (IMPAN)** *p-adic modular forms after Serre I*

**October 5: Błażej Żmija (IMPAN) ***Modular forms modulo p*

SUMMER BREAK

**June 5**

**13:00-14:00 Joanna Kułaga-Przymus (UMK Toruń)**

*Thermodynamic formalism for B-free dynamical systems*

Given B ⊆ N, we consider the corresponding set F_{B} of B-free integers, i.e. n ∈ F_{B} iff no b ∈ B divides n. We define Xη – the B-free subshift – as the smallest subshift containing the indicator η := **1**F_{B} ∈ {0, 1}^{Z} . Such systems are interesting both from the dynamical and number theoretical viewpoint and can manifest various types of behavior. I will concentrate on the entropy and its generalization – topological pressure, with some applications in combinatorics / number theory. The talk will include an overview of the topic, including some of my own results, joint with various subsets of Vitaly Bergelson, Aurelia Dymek, Stanisław Kasjan, Mariusz Lemańczyk, Michał Lemańczyk, Michał Rams, Florian Richter, Daniel Sell and Benjamin Weiss.

**14:15-15:15 Jolanta Marzec-Ballesteros**** (UKW Bydgoszcz)**

*Modular forms in coding theory*

With every linear code one can associate a lattice, and with every lattice - a theta function. It is then, perhaps, not surprising that there is a connection (a homomorphism!) between suitable spaces of codes and modular forms. What is remarkable, is the level of this resemblance: in both worlds there are functions invariant under an action of a group, notions of cusp forms and Hecke operators, also projections and lifts between different geni. The compatibility between these objects will be explained in the first part of the talk. This will prepare the ground for the second part: an adaptation of doubling method in the framework of binary linear codes. The new phenomena that will be presented are the effect of joint work with Thanasis Bouganis.

**May 8 **

**13:00-14:00 Błażej Żmija (Charles University, Prague)**

*Ranks of universal quadratic forms over quadratic fields [slides]*

Let K be totally real number field and *O*_{K} be its ring of integers. We say that a quadratic form with the coefficients in *O _{K}* is universal over K if it is totally positive definite and represents all the totally positive elements of

*O*

_{K}.

For a squarefree positive integer D let R(D) denote the minimal number of variables of a universal quadratic form over Q(\sqrt{D}). The aim of the talk is to present results regarding the typical size of R(D). In particular, we will show that if y>0 is fixed, then for almost all the numbers D (in the sense of natural density) R(D) is greater than D^{1/24-y}.

The talk is based on my joint work with V. Kala and P. Yatsyna.

**14:15-15:15 Anna Goncharuk (Vasyl Karazin National University, Kharkiv) via Zoom**

*Linear differential equation with formal power series inhomogeneity [slides]*

We will consider a linear inhomogeneous differential equation of m-th order with constant coefficients from the ring of p-adic integers. We establish sufficient condition for the uniqueness and existence of the solution of this equation in the ring of formal power series over Z_{p}.

Also we discuss about generalization of this result to other nonarchimedean rings.

In addition, for this type of equations we describe a construction, that is an analogue to the theory of distributions, the notion of convolution and the fundamental solution of the operator corresponding to the equation.

**March 6**

**13:00-14:00 Daniel Vargas Montoya (IMPAN)**

*Algebraicity of hypergeometric functions modulo p *

Recently, it was shown that if f(z) is a hypergeometric function with rational parameters and p is a prime number such that f(z) can be reduced modulo p then the reduction of f(z) modulo p, denoted f mod p, is algebraic over F_p(z) and its algebraicity degree is bounded by p^c where c is an explicit constant independent of p. In this talk, I will show how to construct an explicit polynomial P(z,y) with coefficients in F_p such that P(z ,f mod p)=0. It turns out that, in many examples, such a construction improves the estimate p^c and also gives an estimate of the height of f mod p.

**14:15-15:15 Wojtek Gajda (UAM Poznań) **

*On Mumford-Tate conjecture*

We will discuss some results of a recent work with Marc Hindry. We have checked the Mumford-Tate conjecture for several new classes of abelian varieties over number fields.

**December 5**

**13:00-14:00 Grzegorz Graff (Politechnika Gdańska)**

*On the degree of self-maps of simply-connected 4-manifolds: between topology and number theory*

We describe the restrictions for the degree of self-maps of simply-connected closed 4-manifolds in case their intersection forms represent some classes of symmetric unimodular bilinear forms over Z. It turns out that the topological problem may be translated to an interesting number-theoretic question which we are able to solve in many cases.

