## Number Theory Seminar

Charles University

During the 2022-23 academic year, this seminar is coorganised by Vita Kala and myself.

It runs on Wednesday 14:00 (with some exceptions). We have both online and in-person talks. The last ones take place in the Karlin building, Room K3.

The seminar is intended for students and faculty members, so the level of the talks should be accessible to these people.

We have a mailing list for announcing upcoming talks (and occasionally other related things). If you want to be added to the mailing list, you can contact me or Vita.

Previous talks:

03.05.2023 Tomáš Perutka (Masaryk University)

Title: Adic spaces and condensed sets

In this talk, I am going to introduce the fundamental notion of p-adic geometry, the adic spaces. They emerge from the issues with defining p-adic manifolds: the topology on p-adics is just too weird for the naively defined manifolds to behave nicely. For that reason, adic spaces are defined more in the scheme-like approach. After introducing the basic concepts, I will also talk about the problem of forming the category of quasi coherent sheaves over adic spaces, a problem being resolved by introducing condensed sets. Time permitting, I will talk a bit more about condensed sets and their other applications.

26.04.2023 Daniel Kriz (Institut de mathématiques de Jussieu)

Title: Congruent numbers, Sylvester's conjecture and Goldfeld's conjecture

The Birch and Swinnerton-Dyer conjecture relates the rank of the group of rational solutions of a plane cubic equation (rational points on an elliptic curve) to the order of vanishing of its associated generating function (called the L-function) at the point s=1. The group of rational points can be approximated by the more computable Selmer group, and one needs to prove that the discrepancy, called the Shafarevich-Tate group, is finite. I prove a particular case of this statement for elliptic curves with complex multiplication over an imaginary quadratic field in which a certain prime p is ramified. My main method is Iwasawa theory, which puts the Selmer group into an analytic family of groups parametrized by the p-adic open disk, which on a certain locus in the disk is controlled by L-function values by known cases of the Bloch-Kato conjectures. I show that these L-values themselves fit into a p-adic analytic family by constructing a novel type of p-adic L-function. Using analytic continuation and specializing the families to the center of the disk, I get my statement. Applications include a proof of the 1879 Sylvester conjecture on expressing primes as a sum of two rational cubes, as well as Goldfeld's conjecture on the statistics of the ranks of elliptic curves related to a give curve by a quadratic twist.

19.04.2023 Anna Rio (Politechnic University of Catalonia)

Title: From the Quintic Equation to the Yang-Baxter Equation

Abel-Ruffini theroem states that there is no closed formula solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Galois (1811-1832) took up Abel's work and came up with the key idea that every polynomial has a symmetry group which determines whether it can be solved by radicals.

Developing Galois theory using abstract algebraic structures helps us to see its connections to other parts of mathematics, so that we can study various mathematical objects in algebra and number theory in ways that are not easy to describe in the original language of polynomials.

Galois group interpreted as automorphism group of a field extension provides a natural action of a group algebra as endomorphisms of a field extension.

Hopf Galois theory expands the classical Galois replacing the action of the group algebra by the action of a Hopf algebra.

In the case of separable extensions, the Hopf Galois property admits a group-theoretical formulation well suited for counting and classifying and for explicit descriptions. In this scenario, we play with two finite groups one of them being a regular subgroup of the holomorph of the other one.

On the other hand, the Yang-Baxter equation plays a crucial role in several areas of theoretical physics: quantum groups, knot theory, braided categories and so on. When considerig its set-theoretical solutions we end up with an attached permutation group which is characterized by being the multiplicative group of a left brace. A brace structure involves also a couple of groups and in 2016 it was proven that they hold the same relation as in Hopf Galois theory for separable extensions.

