Number Theory Seminar 2017/18

22.  5. 2018: Pavel Čoupek (Purdue University), Regular primes and Bernoulli numbers

A prime p is regular if it is not a divisor of the class number of the associated cyclotomic field of p-th roots of unity. In this talk, we will survey the classical results of Ernst Kummer on regular primes. Firstly, we will sketch the proof of Fermat’s last theorem for regular-prime exponents. Secondly, we will discuss a criterion for regularity of primes in terms of Bernoulli numbers, which are rational numbers closely connected to special values of the Riemann zeta function.

15.  5. 2018: Pavlo Yatsyna (Royal Holloway, University of London), The Schur-Siegel-Smyth trace problem

The Schur-Siegel-Smyth trace problem remains open for 100 years now. It asks whether 2 is the smallest limit point of the absolute traces of totally positive algebraic integers. This talk will survey some of the known results and open problems relating to it.

24.  4. 2018: Magdaléna Tinková, Universal quadratic forms and indecomposables over biquadratic fields

In this talk, we will focus on indecomposable integers in biquadratic number fields. Such a field contains three quadratic subfields and we will discuss whether their indecomposable integers remain indecomposable in our biquadratic field. We will also show the connection between these elements and universal quadratic forms.

10. – 11. 4. 2018: Karim Johannes Becher (University of Antwerp)

Minicourse, 3 lectures

10. 4. 2018: Karim Johannes Becher (University of Antwerp)

Splitting fields for central simple algebras of exponent 2
Central simple algebras over fields have been studied for over a century, starting with the work of Cayley, Hamilton, Dickson and Wedderburn. Over number fields these algebras are completely classified.
Over an arbitrary base field, even though these algebras are classified abstractly by the so-called Brauer group of the field, their structure is still a mystery. In my talk I will concentrate on central simple algebras of exponent at most two. These are exactly those algebras which carry an involution which is trivial on the base field. A famous theorem by Merkurjev states that every such algebra is equivalent to a tensor product of quaternion algebras. In particular, if every quaternion algebra over the field is split, then there exists no central simple algebra of exponent two over this field. I give an independent elementary proof of this fact. While this proof is based on Zorn’s Lemma, the statement should also have a constructive proof, leading to an explicit bound of the degree of a splitting 2-extension in terms of the degree of the algebra.
3.  4. 2018: Tomáš Kepka, Wolstenholme’s theorem

27.  3. 2018: Anh Dung Le (Bonn University), Topological properties of adeles and ideles

The finiteness of class group is an important result from algebraic number theory, which is usually proved in undergraduate courses by Minkowski’s bound. In this talk we will shed light on this issue from the perspective of adeles and ideles, which will use the machinery of locally compact topological groups.

13.  3. 2018: Vladimír Sedláček (Venice & Brno)

Special factorization method using complex multiplication on elliptic curves
Efficient factorization of composite integers is an old and important problem and cryptographic schemes such as RSA are based on its intractability. In this talk, we will study a very special fast factorization algorithm that works for numbers with a prime factor p satisfying 4p-1=Ds^2, where s \in Z and D belongs to a small predetermined set. The algorithm relies on the construction of an elliptic curve with exactly p points over F_p, for which the theory of complex multiplication is used, but no prior knowledge of it is expected.

6.  3. 2018: Tomáš Hejda, Beta-expansions of rational numbers in quadratic Pisot bases

We study rational numbers with purely periodic greedy $\beta$-expansions. For bases $\beta$ satisfying $\beta^2=a\beta+b$ with $b$ dividing $a$, we give a necessary and sufficient condition for that all rational numbers $p/q\in[0,1)$ with $\gcd(q,b)=1$ have a~purely periodic $\beta$-expansion. A simple algorithm for determining the infimum of $p/q\in[0,1)$  with $\gcd(q,b)=1$ and with not purely periodic $\beta$-expansion is described that works for all quadratic Pisot numbers $\beta$. This work is joint with Wolfgang Steiner (IRIF, Paris).

27.  2. 2018: Jaroslav Hančl, Growth of ideals of combinatorial structures: Partitions and Ordered graphs

An ideal in the combinatorial structure is any set closed to particular operation. We study closely ideals of number partitions and ordered graphs from its enumeration point by defining the counting function of an ideal I that assigns the number of elements of a particular order n in I. The main goal is to characterize possible growths of counting functions in different combinatorial structures.