This is a joint work with Dominik Gulgowski and Piotr Nowak-Przygodzki. Research supported by the National Science Centre, Poland, within the grant Sheng 1 UMO-2018/30/Q/ST1/00228.

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

*Common valuations of division polynomials*

In this talk we discuss the results of joint work with Matteo Verzobio (arXiv.2203.02015) about the common valuation of the division

polynomials of points on elliptic curves. We prove a formula for the cancellation exponent between division polynomials psi and phi

associated with a sequence of points on an elliptic curve defined over a discrete valuation field. The formula is identical with the result

of Yabuta-Voutier for the case of finite extension of p-adic field and generalizes to the case of non-standard Kodaira types for non-perfect

residue fields.

**November 7**

**13:00-14:00 Jakub Byszewski (Jagellonian University, Kraków)**

*Finite-adelically distorted systems*

We introduce a class of dynamical systems, which have a nice formula for the number of periodic points. This class includes all endomorphism of algebraic groups in positive characteristic as well as many examples from topological dynamics. We study the asymptotics and arithmetic properties of the sequences counting their periodic points. As an application, we discuss some results concerning additive cellular automata. The talk is based on joint work with Gunther Cornelissen and Marc Houben.

**14:15-15:15 Jędrzej Garnek (Adam Mickiewicz University, Poznań)**

*p-group Galois cover of curves in characteristic p*

Studying cohomology of a variety with an action of a finite group is a classical and well-researched topic. However, most of the

results consider only the tame ramification case or concern the image of cohomology in the K-theory. During this talk I will

focus on the case of a curve over a field of characteristic p with an action of a finite p-group.

My previous research suggests that in this case the cohomology decomposes as a sum of certain 'local' and 'global' parts.

I will show that this is true under some mild assumptions.

SUMMER BREAK

**June 6, 2022**

**13:00-14:00 Eugenia Rosu (TU Darmstadt and University of Leiden)**

*Twists of elliptic curves with CM*

We consider certain families of sextic twists of the elliptic curve y^2=x^3+1 that are not defined over Q, but over Q[sqrt(-3)]. We compute a formula that relates the central value of their L-functions L(E, 1) to the square of a trace of a modular function evaluated at a CM point. Assuming the Birch and Swinnerton-Dyer conjecture, when the value above is non-zero, we should recover the order of the Tate-Shafarevich group, and we show that the value is indeed an integer square.

**14:15-15:15 Daniel Vargas Montoya (IMPAN)**

* q-strong Frobenius structure*

The notion of strong Frobenius structure is classically studied in the theory of p-adic differential equations. This notion was introduced by B.Dwork in his study of zeta functions. Recently, we propose a new definition of this notion for q-difference operators. The relevance of this definition is supported for two results. The first one deals with confluence and the second one deals with congruence modulo the cyclotomic polynomial. So the first part of the talk is devoted to presenting our definition of q-strong Frobenius structure and the second part we are going to present the two previous results.

**May 9, 2022 **

**13:00-14:00 Gwladys Fernandes (Université de Versailles Saint-Quentin-en-Yvelines) via Zoom**

*Hypertranscendence of solutions of linear difference equations*

The general question of this talk is the classification of differentially algebraic solutions of linear difference equations of the following type:

(*) a_{0}(z)f(z) + a_{1}(z)f(R(z)) + ... + a_{n}(z)f(R^{n}(z))=0,

where, for every i, a_{i}(z) є C(z), R(z) є C(z) and R^{n}(z) is the n-th composition of R(z) with itself. We say that such a function is *differentially algebraic* over C(z) if there exist an non-zero integer n and a non-zero polynomial P є C(z)[X_{0},..., X_{n}] such that P(f(z),..., f^{(n)}(z))=0, where f^{(i)} is the i-th derivative of f with respect to z. Otherwise, it is *hypertranscendental* over C(z).

The classification of differentially algebraic solutions is known for three types of non-linear difference equations : the Schröder's, Böttcher's and Abel's equations : f(qz)=R(f(z)), f(z^{d})=R(f(z)), f(R(z))=f(z)+1, respectively, where q є C^{*}, d є N, d >= 2. A classification of the differential algebraicity of solutions of linear difference equations of the above type (*) is made in an article of B. Adamczewski, T. Dreyfus, C. Hardouin, for these same operators : q-differences z -> qz, mahlerian z -> z^{d}, and shift z -> z+1, by the means of an adapted difference Galois theory.