12.04.2023 László Remete (University of Debrecen) (This talk was held in Zoom)

Title: Monogenity of number fields

Let Q<=K be a field extension of degree n and let O_K be the ring of integers of K. We say that K is monogenic over Q, if O_K is mono-generated as a ring over Z, i.e. O_K=Z[alpha] for some alpha in O_K. In this case (1,alpha,alpha^2,...,alpha^{n-1}) is an integral basis of K and consequently, the index [O_K:Z[alpha]] is one. It is a classical topic of algebraic number theory to decide if a number field is monogenic or not.

The first example of a non-monogenic number field was given by Dedekind. His example is based on the fact that if a prime p in Z does not divide the index of alpha, then the prime ideal decomposition of the ideal pO_K is in one-to-one correspondence with the modulo p factorisation of the minimal polynomial of alpha over Q. It turns out that one can deal with the monogenity of a number field using ramication of rational primes if and only if the field index is not 1. Unfortunatelly, this approach is not complete in the sense that there are non-monogenic number fields with field index 1.

The problem of finding all of the generators of a power integral basis in the number field is equivalent to the problem of solving the index-form equation. It is a Diophantine equation in n-1 variable and of degree \binom{n}{2}, so it is very hard to solve them in general, but this method can be succefull even in the case when the field index is 1.

In this talk I summarise some classical results and methods concerning the mongenity of number fields and some new directions that has been in the scope of the most recent papers.

05.04.2023 Karim Johannes Becher (University of Antwerp)

Title: Universal and positive-universal quadratic forms over fields

The u-invariant of a field is the largest dimension of an anisotropic quadratic form over the field. When the field has characteristic different from 2, the u-invariant can also be characterized as the smallest number d such that all quadratic forms of dimension d over the field are universal. Merkurjev showed that all even natural numbers can occur as the u-invariant of a field. On the other hand, the values 3,5 and 7 are definitely not possible, while (1 and) 9 are possible.

15.03.2023 Błażej Żmija (Charles University)

Title: Ranks of universal quadratic forms over quadratic fields

For a squarefree positive integer D let R(D) denote the minimal rank 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 epsilon>0 is fixed, then for almost all the numbers D (in the sense of natural density) R(D) is greater than D^{1/24 -epsilon}. The talk is based on my joint work with V. Kala and P. Yatsyna.

08.03.2023 Mentzelos Melistas (Charles University)

Title: A divisibility related to the Birch and Swinnerton-Dyer conjecture

The Birch and Swinnerton-Dyer (BSD) conjecture asserts that the size of the group of rational points of an elliptic curve, as well as several other invariants, are related to the behavior of an associated analytic object, the L-function of the curve. After discussing the BSD conjecture for elliptic curves over the rationals, I will focus on the analytic rank zero case and discuss a conjecture of Agashe, which is a consequence of the BSD. I will then present a theorem that proves Agashe's conjecture. Time permitting I will also talk about more recent work of mine, where a problem of similar flavor is solved for semi-stable elliptic curves.

01.03.2023 Nicolas Daans (Charles University)

Title: The Pythagoras number of function fields

The Pythagoras number of a field K is the smallest natural number n such that every sum of squares of elements of K is a sum of n squares of elements of K, or infinity, if such a natural number does not exist. Let us denote the Pythagoras number of K by p(K). Any non-zero natural number (and infinity) is the Pythagoras number of some field.

Very little is known about the behaviour of the Pythagoras number under field extensions, in particular how quickly and freely the Pythagoras number can grow. For example, when L/K is a finite field extension, we only in general know that p(L) is bounded by [L : K]p(K), but in practice, we do not know of any example where p(L) > p(K) + 2 when L/K is a finite field extension. A related open question is whether, when p(K) is finite, then also p(K(X)) is finite, where K(X) is a rational function field over K. We also do not know of any example where p(K(X)) > p(K) + 2.

In this talk, I discuss joint work with Karim Johannes Becher, David Grimm, Gonzalo Manzano-Flores, and Marco Zaninelli, in which we prove for an arbitrary field K that, if p(K(X)) = 2 (the lowest possible value), then p(L) is at most 5 for any finite field extension L of K(X), thereby providing a step in the direction of this open question.

