# Number Theory Seminar 2021/22

Co-organized by Giacomo Cherubini.

**18. 5. Piotr Miska (Kraków), ****On Pythagoras numbers of Henselian rings and fields**

*By n-length of an element a of a ring R we mean the smallest number of n-th powers in R such that their sum is equal to a. If such a number does not exist we set the n-length of a to be infinite. Then we define the n-Pythagoras number p_n(R) of R as the supremum of the set of all finite n-lengths of elements of R. We also consider the n-level s_n(R) of R as the n-length of -1.*

*The subject of n-Pythagoras numbers, especially the classical ones for n=2, is widely studied. In the language of Pythagoras numbers the famous result of Lagrange can be formulated that the 2-Pythagoras numbers of Z and Q are equal to 4. Moreover, we know that if s_2(R) is finite, then s_2(R) ≤ p_2(R) ≤ s_2(R)+2, where the upper bound can be improved to s_2(R)+1 on condition that 2 is invertible in R. In the case of n>2, by the result of Riley, meanwhile, we can conclude that the 3-Pythagoras number of any field is not greater than 3.*

*The aim of the talk is to present some general results on n-Pythagoras numbers and n-levels of Henselian rings and their fields of fractions. As we will see, in most cases the computation of these two values comes down to checking the n-Pythagoras number and n-level of some quotient ring, usually the residual field of the ring. Next, we will apply the general results to rings of power series, coordinate rings of varieties and rings of p-adic numbers.*

**11. 5. Francesco Battistoni (Milan), ****Inequalities for the study of small regulators of number fields**

*Applications of Geometry of Numbers and explicit formulae of Dedekind Zeta functions allowed Astudillo, Diaz y Diaz and Friedman to provide a method which, under certain conditions, detects the number fields with given signature and minimum regulator; however, this method can be improved by looking for the true upper bounds of a inequality used in the procedure. We show how to obtain this true upper bound in the case of totally real fields (thus solving a conjecture by Pohst) and how to get upper bounds very close to the optimal one in the case of fields with one complex embedding. As a consequence, wider lists of fields with one complex embedding and small regulator are obtained. This is a joint work with Giuseppe Molteni (University of Milan).*

**4. 5. Nadir Murru (Trento), ****Periodicity properties for continued fractions in ****Q_p**

**Q_p**

*Continued fractions have been widely studied in the field of p-adic numbers Q_p, but currently there is no algorithm replicating all the good properties that continued fractions have over the real numbers regarding, in particular, finiteness and periodicity. In this talk, we give a brief overview about the various algorithms for defining continued fractions in Q_p, focusing on the Browkin’s approach. After this, we will see some results about the periodicity of Browkin’s continued fractions regarding the length of the preperiod for periodic expansions of square roots and we prove that there exist infinitely many square roots of integers in Q_p that have a periodic expansion with period of length four. Finally, we propose a periodic representation for any quadratic irrational via p-adic continued fractions, even if it is not obtained by a specific algorithm. This periodic representation provides simultaneous rational approximations for a quadratic irrational both in R and Q_p. Moreover given two primes p_1 and p_2, using the Binomial transform, we are also able to pass from approximations in Q_{p_1} to approximations in Q_{p_2} for a given quadratic irrational.*

**27. 4. András Bíró (Budapest), ****Class number one problem for a family of real quadratic fields**

*We consider the class number one problem for such families of real quadratic fields, where the fundamental unit is small. We sketch our solution (given in 2003) of this problem for the Yokoi family n^2+4, and mention a new result which extends this solution to such a family where the fundamental unit may be comparatively larger than in the Yokoi family.*

**20. 4. Mikołaj Frączyk (Chicago), ****Homotopy type of arithmetic locally symmetric spaces**

*Let X be a symmetric space. The collar lemma, also known as the Margulis lemma, says that there exists an epsilon=epsilon(X) such that the epsilon-thin part of a locally symmetric space X/\Gamma looks locally like a quotient by a virtually unipotent subgroup. It turns out that in the arithmetic setting we can improve this lemma by making the epsilon grow linearly in the degree of the number field generated by the traces of elements of Gamma. I will explain why this is the case and present several applications, including the proof that an arithmetic locally symmetric manifold M is homotopy equivalent to a simplicial complex of size bounded linearly in the volume of M and degrees of all vertices bounded uniformly in terms of X. Based on a joint work with Sebastian Hurtado and Jean Raimbault.*