In this talk, we discuss the generalisation of these results to any function R (rational or algebraic over C(z)), in the case where (*) is of order 1. This is a work in progress with L. Di Vizio. Natural applications appear in examples of gererating series of random walks, which satisfy this kind of equation of order 1.

**14:15-15:15 Matthias Storzer (MPIM Bonn) via Zoom**

*Modularity of Nahm sums*

The modularity properties of so-called Nahm sums are known to be related to certain elements in the Bloch group. Nevertheless, a first conjecture about the characterisation of modular Nahm sums in terms of these elements turned out to be false. In this talk, we will review the motivation behind the conjecture and discuss why it fails, which could lead to a refined version.

**April 4, 2022 **

**13:00-14:00 Pierre Colmez (Institut de Mathématiques de Jussieu-Paris Rive Gauche****)**

*The upper half-planes*

We will give a short introduction to the geometric part of the p-adic Langlands program.

**14:15-15:15** **Piotr Miska (Jagellonian University, Kraków) via Zoom**

*(R)-dense and (N)-dense subsets of positive integers and generalized quotient sets*

A subset A of the set of positive integers is (R)-dense if its quotient set R(A)={a/b: a, b in A} is dense in the positive real half-line (with respect to natural topology on real numbers). It is a classical result that the set of prime numbers is (R)-dense. The proof of this fact is based on the property of counting function of prime numbers. Actually, this proof shows something more. Namely, for each infinite subset B of the set of positive integers, the set R(P,B)={p/b: p in P, b in B} is dense in the set of positive real numbers. This motivates to introduce the notion of (N)-denseness. We say that a set A of positive integers is (N)-dense if the set R(A,B) is dense in the set of positive real numbers for every set B of positive integers. During the talk we will consider characterizations of (N)-dense sets and connections between (N)-denseness of a given set.

In 2019 Leonetti and Sanna introduced the notion of direction sets D^k(A)={(a_1/||a||^2, ..., a_k/||a||^2): a=(a_1, ..., a_k) in A^k} that allows us to generalize the property of (R)-denseness. Indeed, A is (R)-dense if and only if D^2(A) is dense in the set of points of unit circle with all the coordinates positive. We will see that denseness of D^k(A) in the set of points of unit sphere with all the coordinates positive is equivalent to denseness of the generalized quotient set R^k(A)={(a_1/a_k, ..., a_{k-1}/a_k): a_1, ..., a_k in A} in the set of points of R^{k-1} with all the coordinates positive.

We will also show some connections between (N)-denseness of a given set A and denseness of sets R^k(A) with the counting function of A and its dispersion.

The talk is based on a joint work with János T. Tóth.

**March 7, 2022 **

**13:00-14:00** **Maciej Ulas (Jagellonian University, Kraków) via Zoom**

*On solutions of certain meta-Fibonacci recurrences*

During the talk I will speak about recent findings concerning the solutions of the recurrence sequence h(n)=h(n-h(n-1))+h(n-2). This is a member of a class of so called meta-Fibonacci (or exotic) sequences. We show that for a broad class of initial conditions the behavior of the solutions is easy, i.e., governed by sequences satisfying linear recurrence with constant coefficients or is closely related to certain functions counting binary partitions of special type. The talk is based on a joint work with Bartosz Sobolewski.

**14:15-15:15 Neelam Saikia (University of Virginia) via Zoom**

*Traces of pth Hecke operators and p-adic hypergeometric functions*

McCarthy defined hypergeometric functions in the p-adic setting using p-adic gamma functions. This function can be described as p-adic analogue of classical hypergeometric function. In this talk we discuss the traces of pth Hecke operators acting on spaces of cusp forms of weight k and level 1 and their relations with p-adic hypergeometric functions. As a consequence of this result we establish relations of Ramanujan’s tau-function and p-adic hypergeometric functions. This is a joint work with Sudhir Pujahari.

