In Spring 2022, the number theory student seminar will meet on Monday from 2:00-3:00 PM via Zoom.
Please contact one of the organizers for the Zoom link.
Here is the webpage for previous number theory student seminar.
Current organizers: Joseph Leung (leung.179@osu.edu), Julian Mejia Cordero (mejiacordero.2@osu.edu), Pan Yan (yan.669@osu.edu)
Speaker: Joseph Leung
Title: Methods on Subconvexity problems: Moment, Delta and their mixture
Abstract: One of the most important objects of interest in analytic number theory is the L-functions. Subconvexity problem, which asks for an estimate of the L-functions on the central value or line, is one of the central problems in the field. To solve the Subconvexity problem, the moment method and the Delta method have been popular tools. In this talk, I will give a brief overview of subconvexity problems and how these two methods are used to solve them. Then, I will talk about how one may apply a mixture of the two methods. To exemplify this idea, we will discuss one classical application by DFI, and a recent work in preparation. The recent one is a joint work with Aggarwal and Munshi.
Speaker: Sohail Farhangi
Title: Generalizations of the Grunwald-Wang Theorem and Applications to Ramsey Theory (slides)
Abstract: The Grunwald-Wang Theorem classifies for a given n, which integers m are not an nth power, but are an nth power modulo every prime p. We obtain a partial generalization of this result by showing that for any odd integer n, and any triple of integers a,b, and c, at least 1 of a,b, or c is an nth power modulo p for every prime p if and only if at least one of a,b, or c is an nth power. We obtain a (somewhat) similar result for even n, and examine some of the obstructions. If time remains after discussing some of the ideas in the proof of this result, we will discuss the motivation. Bergelson and Hindman independently showed that the equation x+y = wz is partition regular over the integers. Our partial generalization of the Grunwald-Wang Theorem is used to give an almost complete classification of the integers a,b,c,m,n for which the equation ax+by = cw^mz^n is partition regular over the integers.
Speaker: Shifan Zhao
Title: Siegel zeros: from Dirichlet L-functions to higher degree L-functions
Abstract: The problem of Siegel zeros started with Dirichlet L-functions. For a family of general L-functions of possibly higher degree, we can also consider their Siegel zeros. Surprisingly, although higher degree L-function come from objects more complicated than Dirichlet characters, they are easier to study when it comes to Siegel zeros. In fact, one can show Siegel zeros do not exist for some family of automorphic L-functions. I will explain the first result obtained by Goldfeld, Hoffstein and Lieman, that the symmetric square L-functions of Maass forms on GL(2) do not admit Siegel zeros. I will then introduce some other results after this breakthrough and compare them with results of Dirichlet L-functions.
Speaker: Jake Huryn
Title: How to build an infinite class field tower
Abstract: In 1964, Golod and Shafarevich settled a long-standing conjecture of Fürtwangler that every class field tower stabilizes by giving a quadratic imaginary number field with infinite 2-class field tower. After a historical motivation of this problem, we will explain how modern cohomological techniques lead to the result of Golod–Shafarevich and refinements.
Speaker: Joseph Leung
Title: A New Variant of the Delta Method and Subconvexity
Abstract: The Delta method have been a popular tool in analytic number theory in recent years, due to several successful attempts in establishing nontrivial bounds in settings such as subconvexity problems. In this talk, I am going to first introduce what the Delta method is and the basic philosophy behind its application. Then I will introduce the classical choices of the delta method followed by a new variant of the Delta method, and compare their differences. Finally, I will talk about the 'standard' way to tackle subconvexity problems using the Delta method, with the hybrid subconvexity problem of $L(1/2, \text{sym}^2 f \otimes g)$ in mind.
Speaker: Zhining Wei
Title: Linear Relations of Siegel Poincare Series and Non-vanshing of the Central Values of Spinor L-functions
Abstract: In this paper, we will first investigate the linear relations of a one parameter family of Siegel Poincar\'e series. Then we give the applications to the non-vanishing of Fourier coefficients of Siegel cusp eigenforms and the central values.
Speaker: Jake Huryn
Abstract: Kummer gave a necessary and sufficient condition for the p-part of the class group of the pth cyclotomic field to be trivial. We discuss a refinement of Kummer’s criterion, the Herbrand–Ribet theorem, and its connections to other parts of number theory, such as modular forms, Galois representations, and Iwasawa theory.