**13. 4. Anna Szumowicz (Caltech), ****Quantitative theory of the trace character**

*Let G be a p-adic reductive group. J.-L. Kim, S. W. Shin and N. Templier conjectured that for every regular element gamma of G, the trace character theta_pi (gamma) grows much faster than deg(pi) when deg(pi) tends to infinity and pi runs over irreducible supercuspidal representations of G. J.-L. Kim, S.W Shin and N. Templier proved the conjecture under some additional conditions. We discuss the further progress on the problem. I will assume no prior knowledge on representation theory of p-adic groups.*

**30. 3. Bartłomiej Bosek (Kraków), ****From 1-2-3 conjecture to Riemann hypothesis**

*We consider some coloring issues related to the famous Erdős Discrepancy Problem. A set of the form A_{s,k}={s,2s,…,ks}, with s,k∈N, is called a homogeneous arithmetic progression. We prove that for every fixed k there exists a 2-coloring of N such that every set A_{s,k} is perfectly balanced (the numbers of red and blue elements in the set A_{s,k} differ by at most one). This prompts reflection on various restricted versions of Erdős’ problem, obtained by imposing diverse confinements on parameters s,k. In a slightly different direction, we discuss a majority variant of the problem, in which each set A_{s,k} should have an excess of elements colored differently than the first element in the set. This problem leads, unexpectedly, to some deep questions concerning completely multiplicative functions with values in {+1,−1}. In particular, whether there is such a function with partial sums bounded from above.*

**16. 3. Matěj Doležálek, ****Subfields of number field extensions and quadratic forms**

*A number of recent results give constructions of number fields of small degrees that do not admit universal quadratic forms of small rank. Given a number field L that is known to have a certain lower bound on the rank of universal quadratic forms, one may try to construct extensions of L that also satisfy this bound. I will present a way of constructing such an extension as the compositum of L and some suitable number field K. The focus will be on ensuring some restrictions on the subfields of KL and examining these restrictions using Galois correspondence, which leads to studying subgroups in direct products of groups. This talk is based on my upcoming bachelor’s thesis.*

**3. 2. Bryce Kerr (Bonn), ****Inverse theorems for power sums**

*This talk is focused on problems which aim to extract structure from sequences of complex numbers which are close to extermal in Turán’s power sum problems. We give some motivation for such problems, sketch some basic results in this direction and conclude with open problems.*

**25. 1. Lorenzo Stefanello (Pisa), ****Local Galois module theory and ramification: An overview**

*The study of the Galois module structure of the valuation ring of a p-adic field has always met a great interest in number theory, and it has stimulated the minds of many great mathematicians. In this expository talk, we will see an overview of the main results of the theory, with a special focus on its interactions with the theory of ramification of p-adic extensions. The knowledge of p-adic fields would be useful but not necessary, as we will discuss the main definitions and properties at the beginning of the talk.*

**4. 1. Jakub Krásenský, ****On quadratic Waring’s problem in totally real number fields**

*The Pythagoras number P(R) of a ring R is the smallest number of squares which suffices to represent any sum of squares. For example, P(Z) = 4, P(O_K) = 3 for K = Q(sqrt(5)) and P(C) = 1. The g-invariants g_R(r) of a ring R are a generalisation of the Pythagoras number, where squares of numbers are replaced by squares of linear forms in r variables, so the resulting sums are quadratic forms in r variables. The study of g-invariants of the ring Z of rational integers is called the quadratic Waring’s problem and goes back to 1930’s. In this talk, I shall present the results of an eponymous paper with Pavlo Yatsyna (which will appear on arXiv on the 3rd of January), which significantly extends the recent results by Chan and Icaza about the g-invariants of O_K where K is a number field.*

**14. 12. Matteo Bordignon (Canberra), ****Some new results on the explicit Pólya-Vinogradov inequality**

*In the talk we will present three recent results regarding the explicit version of the Pólya–Vinogradov inequality: the best explicit version for Pólya–Vinogradov for large primitive characters, one for square-free characters (joint work with Bryce Kerr, MPIM) and at last how these are explicitly related to Burgess’s bound (joint work with Forrest Francis, UNSW).* Abstract

