Number Theory Seminar
2020/2021, Tuesdays 15:40
Upcoming talks
Previous talks
1. 6. Lucile Devin (Chalmers University), Lemke Oliver and Soundararajan bias for sums of two squares
Recently Lemke Oliver and Soundararajan noticed how experimental data exhibits erratic distributions for consecutive pairs of primes in arithmetic progressions, and proposed a heuristic model based on the Hardy–Littlewood conjectures containing a large secondary term, which fit very well the data. We will discuss consecutive pairs of sums of squares in arithmetic progressions, a bias also appears in the experimental data and we develop a similar heuristic model based on the Hardy–Littlewood conjecture for sums of squares to explain it. This is joint work with Chantal David, Jungbae Nam and Jeremy Schlitt.
18. 5. Dimitrios Chatzakos (University of Bordeaux), Refined equidistribution results for lattice points on the hyperbolic plane
Using motivation from results for lattice points on the euclidean plane, we'll discuss some refined equidistribution results for lattice points arising from the action of the modular group on the hyperbolic plane.
4. 5. Kelly Isham (University of California, Irvine), Asymptotic growth of orders in a fixed number field via subrings in Z^n
Let K be a number field of degree n and O_K be its ring of integers. An order in O_K is a finite index subring that contains the identity. A major open question in arithmetic statistics is: what is the asymptotic growth of orders in K? In this talk, we will give the best known lower bounds for this asymptotic growth. The main strategy is to relate orders in O_K to subrings in Z^n via subring zeta functions. Along the way, we will give lower bounds for the asymptotic growth of subrings in Z^n and for the number of fixed index subrings in Z^n.
4. 5. Christopher Frei (TU Graz), Constructing abelian extensions with prescribed norms
Let K be a number field, S a finite set of elements of K, and G a finite abelian group. We explain how to construct explicitly a normal extension L of K with Galois group G, such that all elements of S are norms of elements of L. The construction is based on class field theory and a recent formulation of Tate’s criterion for the validity of the Hasse norm principle. This is joint work with Rodolphe Richard (UCL).
27. 4. Ondrej Bínovský, Rational points on modular curves
A modular curve is a quotient of the upper half-plane by a subgroup of SL_2(Z). The points of a modular curve parametrize elliptic curves enhanced with an additional structure. We will show how an elliptic curve defined over Q with complex multiplication gives rise to a rational point on a certain modular curve, and how this leads to the solution of the class number one problem for imaginary quadratic fields.
20. 4. Martin Čech (Concordia University), Multiple Dirichlet series and a double sum of Jacobi symbols
Given any arithmetic function, Riemann showed how to express its partial sums as an integral that involves a Dirichlet series. We will show how to apply his method to find an asymptotic formula for the double sum of Jacobi symbols. This asymptotic was found by Conrey, Farmer and Soundararajan and is surprising, as it contains a non-smooth function. Riemann's method leads to a double integral involving a double Dirichlet series, whose properties we will investigate.
13. 4. Martin Raška, Sums of squares in quadratic number fields
Quadratic fields Q(sqrt(D)) and their rings of integers O are one of the prototypical objects of study in number theory. In this talk, I will take a look into which totally positive elements of O can be represented as a sum of squares. Especially in which fields all elements of mO+ share this property for some fixed rational integer m, which is the topic of my upcoming Bachelor's thesis. After introduction and some historical background, I will present the approach to this problem as well as my results about its structure. I will also show how it can be tackled algorithmically.
30. 3. Martin Kuděj, Continued fractions with prescribed period
In this talk, we will consider simple continued fractions of quadratic irrationals. After recalling the characterization of (purely) periodic continued fractions, this theory is used to find the form of continued fractions of square roots of positive nonsquare integers and their symmetric part a_1,..., a_k. Next, we will prove Friesen's theorem where for a given symmetric sequence of positive integers a_1,.., a_k, we will find all natural numbers N whose square root has a continued fraction with symmetric part a_1,.., a_k. These positive N will be described as values of certain quadratic polynomial with integer coefficients. We will also include particular examples and applications of such continued fractions for finding all solutions of the Pell's equation. The content of this talk is based on my Bachelor's Thesis, thus the talk will be accessible even to undergraduate students with basic elementary number theory knowledge.
26. 3. Robin Ammon (TU Kaiserslautern), Jordan Blocks of Unipotent Elements in Spin Groups
At the intersection of group theory and algebraic geometry lies the field of linear algebraic groups which studies groups that are at the same time affine varieties. An important class of linear algebraic groups are the spin groups, whose structure is partly controlled by their so-called unipotent elements. In this talk, I will give an overview over my Master’s thesis whose main objective was to study these elements by determining their Jordan normal form. After an introduction to the topic and the central objects, I will explain the approach to the problem as well as the ideas behind some key arguments that lead to an algorithmic solution. Finally, I will present some of the computational data along with a few theoretical results on the Jordan blocks of unipotent elements that have been obtained based on the calculations and the structure of the algorithm.