19. 12. 2017: Magdaléna Tinková, Arithmetics in number systems with cubic base

In 1957, A. Rényi introduced so-called greedy expansions, one way how to express numbers using a base and a finite set of digits. We will show how to find bounds on the number of fractional digits appearing when we add and subtract greedy expansions, namely for the case of cubic bases.

12. 12. 2017: Jakub Krásenský (FJFI ČVUT), Soustavy s “řídkými” abecedami

Přednáška se bude týkat pozičních číselných soustav na mřížkách. Za základ takové soustavy bereme nějaké lineární zobrazení (endomorfismus mřížky) a abecedou je konečná množina vektorů. Numeračním systémem (GNS) nazýváme soustavu, která umožňuje vyjádřit každý prvek mřížky právě jedním způsobem. Výsledek od Germána a Kovácse z roku 2007 ukazuje, že pro základ s dostečně velkými vlastními čísly vždy existuje abeceda, která spolu s ním tvoří GNS. Tento výsledek zesílíme a ukážeme, že za jistých doplňujících předpokladů takových abeced existuje nekonečně mnoho a všechny jejich nenulové prvky leží libovolně daleko od nuly.
6. 12. 2017: Dominik Lachman, Bruhat-Tits buildings at the KAFKA seminar
28. 11. 2017: Tomáš Vávra, Periodic representations in number systems with an algebraic base
After giving an introduction to $(\beta,A)$-representations, we will show that if $\beta\in\mathbb C$, $|\beta|>1$ is an algebraic number, then there exists an (integer) alphabet $A$ such that each element of the field extension $\mathbb Q(\beta)$ admits an eventually periodic $(\beta,A)$-representation. We will also show how the question whether a pair $(\beta,A)$ has this property is linked to fractal geometry.
21. 11. 2017: Pavlo Yatsyna (Royal Holloway, University of London)
Universal quadratic forms and interlacing polynomials
It is a classical result in number theory that any natural number can be represented as a sum of four squares. Over the ring of integers of $\mathbb{Q}(\sqrt{5})$, every totally positive integer can be written as a sum of three squares. For a general quadratic form, Blomer and Kala recently showed that the number of variables required in a real quadratic number field is unbounded. I will present a similar result, based on studying interlacing polynomials, which allows the degree of a field extension to be arbitrarily large.

14. 11. 2017: Pavlo Yatsyna (Royal Holloway, University of London)

Salem numbers of trace −2, and a conjecture of Estes and Guralnick
Estes and Guralnick conjectured necessary and sufficient conditions for a polynomial to appear as the minimal polynomial of a symmetric matrix with rational integer coefficients. They confirmed their conjecture for polynomials of degree up to 4. In this talk, I will show that there are counterexamples to Estes—Guralnick’s conjecture for all degrees strictly larger than 5. One of the ingredients in the proof is to show that there are Salem numbers of degree 2d and trace −2 for every d≥12.

31. 10. 2017: Martin Čech, Different approaches toward the proof of the prime number theorem

We will show how complex analysis is used in number theory and discuss different approaches to the proof of the prime number theorem. These approaches will include the classical one introduced by Riemann, Hadamard and de la Vallée Poussin at the end of 19th century, and a more modern “pretentious” approach based on Halász’s theorem proved in 1970’s and recently further developed by Granville, Soundararajan and others.

24. 10. 2017: Martin Čech, What are arithmetic functions and how to estimate them?

Many questions in number theory, such as what is the average number of divisors of a natural number, can be stated in terms of arithmetic functions. Giving precise answers to these questions is very hard, which is the reason why analytic number theory studies their estimates. In the lecture, we are going to study basic properties of arithmetic functions and elementary techniques of estimating their rate of growth and the errors in the estimates.

17. 10. 2017: Kristýna Zemková, Composition of quadratic forms over number fields

The correspondence between ideals in a quadratic number field and quadratic forms with integral coefficients dates back to Gauss and Dedekind. But what about quadratic forms whose coefficients are algebraic integers? In the talk I will present my recent result on generalization of this correspondence to some number fields.
10. 10. 2017: Víťa Kala, Arithmetics of number fields and universal quadratic forms
This will be an introductory talk (accessible to students) to several exciting topics of current research.
The arithmetics of number fields has long played a key role throughout number theory, for example in solving diophantine equations. I will discuss some recent results on the additive structure of rings of integers of real quadratic fields and their relation to the study of quadratic forms (joint work with Valentin Blomer and Tomas Hejda).