**30. 11. Daan van Gent (Leiden), ****Indecomposable algebraic integers**

*In algebraic number theory, the finiteness of the ideal class group of an order in a number field is generally proved via a lattice argument: the order forms a lattice and every ideal class contains an integral ideal with a small enough non-zero element. In this talk we will consider \overline{Z}, the ring of algebraic integers, which is a lattice in a similar sense, and we will treat this lattice as intrinsically interesting. We will state several open problems and some partial results, mainly with regards to the Voronoi polyhedron of \overline{Z}. This motivates the study of the indecomposable elements of \overline{Z}.*

**23. 11. Pawel Gladki (Katowice), ****On the categories of superpowersets and superpowergroups **

*In this talk we shall introduce the notion of superpowersets and superpowergroups, which are objects that provide a new approach to studying powersets of groups, as well as hypergroups (that is group-like objects with multivalued binary operation) and some fuzzy algebras. As an application, we shall show how the classical quadratic form theory over fields can be axiomatized using these tools, and how some rather difficult to phrase questions such as the question of Witt equivalence of fields, can be easily stated in the language of superpowergroups.*

**16. 11. Alessandro Fazzari****, ****On the behaviour of the Riemann zeta function on the critical line**

*The Riemann zeta function is of major interest in number theory, even more when we look at its behaviour on the so-called critical line. In the talk we will focus on the value-distribution of zeta there, exploring both the classical Selberg’s central limit theorem with its meaningful consequences and some newer results, still subject of research.*

**9. 11. Giacomo Cherubini, ****Introduction to the Schur-Siegel-Smyth trace problem**

*Consider a polynomial f with integer coefficients, degree n and trace t. If f has only positive real roots, then t is larger than n. Therefore, for any given number r greater than 1, we could consider the following question: as n and t vary, do we have infinitely many irreducible polynomials with trace t and degree n, such that t = r n ? The Schur-Siegel-Smyth trace problem asserts that if r is strictly smaller than 2 there should be only finitely many such polynomials. In the talk I will give an overview of known results and basic ideas to study this problem.*

**19. 10. Daniel Gil-Muñoz, ****An overview of Hopf-Galois theory**

*Hopf-Galois theory is a generalization of Galois theory with the use of Hopf algebras, in the sense that the class of Galois extensions is enlarged to a class of Hopf-Galois extensions on which the acting object is not a Galois group but a Hopf algebra. This notion was introduced by S. U. Chase and M. E. Sweedler in 1969 and has led to a significant body of research thereafter. This talk aims to introduce naturally the notion of Hopf-Galois extension from the point of view of Galois theory, and shows some of the main arising results. First, I shall focus on the case of separable Hopf-Galois extensions, where a surprising connection with the theory of skew braces and the Yang-Baxter equation has been found. In the last part of the talk, I will show the application of Hopf-Galois theory to Galois module theory, which investigates the module structure of the ring of algebraic integers of an extension of local or global fields.*

**12. 10. Bára Tížková, ****Number of variables of universal quadratic forms via traces of algebraic integers**

*In this accessible talk, I will take a look at the main ideas behind the proof of the following theorem: in each degree 2n, there are infinitely many totally real number fields that require universal quadratic forms to have arbitrarily large rank. This theorem was the topic of my Bachelor’s thesis and was proved by my supervisor Vítězslav Kala as an extension of his previous results concerning quadratic fields. The key step in the proof is to estimate the trace of an algebraic integer; therefore, after an introduction to the universal quadratic forms, I will give an overview of some relevant theory of traces and discriminants.*

**5. 10. Sudhir Pujahari (Warsaw), ****Arithmetic and statistics of sums of eigenvalues of Hecke operators**

*In the first part of the talk, we will see a brief introduction to the theory of equidistribution and Sato-Tate conjecture. In the second part of the talk we will study the distribution of gaps between eigenvalues of Hecke operators in both horizontal and vertical setting. In the final part of the talk, using group theory we will study the number of distinct prime factors of sums of eigenvalues of Hecke eigenforms. Moreover, we will see an all purpose Erdös-Kac theorem and use it to derive the classical Erdös-Kac theorem which states that the number of distinct prime factors of a random natural number are normally distributed. We will also obtain Erdös-Kac type results for several number theoretic objects including sums of eigenvalues of Hecke operators. This is joint work with M. Ram Murty and V. Kumar Murty.*