Spring 2024
The OSU Student Number Theory Seminar for Spring 2024 semester is held biweekly on Friday 12:40-1:35pm, MW152. Anyone interested in number theory (broadly interpreted) is welcome to attend!
Friday 4/19: Shifan Zhao
Title: Low-lying zeros of spinor L-functions attached to Siegel modular forms
Abstract: It is widely believed that zeros of automorphic L-functions correspond to eigenvalues of random matrices. Evidence of such a connection is the Katz-Sarnak heuristic, which predicts that the distribution of low-lying zeros of families of automorphic L-functions are governed by certain classical compact groups depending on the family. In this talk, I will present recent progress on low-lying zeros of spinor L-functions attached to Siegel modular forms. I will give a brief introduction on the concepts mentioned above, state the results and give a proof sketch. Time permitting, we will talk about standard L-functions as well.
Friday 4/12: Jingxuan Geng
Title: The General $\sqrt{2}$ Phenomenon in Partial Euler Products
Abstract: Thanks to the Weil bound, the L-function $L(E,s) = \prod_p L_p(E,s)$ of an elliptic curve $E$ over $\mathbb{Q}$ converges on the half plane $\Re(s) > 3/2$. Assuming the (now proven) modularity conjecture, $L(E,s)$ extends analytically to the entire plane, and the behavior of $L(E,s)$ at $s=1$ was conjectured to be related to the arithmetic data of $E$. Surprisingly, Goldfeld showed that $L(E,1) = \sqrt{2} \ \prod_p L_p(E,1)$ if the product of RHS converges. We will first show proof of the unexpected fact. Then we will discuss a more general result of $\sqrt{2}$ phenomenons for a large family of L-functions. After that, we will discuss its relation to the Birch and Swinnerton-Dyer conjecture by reformulating the BSD conjecture as a formal Tamagama number conjecuture.
Friday 3/29: Tianyu Zhao
Title: Bounds on the Riemann Zeta Function and its Argument on the Critical Line
Abstract: The Riemann zeta function is one of the most studied objects in analytic number theory, especially due to the famous Riemann Hypothesis. In particular, we are interested in the size of zeta and its argument on the critical line, on which all nontrivial zeros are supposedly located. In this talk, we first give a general survey of known results (upper bounds, lower bounds, and mean values), and then we will focus on methods for finding large values.
Friday 3/1: Yifei Zhang
Title: Existence of Nontrivial Solution to ax^2+by^2+cz^2 over Q
Abstract: We will give a necessary and sufficient condition on a,b, and c so that the Diophantine equation in the title has a nontrivial solution over Q, using Hasse-Minkowski theorem: a glorious success of local-to-global principle.
Friday 2/16: Nick Geis
Title: Counting Sign Changes of Partial Sums of Random Multiplicative Functions
Abstract: Let $f$ be a Rademacher random multiplicative function and let $M_f(x)$ denote the partial sum of $f$ up to $x$. We will prove that $M_f(x)$ changes signs infinitely often almost surely as well as provide a way to count, almost surely, the number of such sign changes. The aim of this talk is to be accessible to students of any year and in any number theory discipline.
Friday 2/2: Stefan Nikoloski
Title: Universal Deformation Rings
Abstract: The lifting properties of a group representation are governed by the universal lifting ring. In this talk we will introduce the notion of a universal lifting/deformation ring. We will also compute some examples of such rings. Eventually we will prove the main result about their structure.
Friday 1/19: Will Newman
Title: A Diophantine Equation
Abstract: In this talk, I use tools from algebraic number theory to find all integer solutions (x,y) to x^3-dy^3=1 for a choice of integer d.