WINTER BREAK

**December 6, 2021**

**13:00-14:00 Jakub Byszewski (Jagellonian University, Kraków)**

*Tiling a rectangle by rectangles*

In 1903 Dehn showed that a rectangle can be tiled by squares if and only if the ratio of its sides is a rational number. In 1994--95 Freiling--Rinne and Laczkovich--Szekeres solved a converse problem, namely, for which real numbers r can one tile a square by rectangles with side ratio r; the answer is much more interesting---the property holds if and only if r is an algebraic number all of whose conjugates have positive real part. Soon afterwards the result was generalised to tilings of an arbitrary rectangle by rectangles with a given side ratio. Unfortunately, in this case the corresponding condition is somewhat mysterious, and it was not clear whether it was algorithmically decidable.

In the talk, we give a geometric interpretation of this condition and we show that it is decidable. We use these methods to generalise the above results to tilings of squares by finitely many rectangles of possibly different side ratio under the extra assumption that all these ratios are algebraic numbers. We also classify all pairs of rectangles which are mutually tileable (there are nontrivial examples!).

**14:15-15:15 Bidisha Roy (IMPAN)**

*On ranks of quadratic twists of a Mordell curve*

Ranks of elliptic curves is a classical topic and it has a vast literature in algebraic number theory. In this talk, we will consider the quadratic twists of the Mordell curve E : y^{2} = x^{3} − 1. For a square-free integer k, the quadratic twist is given by E_{k} : y^{2} = x^{3} − k^{3}. In the first part of this talk, we will see that there exist infinitely many k with more than one prime factors such that the rank of E_{k} is 0. Next, we will conclude by witnessing an infinite

family of curves {E_{k} } such that the rank of each E_{k} is positive.

**November 8, 2021**

**13:00-14:00 Krystian Gajdzica (Jagellonian University, Kraków)**

*Arithmetic properties of the restricted partition function*

Let *A*=(a_{n})_{n>0} be a sequence of positive integers. The function p* _{A}*(n,k) enumerates the number of multi-color partitions of n into parts in {a

_{1}, a

_{2 }, ... ,a

_{k}}. We present several arithmetic properties of the sequence (p

_{A}(n,k) mod m)

_{n>0 }for an arbitrary fixed integer m >= 2, and apply them to the special cases of

*A*. In particular, for a given parameter k, we investigate both the upper bound for the odd density of p

_{A}(n,k) and the lower bound for the density of { n > 0 : p

_{A}(n,k) != 0 mod m }. Moreover, we perform some new results related to restricted m-ary partitions and state a few open questions at the end of the talk.

**14:15-15:15 Masha Vlasenko (IMPAN, Warsaw)**

*Integrality of instanton numbers*

Instanton numbers of Calabi--Yau threefolds are defined by Gromov--Witten theory. They 'count' curves of fixed degree on the manifold. The actual definition involves integration over the moduli space of curves, which gives a priori rational numbers. Integrality of instanton numbers of Calabi--Yau threefolds is an arithmetic counterpart of the mirror symmetry conjecture. The mirror theorem allows to express them in terms of solutions of a differential equation on the mirror manifold. However, the integrality of instanton numbers is not clear from this expression either. In 2003 Jan Stienstra outlined an approach to integrality using the p-adic Frobenius structure on the differential equation. In a recent series of papers with Frits Beukers we propose an explicit and rather elementary construction of the Frobenius structure, which allows us to prove integrality of instanton numbers in several key examples of mirror symmetry. In this talk I will speak about the beginnings of mirror symmetry and explain the ideas of our construction.

**October 4, 2021**

**13:00-14:00 Bartosz Naskręcki (Adam Mickiewicz University, Poznań)**

*Higher moments of elliptic curves*

Abstract: In this talk we will discuss higher moments of parametric families of elliptic curves. We investigate in particular the second moment of one-parametric families. After the work of Michel, it is known that these moments have motivic origin and we will study second moment related to a large family of cubic pencils, indicating the connection between the second moment and the point count on a certain genus 2 curve. Under the general Sato-Tate conjecture for curves we prove that the second moment has the expected negative bias, proving the conjecture of Steven Miller in this context. Finally, we will mention possible generalizations and other explicit motivic constructions related to second moments.

