FGC-HRI-IPM Joint Number Theory Seminar
When: Mondays
Where: Zoom
Seminar mailing list: google group
Talks
Speaker: Yen-Tsung Chen
Title: On the partial derivatives of Drinfeld modular forms of arbitrary rank
Abstract: In the 1980's, the study of Drinfeld modular forms for the rank 2 setting was initiated
by Goss. Recently, by the contributions of Basson, Breuer, Häberli, Gekeler, Pink et. al., the theory of
Drinfeld modular forms has been successfully generalized to the arbitrary rank setting. In this talk,
we introduce an analogue of the Serre derivation acting on the product of spaces of Drinfeld modular
forms of rank r>1, which also generalizes the differential operator introduced by Gekeler in the rank two case.
This is joint work with Oğuz Gezmiş.
Date: January 16, 2023
Slides
Video link
Passcode: *wzs3Dh8
Speaker: Enis Kaya, KU Leuven
Title: Computing Schneider p-adic heights on hyperelliptic Mumford curves
Abstract:There are several definitions of p-adic height pairings on curves in the literature,
and algorithms for computing them play a crucial role in, for example, carrying out the quadratic Chabauty
method, which is a p-adic method that attempts to determine rational points on curves of genus at least two.
The p-adic height pairing constructed by Peter Schneider in 1982 is particularly important because the
corresponding p-adic regulator fits into p-adic versions of Birch and Swinnerton-Dyer conjecture.
In this talk, we present an algorithm to compute the Schneider p-adic height pairing on hyperelliptic Mumford curves.
We illustrate this algorithm with a numerical example computed in the computer algebra system SageMath.
This talk is based on a joint work in progress with Marc Masdeu, J. Steffen Müller and Marius van der Put.
Date: December 19, 2022
Video link
Passcode: 0L?i5WJB
Speaker: Amir Akbary (University of Lethbridge)
Title: Value-distribution of automorphic L-functions
Abstract: After a brief introduction on the value-distribution of arithmetic functions and
L-functions, we give an overview of our joint work with Alia Hamieh (University of Northern British Colombia)
on the value-distribution of logarithmic derivative of certain automorphic L-functions. Among other things,
we describe an upper bound for the discrepancy of the distribution of the values (at a point on the edge of the critical strip)
of the twists of a fixed automorphic L-function with quadratic Dirichlet characters. Our result can be considered as an
automorphic analogue of a result of Lamzouri, Lester, and Radziwill for the logarithm of the Riemann zeta function. Our
estimate is conditional on certain expected bounds on the local parameters of L-functions which is known to be true for GL(1) and GL(2).
Date: November 21, 2022
Slides
Video link
Passcode: 8jK%Wd^Y
Speaker: Haydar Goral, IYTE
Title: Lehmer’s conjecture via model theory
Abstract: In this talk, we first introduce the height function and the Mahler measure on the field
of algebraic numbers. We state and give a survey on Lehmer’s conjecture for the Mahler measure, which is still
an open problem. Then, we consider the field of algebraic numbers with elements of small Mahler measures in
terms of model theory, and we link this theory with Lehmer’s conjecture. Our approach is based on Van der Waerden's theorem from additive combinatorics.
Date: November 7, 2022
Video link
Passcode: 48KJz#$&
Speaker: Mohammad Sadek, Istanbul
Title: How often do polynomials hit squares?
Abstract: Given a polynomial with rational coefficients, one may investigate the possible values
that may be attained by these polynomials over the set of rational numbers. For centuries, number theorists have
been giving due attention to square rational values assumed by rational polynomials. It turns out that seeking an
answer to this question connects number theory and geometry. Answering this question for a polynomial in one variable
will lead us to study the arithmetic of certain algebraic curves. We will spend some time explaining the geometry beneath
the question when the degree of the polynomial is at least 3. For polynomials in more than one variable, the geometric
structure is remote. In the latter case, we will present some of the old and recent developments in the theory shedding some light on some classical Diophantine questions.
Date: October 24, 2022
Video link
Passcode: eq9*WSUW