22.02.2023 Jakub Krásenský (Charles University)

Title: Sums of integral squares in number fields

For any ring R, its Pythagoras number is the smallest number P(R) such that any sum of squares in R can be written as a sum of at most P(R) squares. Lagrange's celebrated four-square theorem can be stated as P(Z)=4. We will look for analogous results in other number fields; the most interesting ones turn out to be totally real fields, i.e. such that the image of all their complex embeddings is in fact a subset of the real numbers.

While the Pythagoras number of a number field is easy to determine, the Pythagoras number of its ring of integers is usually unknown. We will discuss the available results and the basic ideas behind them. In particular, to obtain upper bounds for Pythagoras numbers, we introduce the so-called g-invariants, which are similar to the Pythagoras number, but squares of numbers are replaced by squares of linear forms. The study of g-invariants, sometimes called the quadratic Waring's problem, is far from solved even in the case of rational integers; however, we will see some nontrivial results.

15.02.2023 Pawel Gladki (University of Silesia)

Title: Natural homomorphism of Witt rings of a certain cubic order

Let K be a number field and O_K its ring of integers. A famous result by Knebusch asserts that the natural homomorphism of Witt rings W O_K --> W K is injective. This, however, fails to be true if we replace O_K with an arbitrary ring R whose field of fractions is equal to K We shall consider one particular class of such rings here, namely orders, that is subrings O of O_K which are also Z-modules of rank n = [K :Q]. The case of orders in quadratic number fields is relatively well understood with both examples of natural homomorphisms of Witt rings being injective and not. In this talk we shall take a closer look at orders in cubic number fields. While orders in quadratic number fields are easy to describe and classify, cubic orders are considerably more difficult to handle. Nevertheless, we manage to exhibit an example of a cubic order O whose Witt ring W O naturally embedds into the Witt ring of its field of fractions.

Subsequent lecture: From Witt rings of rings to Witt groups of exact categories with duality.

14.12.2022 Álvaro Lozano-Robledo (University of Connecticut)

Title: What is a Galois representation?

We will introduce the basic properties of the absolute Galois group of Q, and its representations (which are called Galois representations). Then, we will produce examples of Galois representations, concentrating on those coming from elliptic curves. In particular, we will discuss a recent classification of l-adic Galois representations attached to elliptic curves by Rouse, Sutherland, and Zureick-Brown (in the non-CM case) and by the speaker (in the CM case).

07.12.2022 Maciej Ulas (Jagiellonian University) (This talk was held in Zoom)

Title: Values of binary partition function represented by a sum of three squares

Let m be a positive integer and b_{m}(n) be the number of partitions of n with parts being powers of 2, where each part can take m colors. We show that if m = 2k − 1, then there exists the natural density of integers n such that b_{m}(n) can not be represented as a sum of three squares and it is equal to 1/12 for k = 1, 2 and 1/6 for k ≥ 3. In particular, for m = 1 the equation b_{1}(n) = x^2 + y^2 + z^2 has a solution in integers if and only if n is not of the form 2^{2k−1}(8s + 2t_s + 3) + i for i = 0, 1 and k, s are positive integers, and where t_n is the nth term in the Prouhet-Thue-Morse sequence. Similar characterization is obtained for the solutions in n of the equation b_{2m−1}(n) = x^2 + y^2 + z^2. The talk is based on a joint paper with Bartosz Sobolewski.

30.11.2022 Fabio Ferri (University of Exeter)

Title: On reduction steps for Leopoldt's conjecture

Let p be a rational prime and let L/K be a Galois extension of number fields with Galois group G. Under some hypotheses, we show that Leopoldt's conjecture at p for certain proper intermediate fields of L/K implies Leopoldt's conjecture at p for L; a crucial tool will be the theory of idempotent relations in Q[G]. We also consider relations between the Leopoldt defects at p for intermediate extensions of L/K. We finally show that our results combined with some techniques introduced by Buchmann and Sands allow us to find infinite families of nonabelian totally real Galois extension of Q satisfying Leopoldt's conjecture for certain primes. This is joint work with Henri Johnston.