**14:15 - 15:15 Jolanta Marzec (Kazimierz Wielki University, Bydgoszcz) **

*Construction of Poincaré-type series by generating kernels*

The Poincaré-type series mentioned in the title refer to "nice" vector valued functions defined on the complex upper half-plane which transform in a suitable way with respect to a multiplier system of real weight k under the action of a Fuchsian group of the first kind. As we will explain, they are very closely related to certain automorphic kernels which admit a spectral expansion with respect to the eigenfunctions of the hyperbolic weighted Laplacian of weight k. Following an approach of Jorgenson, von Pippich and Smajlović (where k=0), we use spectral expansion associated to the Laplacian to first construct wave distribution and then identify conditions on its test functions under which it represents automorphic kernels. As we will see, one of advantages of this method is that the resulting series may be naturally meromorphically continued to the whole complex plane.

This talk presents joint work with Y. Kara, M. Kumari, K. Maurischat, A. Mocanu and L. Smajlović.

SUMMER BREAK

**June 28, 2021** ONLINE SEMINAR

**Vasily Golyshev (Moscow)**

*Modularity proofs via fibered motives*

I will explain how techniques of fibered hypergeometric motives can be used to provide `opportunistic' modularity proofs for conifold fibers in Calabi-Yau families. This is a report on joint work with Don Zagier, and work in progress with Kilian Bönisch and Albrecht Klemm.

**June 21, 2021** ONLINE SEMINAR

**Lars Kühne (University of Copenhagen)**

*Equidistribution and Uniformity in Families of Curves [video]*

In the talk, I will present an equidistribution result for families of (non-degenerate) subvarieties in a (general) family of abelian varieties. This extends a result of DeMarco and Mavraki for curves in fibered products of elliptic surfaces. Using this result, one can deduce a uniform version of the classical Bogomolov conjecture for curves embedded in their Jacobians, namely that the number of torsion points lying on them is uniformly bounded in the genus of the curve. This has been previously only known in a few select cases by work of David–Philippon and DeMarco–Krieger–Ye. Finally, one can obtain a rather uniform version of the Mordell-Lang conjecture as well by complementing a result of Dimitrov–Gao–Habegger: The number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Again, this was previously known only under additional assumptions (Stoll, Katz–Rabinoff–Zureick-Brown).

**June 14, 2021** ONLINE SEMINAR

**Jason Bell (University of Waterloo)**

*Effective isotrivial Mordell-Lang in positive characteristic [slides][video]*

The Mordell-Lang conjecture (now a theorem, proved by Faltings, Vojta, McQuillan, …) asserts that if *G* is a semiabelian variety *G* defined over an algebraically closed field of characteristic zero, *X* is a subvariety of *G*, and Γ is a finite rank subgroup of *G,* then *X *∩ Γ is a finite union of cosets of Γ. In positive characteristic, the naive translation of this theorem does not hold, however Hrushovski, using model theoretic techniques, showed that in some sense all counterexamples arise from semiabelian varieties defined over finite fields (the isotrivial case). This was later refined by Moosa and Scanlon, who showed in the isotrivial case that the intersection of a subvariety of a semiabelian variety *G* with a finitely generated subgroup Γ of *G* that is invariant under the Frobenius endomorphism *F*:*G→G* is a finite union of sets of the form *S*+*A*, where *A* is a subgroup of Γ and *S* is a sum of orbits under the map *F*. We show how how one can use finite-state automata to give a concrete description of these intersections Γ ∩ *X* in the isotrivial setting, by constructing a finite machine that identifies all points in the intersection. In particular, this allows us to give decision procedures for answering questions such as: is *X *∩ Γ empty? finite? does it contain a coset of an infinite subgroup? In addition, we are able to read off coarse asymptotic estimates for the number of points of height ≤*H* in the intersection from the machine. This is joint work with Dragos Ghioca and Rahim Moosa.

**June 7, 2021** ONLINE SEMINAR

**Michael J. Schlosser (University of Vienna)**

*On the infinite Borwein product raised to a real power [video][slides]*

We study the $q$-series coefficients appearing in the expansion of $\prod_{n\ge 1}[(1-q^n)/(1-q^{pn})]^\delta$, the infinite Borwein product for an arbitrary prime $p$, raised to an arbitrary positive real power $\delta$. Application of the Hardy-Ramanujan-Rademacher circle method gives an asymptotic formula for the coefficients. For $p=3$ we give an estimate of their growth which enables us to partially confirm an earlier conjecture we made concerning an observed sign pattern of the coefficients when the exponent $\delta$ is within a specified range of positive real numbers. We then take a closer look at the cube of the infinite Borwein product, for arbitrary $p$ (now a positive integer), and establish some vanishing and divisibility properties of the respective coefficients.

