Tomáš Vávra

Number Theory Seminar

Winter semester 2018/2019

• 9 October 2018 Ezra Waxman: Variance of Gaussian Primes across Sectors.
A Gaussian prime is a prime element in the ring of Gaussian integers Z[i]. As the Gaussian integers lie on the plane, interesting questions about their geometric properties can be asked which have no classical analogue among the ordinary primes. Specifically, to each Gaussian prime a + bi, we may associate an angle whose tangent is the ratio b/a. Hecke proved that these Gaussian primes are equidistributed across sectors of the complex plane, by making use of (infinite) Hecke characters and their associated L-functions. In this talk I will present a conjecture, motivated by a random matrix model, for the asymptotic variance of Gaussian primes across sectors. I will also discuss ongoing work towards a more refined conjecture, which picks up lower-order-terms. Finally, I will introduce a function field model for this problem, which will yield an analogue to Hecke’s equidistribution theorem. By applying a result of N. Katz concerning the equidistribution of “super even“ characters (the function field analogues to Hecke characters), I will provide a result for the variance of function field Gaussian primes across sectors; a computation whose analogue in number fields is unknown beyond a trivial regime (Joint work with Zeev Rudnick).


• 16 October 2018 Tomáš Vávra: Continued fractions with noninteger coefficients.
We consider continued fractions, whose coefficients take values in certain subsets of algebraic integers. J. Bernat showed that the choice of \beta-integers with \beta being the golden ratio leads to finite continued fraction expansion of the whole extension Q(beta). Using a different method, we will show analogous results for sets of coefficients arising from the so-called cut-and-project scheme.


• 23 and 30 October 2018 Ondrej Bínovský: Heegner's solution of the class number problem for imaginary quadratic fields.
In his 1801 Disquisitiones Arithmeticae, Gauss conjectured that there are only finitely many imaginary quadratic fields with a given class number h. The special case when h = 1 was first solved by Kurt Heegner in 1952. Heegner proved that there are exactly 9 imaginary quadratic fields with class number 1, namely Q( \sqrt(−m)) for m ∈ {3,4,7,8,11,19,43,67,163}. His proof employs modular functions and the theory of complex multipli- cation. We will explain how can these transcendental methods be applied to obtain results on imaginary quadratic fields, and give an exposition of Heegner’s proof.


• 6 November: NO SEMINAR
  
• 13 November Daniel El-Baz (Max Planck Institute for Mathematics): Effective equidistribution of rational points on certain expanding horospheres.
In a 2016 paper, Manfred Einsiedler, Shahar Mozes, Nimish Shah and Uri Shapira used techniques from homogeneous dynamics to establish the equidistribution of primitive rational points on expanding horospheres in the space of unimodular lattices in at least 3 dimensions. Due to the nature of their proof, relying in particular on Marina Ratner's measure-classification theorem, their result does not come with a quantitative error term. I will discuss a joint work with Bingrong Huang and Min Lee, in which we pursue an analytic number-theoretic approach to give a rate of convergence for a specific horospherical subgroup in any dimension. This extends work of Min Lee and Jens Marklof who dealt with the 3-dimensional case in 2017.


• 20 November Ondrej Bínovský: Heegner's solution of the class number problem for imaginary quadratic fields.
finishing the talks of October 23 and 30


• 27 November Arno Fehm (TU Dresden): A p-adic analogue of Siegel's theorem on sums of four squares.
Every positive rational number is the sum of four squares by a well-known theorem of Euler. As predicted by Hilbert and proven by Siegel, this generalizes to arbitrary number fields K when one replaces 'positive' by 'totally positive', i.e. positive with respect to every embedding of K into the reals. I will motivate and present a p-adic analogue of this, which gives a constructive description of those elements of K that are totally p-adically integral, i.e. p-adic integers for each embedding of K into the p-adic numbers. The proof of this result involves the Brauer-Hasse-Noether local-global principle for central simple algebras. Joint work with Sylvy Anscombe and Philip Dittmann.


• 4 December Julio Andrade (University of Exeter): The Hybrid Euler-Hadamard formula in Number Fields and Function Fields.
This talk is divided into two parts. In the first part, I will describe the hybrid Euler-Hadamard formula for the Riemann zeta-function and how it connects with the moments of the Riemann zeta-function and the pair-correlation of their zeros. This is a joint work with Kevin Smith. In the second part, I will describe how the original Euler-Hadamard formula can be adapted for a family of L-functions in function fields and how it can be used to study moments on the critical point of such family of L-functions.


• 11 December Pavlo Yatsyna (University of London): Equiangular lines in Euclidean spaces.
We present a result about the equiangular line systems, that is, sets of unit vectors (lines) in the Euclidean space of dimension d, such that an absolute value of the inner product of any two distinct lines is the same. In this talk, we explain why 50 lines in dimension 17 cannot exist.