Number Theory Seminar 2018/19

Co-organized by Magda Tinková and Tomáš Vávra.

21. 5. Nihal Bircan Kaya (Cankiri, Turkey), On Sequences of Integers of Quadratic Fields and Relations with Artin’s Primitive Root Conjecture

I will consider the integers alpha of the quadratic field Q(sqrt d) where d is a square-free integer. Using the embedding into GL(2,R) we obtain bounds for the first n such that alpha^n = 1 mod p.

More generally, if O_f is a number ring of conductor f, we study the first integer n=n(f) such that alpha^n lies in O_f. We obtain bounds for n(f) and for n(fp^k). We allow any norm N(alpha)<>0. The case where  alpha is the fundamental unit in a real quadratic number field is of special interest. We also study a certain probability distribution suggested by the numerical results.

In the second part of my talk I will indicate in details how my results relate to Artin primitive root type problems over quadratic fields.

9. 5. Marine Rougnant (Besançon), p-rationality of number fields of low degree

A prime in a Galois extension can remain prime or not, and such information is governed by ramification theory. Specifically, for a fixed prime p, we can consider the maximal extension unramified outside p. The Galois group of this extension is linked to the conjecture of Gras on p-rational fields.

We propose to explore this conjecture in the case of some low degree fields: first theoretically, using the abc conjecture, then experimentally, with PARI/GP computations. We will see that these two points of view support the conjecture.

30. 4. Jakub Krásenský, Universal quadratic forms over biquadratic fields
23. 4. Kristína Mišlanová, Matice Legendreových symbolov [Matrix of Legendre symbols, in Slovak]

Pri skúmaní znamienkových matíc Dummit-Dummit-Kisilevsky nedávno definovali jednu ich triedu ako matice Legendreových symbolov modulo rôzne prvočísla  (matice kvadratických zvyškov) a aj ich zovšeobecnenie v podobe matíc kubických zvyškov definovaných pomocou kubických mocninných symbolov. Na základe tohto článku ukážeme ako sú tieto matice definované a popíšeme ich charakterizáciu, predovšetkým blokový tvar týchto matíc. Ďalším krokom je zovšeobecnenie tejto charakterizácie pri rozšírení definícii na voľbu neprimárnych prvočiniteľov.

17. 4. (seminar KAFKA) Víťa Kala, Lifting problem for universal quadratic forms
16. 4. Tomáš Hejda, Ternary universal quadratic forms in arithmetic sequences

We say that a diagonal ternary positive form Q(x,y,z)=ax^2+by^2+cz^2 over Z is (k,l)-universal (for positive k and l with l<k) iff all natural numbers of the form n=km+l are represented by Q. We say that Q is almost (k,l)-universal iff all but finitely many such numbers are represented. We show that almost universal forms exist for all k and non-zero l. Then we restrict to the case k=p is prime and we give numerical and statistical arguments why one should conjecture whether there are finitely or infinitely many (p,l)-universal forms.

2. 4. Hartmut Monien (Bonn), Inverse Galois theory: new results for genus zero sporadic groups
2. 4. Matěj Doležálek (Humpolec), Quaternions, four-square theorems and their analogues

Lagrange’s four-square theorem states that any positive integer can be represented as the sum of four squares of integers, whereas Jacobi’s four-square theorem gives the exact number of such representations for a given positive integer. In this talk, we will show how one can prove these theorems by examining some algebraic and ring-theoretic properties of quaternions, as well as how this approach can be modified to prove the analogous theorems for certain other quadratic forms.

26. 3. Hana Dlouhá, Algebraic properties of multidimensional continued fractions

In 1839 Hermite posed to Jacobi the problem of finding a method for representing real numbers by sequences of nonnegative integers, such that the periodic representations would correspond to the algebraic properties of the numbers (especially to the degree). Continued fractions completely solve this problem for quadratic irrationalities, but for numbers of degree >2 it showed to be a very hard problem. Starting with Jacobi, there were published many modifications of the classical continued fraction algorithm, called multidimensional continued fractions, that attempts to solve this question.

In this talk we give a summary of the vectorial multidimensional continued fractions and its algebraic properties.

19. 3. Magdaléna Tinková, Indecomposable integers in real quadratic fields of odd discriminant

In this talk, we will discuss so-called indecomposable integers in real quadratic fields. In 2016, Jang and Kim stated a conjecture about these elements, which, in some cases, was later disproved by Kala. We will show some results related to this conjecture and briefly sketch the method which leads to finding counterexamples in all the cases of quadratic fields. This is joint work with Paul Voutier.

12. 3. Kristýna Zemková (Dortmund), Non-normal number fields

Considering the cyclotomic number field Q(zeta_k) (where zeta_k is a primitive k-th root of unity), what can be said about its quadratic extensions? In particular, we are interested in the extensions of the form Q(sqrt{a+bzeta_k}).  For certain pairs of integers a, b and k>2 (k<>6), we show that such a field is not normal over Q. (Joint work with Pieter Moree and Carlo Pagano.)

8. 3. Prague-Dresden Number Theory Day
26. 2. Ezra Waxman, Analytic number theory in function fields

Let F_q denote the finite field with q elements, and let F_q[T] denote the ring of polynomials in modern number theory is the deep connection between function fields (such as F_q(T), the fraction field of F_q[T]) and number fields (such as the field of rational numbers Q). In particular, the statistical behavior of prime polynomials in the ring F_q[T] mirrors that of the ordinary prime numbers. In this talk, I will elaborate on this rich analogy between function fields and number fields. In particular, I will provide a proof of the prime polynomial theorem (i.e. the function field analogue of the prime number theorem), and discuss several open problems related to the statistical distribution of prime numbers that have been successful resolved in the function field setting.

11. 12. Pavlo Yatsyna (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.

4. 12. Julio Andrade (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.

27. 11. Arno Fehm (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.

20. 11. Ondrej Bínovský, Heegner’s solution of the class number problem for imaginary quadratic fields
13. 11. Daniel El-Baz (Bonn), 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.

23. 10. + 30. 10. 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 multiplication. 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.

16. 10. 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.

9. 10. 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).

Pangi valley