23.11.2022 Óscar Rivero (Warwick University) (This talk was held in Zoom)

Title: From exceptional zeros to a p-adic Harris--Venkatesh conjecture

Beginning in the 80s with the celebrated work of Mazur, Tate and Teitelbaum, the study of exceptional zeros for p-adic L-functions has become a very fruitful area in number theory. In this talk, we begin by giving a historical survey of several applications of this theory, which include certain cases of the p-adic Birch and Swinnerton-Dyer conjecture and the Gross--Stark conjectures. We connect this with a result obtained during my PhD in a joint work with V. Rotger, and which can be seen as a Gross--Stark formula for the adjoint of a weight one modular form. Finally, we describe a tantalizing connection between our work and a deep conjecture of Harris and Venkatesh, which explains the presence of the same system of Hecke eigenvalues in multiple degrees of cohomology.

16.11.2022 Matteo Bordignon (Charles University)

Title: An explicit version of Chen's Theorem

Drawing inspiration from the work of Nathanson and Yamada we will show that every even integer larger than exp (exp (34.5)) can be written as the sum of a prime and the product of at most two primes.

02.11.2022 Siu Hang "Gordon" Man (Charles University)

Title: A density theorem of Sp(4)

We prove a density theorem that bounds the number of automorphic forms of level q for the group Sp(4) that violates the Ramanujan conjecture relative to the amount by which they violate the conjecture, which goes beyond Sarnak’s density hypothesis. The proof relies on a relative trace formula of Kuznetsov type, and non-trivial bounds for certain Sp(4) Kloosterman sums.

19.10.2022 Giacomo Cherubini (Charles University)

Title: Real quadratic fields with large class number

We know that every integer can be factored in a unique way as a product of primes. This is no longer true over number fields and the class number indicates "how badly unique factorization fails": if the class number is one then we have unique factorization, while anything bigger than one means we don’t. A long-standing open conjecture of Gauss states that there are infinitely many real quadratic fields with class number one. In the opposite direction, one can prove that there are infinitely many real quadratic fields with class number as large as possible. In this talk I will explain what ”as large as possible” means and a few ideas on how the result can be proved. This is joint work with Fazzari, Granville, Kala and Yatsyna.

12.10.2022 Fabien Pazuki (University of Copenhagen)

Title: Northcott numbers and applications.

A set of algebraic numbers with bounded degree and bounded height is a finite set, by Northcott's theorem. The set of roots of unity is of height zero, but is infinite. What about other sets of algebraic numbers? When is a set of bounded height still infinite? A way to approach this question is through the Northcott number of these sets. We will study some of their properties, discuss links to Julia Robinson's work on undecidability, and explain other applications towards height controls in Bertini statements. The talk is based on joint work with Technau and Widmer.

05.10.2022 Pietro Mercuri (University La Sapienza, Rome)

Title : Rogers-Ramanujan type identities arising from a partition identity of Alladi, Andrews and Gordon

We start recalling the basic definitions of integer partitions and generating functions. After this, we show how, using a generalization of integer partitions, we obtain an infinite family of Rogers-Ramanujan type analytical identities (i.e., identities between an infinite product and an infinite sum) between generating functions of suitable sets of integer partitions.

20.09.2022 Ethan Lee (University of New South Wales Canberra)

The number of integral ideals in a number field

The widely useful floor function [x], which counts the number of (positive) integers in the field of rational numbers up to x, is simple to approximate. Analogously, the number fields generalisation of [x] (called the ideal-counting function) is also useful, but it is not so simple to approximate. In this talk, I introduce some historical results and describe my recent contribution to the literature (an explicit estimate for the ideal-counting function).