This is joint work with Nian Hong Zhou.

**May 31, 2021 ** ONLINE SEMINAR

**Daniel Vargas-Montoya (University Lyon 1)**

*Algebraicity modulo p of G functions, hypergeometric series and strong Frobenius structure*

*[slides][video] (we apologize for the background noise caused by the low quality of the internet connection)*

B. Dwork in his work about zeta function of a hypersurface over finite fields introduced the notion of strong Frobenius structure. In this talk we are going to take up this notion for the study of algebraicity modulo p of Siegel G functions, where p is a prime number.

Firstly, we are going to see that if f(t) is a power series (or Siegel G function) with coefficients in the ring of integers Z and if f(t) is solution of a differential operator L having strong Frobenius structure for p of period h, then the reduction of f modulo p is algebraic over F_{p}(t) and its algebraicity degree is bounded by p^{n^2h}, where n is the order of L and F_{p} is the field of p elements. Secondly, we are going to show that, under reasonable hypotheses, rigid differential operators have a strong Frobenius structure for almost every prime number p.

Finally, we are going to illustrate our results with several examples coming of hypergeometric series of type _{n}F_{n-1}.

**May 24, 2021** ONLINE SEMINAR

**Éric Delaygue (University Lyon 1)**

*On primary pseudo-polynomials and around Rusza's Conjecture [slides]*

Every polynomial P(X) with integer coefficients satisfies the congruences P(n+m)=P(n) mod m for all integers n and m. An integer valued sequence is called a pseudo-polynomial when it satisfies these congruences. Hall characterized pseudo-polynomials and proved that they are not necessarily polynomials. A long standing conjecture of Ruzsa says that a pseudo-polynomial a(n) is a polynomial as soon as limsup |a_n|^{1/n} < e. A primary pseudo-polynomial is an integer valued sequence a(n) such that a(n+p)=a(n) mod p for all integers n ≥ 0 and all prime numbers p. The same conjecture has been formulated for them, which implies Ruzsa’s, and this talk will revolve around this conjecture.

**May 17, 2021** ONLINE SEMINAR

**Marco Streng (Universiteit Leiden****)**

*Obtaining modular units via a recurrence relation [video][slides]*

The modular curve Y1(N) parametrises pairs (E,P), where E is an elliptic curve and P is a point of order N on E. One tool for studying this curve is the group of modular units on it, that is, the group of algebraic functions with no poles or zeroes.

We first review how a recurrence relation (related to elliptic divisibility sequences) gives rise to defining equations for the curves Y1(N). We then show that the same recurrence relation also gives explicit algebraic formulae for a basis of the group of units on Y1(N).

This proves a conjecture of Maarten Derickx and Mark van Hoeij.

**May 10, 2021** ONLINE SEMINAR

**Robert Slob (Ulm University) **

*Primitive divisors of sequences associated to elliptic curves over function fields [video][slides]*

In the first part of the talk, we give a gentle introduction into the subject of divisibility sequences over the rational numbers and discuss the notion of a primitive divisor/Zsigmondy bound. We then explain how these notions can be extended to number fields and function fields, and how to obtain a divisibility sequence from a non-torsion point on an elliptic curve over any of these fields. There will also be plenty of nice examples.

In the second part of the talk, we discuss the typical methods that are used to prove the existence of a Zsigmondy bound for a divisibility sequence obtained from a non-torsion point on an elliptic curve E over a number or function field K. Let P be this non-torsion point in E(K), and suppose Q is a torsion point in E(K). We can also associate a sequence of divisors {D_{nP+Q}} on K to the sequence of points {nP+Q}. In my preprint, we proved the existence of a Zsigmondy bound for this sequence {D_{nP+Q}} for K a function field (under some minor conditions), extending the analogous result of Verzobio over number fields. I will provide the crucial ideas to apply the existing methods of the case {nP} to my case {nP+Q}. Additionally, I will highlight the differences with the number field case.

**April 26, 2021** ONLINE SEMINAR

**Valentijn Karemaker (University of Utrecht)**

*Polarisations of abelian varieties over finite fields via canonical liftings [video][slides]*

In this talk we will give a widely applicable and computable description of polarisations of abelian varieties over finite field. More precisely, we will describe all polarisations of all abelian varieties over a finite field in a fixed isogeny class corresponding to a squarefree Weil polynomial, when one variety in the isogeny class admits a canonical lifting to characteristic zero. The computability of the description relies on applying categorical equivalences between abelian varieties over finite fields and fractional ideals in étale algebras. This is joint work with Jonas Bergström and Stefano Marseglia.