23. 3. Stevan Gajović (University of Groningen), The probability that a random polynomial with p-adic coefficients has a fixed number of roots in Qp
Let f be a random polynomial in Zp[x] of degree n. We determine the density of such polynomials f that have exactly r roots in Qp. We also determine the expected number of roots of monic polynomials f in Zp[x] of degree n, and more generally, the expected number of sets of exactly d elements consisting of roots of such f. We show that these densities are rational functions in p, and discuss the remarkable symmetry phenomenon that occurs. We give the asymptotic results when p tends to infinity. This is joint work Manjul Bhargava, John Cremona, and Tom Fisher.
16. 3. Hanson Smith (University of Connecticut), Monogeneity and Torsion
A number field is monogenic (over Q) if the ring of integers admits a power integral basis, i.e., a Z-basis of the form {1, alpha, alpha^2,..., alpha^{n-1}}. The first portion of the talk will be spent revisiting some classical examples of monogeneity and non-monogeneity. We will pay particular attention to obstructions to monogeneity and relations to other arithmetic questions. With some of the classical context for monogeneity and power integral bases in hand, we will investigate the monogeneity of division fields of elliptic curves. This will culminate in two results: one describing non-monogeneity "horizontally" and the other non-monogeneity "vertically". We will finish with some of the difficulties of generalizing to abelian varieties of dimension greater than one, highlighting a partial generalization.
9. 3. Víťa Kala, Universal quadratic forms over number fields
This will be an introductory talk (accessible to students) to several exciting topics of current research. In particular, I will discuss some important properties of rings of integers in number fields (e.g., Z[sqrt 2]) and the structure of universal quadratic forms, i.e., those that represent all "totally positive" elements.
15. 12. Pavel Čoupek (Purdue University), Geometric quadratic Chabauty over number fields
The Chabauty-Coleman method and quadratic Chabauty are p-adic methods that aim to determine the set of all rational points of a smooth projective curve of genus at least 2, assuming certain inequalities involving ranks of its Jacobian. After review of some of the algebro-geometric background I will describe a geometric approach to quadratic Chabauty developed by Edixhoven and Lido over the rational numbers. Then I will discuss how to extend the method to the case of a general number field. This last part is a work in progress in collaboration with David Lilienfeldt, Luciena X. Xiao and Zijian Yao.
8. 12. Alexander Slávik, Consonances, dissonances, and some fractions
Why do some tones fit together, while other combinations sound unpleasant? How did these views evolve over time? A very informal talk with a bit of arithmetics plus a lot of music with examples and experiments.
24. 11. Giacomo Cherubini, Introduction to the hyperbolic circle problem
I will review the classical Gauss circle problem, which asks to count integer points in given circles of expanding radius. After that, I will explain the analogous problem in the hyperbolic plane and describe what is currently known.
10. 11. Tomáš Vávra (University of Waterloo), Distinct unit generated number fields and finiteness in number systems
A distinct unit generated field is a number field K such that every algebraic integer of the field is a sum of distinct units. In 2015, Dombek, Masáková, and Ziegler studied totally complex quartic fields, leaving 8 cases unresolved. Because in this case there is only one fundamental unit u, their method involved the study of finiteness in positional number systems with base u and digits arising from the roots of unity in K. First, we consider a more general problem of positional representations with base beta with an arbitrary digit alphabet D. We will show that it is decidable whether a given pair (beta, D) allows eventually periodic or finite representations of elements of O_K. We are then able to prove the conjecture that the 8 remaining cases indeed are distinct unit generated.
3. 11. Pavlo Yatsyna, Lattices, number fields and related problems
Amidst the plethora of approaches to extremal problems in number theory - we consider lattices. With the field being totally real, it is natural to study positive definite quadratic forms. Many of the classical problems have their canonical counterparts in this setting. In the talk, I will address a selection of associated known results and conjectures.
13. 10. Giacomo Cherubini, Automorphic methods in number theory
I will give an overview of some basic tools used in analytic number theory, with a focus on modular forms and automorphic forms. As an application, I will mention how to determine the average of class numbers of indefinite quadratic forms when they are partly separated from the regulator.
6. 10. Matěj Doležálek, Quaternions, geometry of numbers, and universal quadratic forms
One of the ways to prove the classical four-square theorems of Lagrange and Jacobi is to examine some algebraic properties of certain rings of quaternions. We shall discuss how this approach may be adapted to study quadratic forms over real number fields K whose O_K is a unique factorization domain. Using a quaternionic ring H such that the norm of an element is expressed by the quadratic form, the main challenge is showing that H is a principal ideal domain. This can be achieved by using geometry of numbers to prove that fractional ideals of H with prime denominators contain elements of a small norm.
Recommended talks
Francesco Veneziano, Finiteness and periodicity of continued fractions over quadratic number fields