Please check back for new videos in Number Theory / Arithmetic Geometry.
Abstract: In order to understand problems in dynamics which are sensitive to arithmetic properties of return times to regions, it is desirable to generalize classical results about rotations on the circle to the setting of rotations on adelic tori. One such result is the classical three gap theorem, which is also referred to as the three distance theorem and as the Steinhaus problem. It states that, for any real number, a, and positive integer, N, the collection of points na mod 1, where n runs from 1 to N, partitions the circle into component arcs having one of at most three distinct lengths. Since the 1950s, when this theorem was first proved independently by multiple authors, it has been reproved numerous times and generalized in many ways. One of the more recent proofs has been given by Marklof and Strömbergsson using a lattice based approach to gaps problems in Diophantine approximation. In this talk, we use an adaptation of this approach to the adeles to prove a natural generalization of the classical three gap theorem for rotations on adelic tori. This is joint work with Alan Haynes.
Link to paper: https://arxiv.org/pdf/2107.05147.pdf
Subject codes: 11J71, 11S82, 37P05, 37A45
Speaker's Contact Information:
Email: atdas@cougarnet.uh.edu
Website: https://www.math.uh.edu/~atdas/
Abstract: We consider a multivariate polynomial Q_n defined over the unitary cube, endowed with a symmetrical behaviour, and we look for its maximum. This is required in order to obtain an estimate for the discriminant of a totally real number field in terms of the regulator and the degree. Instead of studying local analytical properties, we obtain the result by means of a global approach, which considers the sign of the variables and transforms the problem into the research of signs patterns for an array: in particular, there is a procedure which transforms any considered array into a specific configuration which provides the desired maximum value. This confirms a conjecture by Pohst on the value of the maximum, which was precisely proved just up to n=11.
Link to paper: https://arxiv.org/pdf/2101.06163.pdf
Subject codes: 11R80, 11Y40
Speaker's Contact Information:
Email: francesco.battistoni@univ-fcomte.fr
Website: https://sites.google.com/view/fbattistoni
Abstract: Consider a sector in the complex plane which is the region between two concentric circles centered at the origin and which is cut by two rays emanating from the origin. Such a sector is "narrow" if the distance between the circles and/or the angle between the rays is small. Similar to how one investigates the distribution of rational primes in short intervals, we can investigate the distribution of prime ideals in Z[i] in narrow sectors. In this work, we show how one can adapt Heath-Brown's method for counting rational primes in short intervals to obtain an asymptotic count for the number of Gaussian primes in narrow sectors.
Link to paper: https://arxiv.org/pdf/2008.11325.pdf
Subject codes: 11N05, 11N25, 11N32
Speaker's Contact Information:
Email: jstucky95@ksu.edu
Website: https://www.math.ksu.edu/~jstucky95/
Abstract: Additive combinatorics studies the presence of patterns such as arithmetic progressions in subsets of groups and rings. These structures can be examined using ideas from combinatorics, number theory, analysis and dynamics. In this talk, we estimate the number of certain polynomial configurations in subsets of finite fields. An example of configurations that we look at is x, x+y^2, x+2y^2, a three-term arithmetic progression with square differences. As a corollary, we derive upper bounds for the size of the subsets of finite finites lacking these configurations. This is connected to the polynomial extension of Szemerédi theorem proved by Bergelson and Leibman.
Link to paper: https://arxiv.org/pdf/1907.08446.pdf, https://arxiv.org/pdf/2001.05220.pdf
Speaker's Contact Information:
Email: boryskuca@gmail.com
Abstract: I will describe some of the results of a recent paper joint with Kathrin Bringmann and Ben Kane. First introducing the notion of t-core partitions, the motivation and some history of their study. I'll then give a brief description of our new results that relate certain 7-cores with Hurwitz class numbers and a genus of quadratic forms in the class group.
Link to paper: https://arxiv.org/abs/2005.07020
Speaker's Contact Information:
Email: jmales@math.uni-koeln.de
Website: http://www.mi.uni-koeln.de/~jmales/
Abstract: The Mehta integral is the canonical partition function for 1-dimensional log-Coulomb gas in a harmonic potential well. Mehta and Dyson showed that it also determines the joint probability densities for the eigenvalues of Gaussian random matrix ensembles, and Bombieri later found its explicit form. We introduce the p-adic analogue of the Mehta integral as the canonical partition function for a p-adic log-Coulomb gas, discuss its underlying combinatorial structure, and find its explicit formula and domain.
Link to paper: https://arxiv.org/abs/2001.03892
Speaker's Contact Information:
Email: jwebster@uoregon.edu
Website: https://pages.uoregon.edu/jwebster/
Abstract: A theorem of Brauer shows that there exists a positive integer B(r, k) such that any colouring of the set {1, 2, ..., B(r, k)} with r colours produces a monochromatic k-term arithmetic progression which receives the same colour as its common difference. We obtain a quantitative version of this result. In particular, we show that the quantity B(r, k) can be taken to be double exponential in the number of colours r, and quintuple exponential in the length k. This talk is based on joint work with Sean Prendiville.
Link to paper: https://doi.org/10.1112/blms.12327
Subject Codes: 11B30, 05D10
Speaker's Contact Information:
Abstract: This talk will serve as an exposition of a recent preprint investigating the division fields of elliptic curves. In this work we show that for various positive integers n there exist of infinite families of elliptic curves over Q with n-division fields, Q(E[n]), that are not monogenic, i.e., the ring of integers does not admit a power integral basis. We parametrize some of these families explicitly. Moreover, we show that every E/Q without CM has infinitely many non-monogenic division fields. Our main technique combines a global description of the Frobenius obtained by Duke and Tóth with a simple algorithm based on ideas of Dedekind.
Link to paper: https://arxiv.org/abs/2007.12781
Subject Codes: 11G05, 11R04
Speaker's Contact Information:
Email: hanson.smith@uconn.edu
Website: http://math.colorado.edu/~hwsmith/
Abstract: The Brauer group of a variety can detect both algebraic and arithmetic properties of the underlying object. In particular, the Brauer-Manin obstruction that lies in the Brauer group can obstruct the existence of rational points. In this talk, I will discuss an algorithm to compute the prime torsion of the Brauer group of an elliptic curve E explicitly over various ground fields k. This algorithm gives generators and relations of the torsion subgroup as tensor products of symbol algebras over the function field of the elliptic curve. As a consequence of the algorithm, I will give an upper bound on the symbol length of the prime torsion of Br(E)/Br(k).
Link to paper: https://arxiv.org/abs/1909.05317
Subject Codes: 16K50, 14H52, 14F22
Speaker's Contact Information:
Email: cu9da@virginia.edu
Abstract: We discuss a construction of nearly overconvergent modular forms for the purpose of interpolating differential operators which act on them. Motivation for the construction comes from the theory of p-adic L-functions.
Link to paper: https://arxiv.org/abs/2006.02543
Speaker's Contact Information:
Email: jaycock@uoregon.edu
Website: pages.uoregon.edu/jaycock
Abstract: Finding integer solutions to norm form equations is a classic Diophantine problem. Using the units of the associated coefficient ring, we can produce sequences of solutions to these equations. It turns out that these solutions can be written as tuples of linear homogeneous recurrence sequences, each with characteristic polynomial equal to the minimal polynomial of our unit. We show that for certain families of norm forms, these sequences are linear divisibility sequences.
Link to paper: https://arxiv.org/abs/2007.07392
Contact Information:
Email: ebellah@uoregon.edu
Website: pages.uoregon.edu/ebellah