**April 19, 2021** ONLINE SEMINAR

**Rusen Li (Shandong University)**

*Summation formulas of q-hyperharmonic numbers [video][slides]*

In 1990, Spieß gave some identities including the types of $\sum_{\ell=1}^n\ell^k H_\ell$, $\sum_{\ell=1}^n\ell^k H_{n-\ell}$ and $\sum_{\ell=1}^n\ell^k H_\ell H_{n-\ell}$. In this talk, based upon a certain type of $q$-harmonic numbers $H_n^{(r)}(q)$, several formulas of $q$-hyperharmonic numbers are derived as $q$-generalizations. The main tools used in the talk are Abel’s identity and a q-version of the relation by Spieß.

This is based on a joint work with Takao Komatsu.

**April 12, 2021** ONLINE SEMINAR

**Anna Szumowicz (Caltech)**

*Cuspidal types on GL _{p}(O) [video][slides]*

Let *F* be a non-Archimedean local field and let *O* be its ring of integers. We describe the cupidal types on GL_{p}(*O*) (where p is a prime number) using Clifford theory. This gives some information and invariants attached to cuspidal types called orbits. We give an example which shows that the orbit of a representation does not give enough information to determine whether a representation is a cuspidal type or not.

**March 29, 2021** ONLINE SEMINAR

**Boris Adamczewski (Institut Camille Jordan & CNRS)**

*Furstenberg's conjecture, Mahler's method, and finite automata [video]*

It is commonly expected that expansions of numbers in multiplicatively independent bases, such as 2 and 10, should have no common structure. However, it seems extraordinarily difficult to confirm this naive heuristic principle in some way or another. In the late 1960s, Furstenberg suggested a series of conjectures, which became famous and aim to capture this heuristic. The work I will discuss in this talk is motivated by one of these conjectures. Despite recent remarkable progress by Shmerkin and Wu, it remains totally out of reach of the current methods. While Furstenberg’s conjectures take place in a dynamical setting, I will use instead the language of automata theory to formulate some related problems that formalize and express in a different way the same general heuristic. I will explain how the latter can be solved thanks to some recent advances in Mahler’s method; a method in transcendental number theory initiated by Mahler at the end of the 1920s. This a joint work with Colin Faverjon.

**March 22, 2021** ONLINE SEMINAR

**Enis Kaya (University of Groningen)**

*Explicit Vologodsky integration for hyperelliptic curves [video][slides]*

Let *X* be a curve over a p-adic field with semi-stable reduction and let ω be a meromorphic 1-form on X. There are two notions of p-adic integration one may associate to this data: the Berkovich–Coleman integral which can be performed locally; and the Vologodsky integral with desirable number-theoretic properties. In this talk, we present a theorem comparing the two, and describe algorithms for computing these integrals in the case that X is a hyperelliptic curve. We also illustrate our algorithm with a numerical example computed in Sage. This talk is partly based on joint work with Eric Katz.

**March 15, 2021** ONLINE SEMINAR

**Carlo Verschoor (University of Utrecht)**

*Bailey Type Factorizations of Horn Functions [video][slides]*

A well-known identity by Bailey states that Appell’s F4 function can be written as the product of two Gauss hypergeometric functions under a suitable specialization of its parameters. Other identities of this type are known for Appell’s F2 and F3, which are closely related to Bailey’s identity. The aim of this talk is to show that the same can be done for Horn’s H1, H4 and H5 functions.

**March 8, 2021** ONLINE SEMINAR

**Sudhir Pujahari (University of Warsaw)**

*Arithmetic and statistics of sums of eigenvalues of Hecke operators [video]*

In the first part of the talk, we will study about the distribution of gaps between eigenvalues of Hecke operators in both horizontal and vertical settings. As an application of this we will obtain a strong multiplicity one theorem and evidence towards Maeda conjecture. The horizontal setting is a joint work with M. Ram Murty. In the second part of the talk, using recent developments in the theory of l-adic Galois representations we will study the normal number of prime factors

of sums of Fourier coefficients of eigenforms. Moreover, we will see the distribution of distinct prime factors of sums of Fourier coefficients of eigenforms. The final part is a joint work with M. Ram Murty and V. Kumar Murty.

**March 1, 2021** ONLINE SEMINAR

**Mikołaj Frączyk (University of Chicago)**

*Sarnak’s density hypothesis in horizontal families [video]*

Let G be a real semi simple Lie group with an irreducible unitary representation π. The non-temperedness of π is measured by the real parameter p(π) which is defined as the infimum of p≥2 such that π has non-zero matrix coefficients in L_{p}(G). Sarnak and Xue conjectured that for any arithmetic lattice Γ⊂G and principal congruence subgroup Γ(q)⊂Γ, the multiplicity of π in L_{2}(G/Γ(q)) is at most O(V(q)^{2}^{/p(π)+ϵ}) where V(q^{a}×SL(2,ℂ)^{b}. These families of lattices, which we call horizontal, are given as unit groups of maximal orders of quaternion algebras over number fields.

WINTER BREAK

**January 25, 2021 **ONLINE SEMINAR** **

**Oleksiy Klurman (University of Bristol)**

*Monotone chains of Hecke cusp forms [video]*

We discuss a general joint equidistribution result for the Fourier coefficients of Hecke cusp forms. One simple consequence of such a result is that there exist infinitely many integers n (in fact an upper density of this set is positive) such that τ(n)<τ(n+1)<τ(n+2) where τ is the Ramanujan τ-function. This is based on a joint work with A. Mangerel (CRM).

**January 18, 2021 **ONLINE SEMINAR** **

**Netan Dogra (King's College London)**

*Some new results in the nonabelian Chabauty method [video]*

Abstract: In this talk I will discuss the nonabelian Chabauty method, which seeks to use p-adic analytic functions to determine the finite sets of rational points on higher genus curves, and some new cases where it can be used to determine the solutions to Diophantine equations.

**January 11, 2021 **ONLINE SEMINAR

**Wojtek Wawrów (London School of Geometry and Number Theory)**

*Ranks of Jacobians over rational numbers*

The problem of finding out the possible Mordell-Weil ranks of abelian varieties of given dimension over a fixed number field has a long history. In this talk we shall survey various results which have appeared over the years bounding from below the maximal rank that those varieties can have, with particular emphasis on Jacobians of special families of curves over the field of rational numbers.** **

**December 21, 2020 **ONLINE SEMINAR** **

**Bidisha Roy (IMPAN)**

*Torsion groups of elliptic curves over number fields*

Computing torsion groups of elliptic curves defined over number fields is a classical topic and it has a vast literature in algebraic number theory. Any elliptic curve of the form y^{2} = x^{3} + c is called a Mordell curve. Mordell curves are well studied elliptic curves with complex multiplication.

In this talk, for Mordell curves defined over the field of rational numbers we will discuss the classification of torsion groups over cubic and sextic fields. Also, we present the classification of torsion groups of Mordell curves defined over cubic fields. For Mordell curves over sextic fields, we provide all possible torsion groups. In the second part, we briefly discuss torsion groups of Mordell curves over higher degree number fields.

**December 14, 2020 ** ONLINE SEMINAR

**Tim Dokchitser (University of Bristol)**

*Curves with tame torsion [video][slides]*

In this talk I will sketch the proof of the fact that in every genus and for every prime p there are curves over Q with tame p-torsion. In genus 1, this is something this is quite easy to deduce from the theory of the Tate curve, and I will explain how an explicit version of the theory of hyperelliptic Mumford curves gives this in arbitrary genus. This is joint work with Matthew Bisatt.

**December 7, 2020 **ONLINE SEMINAR

**Gunther Cornelissen (Utrecht University)**

*An analogy between number theory and spectral geometry* * [notes][video]*

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, 2020 **ONLINE SEMINAR

**Robert Osburn (University College Dublin)**

*Generalized Fishburn numbers, torus knots and quantum modularity [slides]*

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, 2020 **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, 2020 **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, 2020 **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, 2020 **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, 2020 **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, 2020 **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, 2020 **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, 2020 **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, 2020** 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, 2020** 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, 2020** 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, 2020** 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, 2020** 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, 2020** 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, 2020 ** SEMINAR CANCELLED

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

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

**March 23, 2020** SEMINAR CANCELLED

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

*Universal optimality of the E8 and Leech lattices *

**March 9, 2020**

**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, 2020**

**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.