Iwasawa theory of unramified geometric Z_p-extensions of function fields

Speaker: 

Bryden Cais

Institution: 

University of Arizona

Time: 

Thursday, May 29, 2025 - 4:00pm

Host: 

Location: 

RH 306

In this talk, I will describe a novel Iwasawa theory for unramified Z_p-extensions of global function fields over an algebraically closed field of characteristic p. In this context, the p-adic slopes of Frobenius acting on the first crystalline cohomology of the associated Z_p-tower of algebraic curves provide a new kind of Iwasawa-theoretic object to study, and I will present evidence for a recent conjecture about the limiting behavior of these slopes.

Jacobians of Graphs via Edges and Iwasawa Theory

Speaker: 

Jon Aycock

Institution: 

UCSD

Time: 

Thursday, May 29, 2025 - 3:00pm to 4:00pm

Location: 

RH 306

The Jacobian (or sandpile group) is an algebraic invariant of a graph that plays a similar role to the class group from number theory. There are multiple recent results controlling the sizes of these groups in Galois towers of graphs that mimic the classical results in Iwasawa theory, though the connection to the values of the Ihara zeta function often requires some adjustment. In this talk we will give a new way to view the Jacobian of a graph that more directly centers the edges of the graph, construct a module over the relevant Iwasawa algebra that nearly corresponds to the interpolated zeta function, and discuss where the discrepancy comes from.

Characteristic polynomial of random tridiagonal matrices

Speaker: 

Daniele Garzoni

Institution: 

USC

Time: 

Thursday, April 17, 2025 - 3:00pm to 4:00pm

Location: 

RH 306

In the talk, we will discuss the irreducibility and the Galois group of random polynomials over the integers. After giving motivation (coming from work of Breuillard--Varjú, Eberhard, Ferber--Jain--Sah--Sawhney, and others), I will present a result, conditional on the extended Riemann hypothesis, showing that the characteristic polynomial of certain random tridiagonal matrices is irreducible, with probability tending to 1 as the size of the matrices tends to infinity. 

The proof involves random walks in direct products of SL_2(p), where we use results of Breuillard--Gamburd and Golsefidy--Srinivas. 

Joint work with Lior Bary-Soroker and Sasha Sodin.

On the diagonal and Hadamard grades of hypergeometric functions

Speaker: 

Joe Kramer-Miller

Institution: 

Lehigh University

Time: 

Thursday, May 22, 2025 - 3:00pm to 4:00pm

Host: 

Location: 

RH306

Diagonals of multivariate rational functions are an important class of functions arising in number theory, algebraic geometry, combinatorics, and physics. For instance, many hypergeometric functions are diagonals as well as the generating function for Apery's sequence. A natural question is to determine the diagonal grade of a function, i.e., the minimum number of variables one needs to express a given function as a diagonal. The diagonal grade gives the ring of diagonals a filtration. In this talk we study the notion of diagonal grade and the related notion of Hadamard grade (writing functions as the Hadamard product of algebraic functions), resolving questions of Allouche-Mendes France, Melczer, and proving half of a conjecture recently posed by a group of physicists. This work is joint with Andrew Harder.

 

Rational Points on a Family of Genus 3 Hyperelliptic Curves

Speaker: 

Roberto Hernandez

Institution: 

Emory Univeristy

Time: 

Thursday, March 13, 2025 - 3:00pm to 4:00pm

Location: 

RH 306

Let $C$ be a smooth projective curve of genus $g \geq 2$. By Faltings Theorem, we know that there are only finitely many rational points on $C$. We compute the rational points on a family of genus 3 hyperelliptic curves which are curves of the form $y^2 = f(x)$ where $f(x)$ is a polynomial of degree $2g+1$ or $2g+2$ via the method of Dem’janenko-Manin.

How do generic properties spread?

Speaker: 

Jerry Yu Fu

Institution: 

Caltech

Time: 

Thursday, February 27, 2025 - 3:00pm to 4:00pm

Location: 

RH 306

Given a family of algebraic varieties over an irreducible scheme, a natural question to ask is what type of properties of the generic fiber, and how do those properties extend to other fibers. For example, the Hilbert irreducibility theorem states that a dominant map from an irreducible variety X defined over a number field to some projective space which is generically of degree d provides a Zariski dense set of degree d points on X. One can also get quantitative estimates for size of the complement which does not carry the generic property.

We will explore this topic from an arithmetic point of view by looking at several scenarios. For instance, suppose we have a 1-dimensional family of pairs of elliptic curves over a number field,  with the generic fiber of this family being a pair of non-isogenous elliptic curves. One may ask how does the property of "being (non-)isogenous" extends to the special fibers. Can we give a quantitative estimation for the number of specializations of height at most B, such that the two elliptic curves at the specializations are isogenous? 

The distribution of conjugates of an algebraic integer

Speaker: 

Alexander Smith

Institution: 

UCLA

Time: 

Thursday, February 20, 2025 - 3:00pm to 4:00pm

Location: 

RH 306

For every odd prime p, the number 2 + 2cos(2 pi/p) is an algebraic integer whose conjugates are all positive numbers; such a number is known as a totally positive algebraic integer. For large p, the average of the conjugates of this number is close to 2, which is small for a totally positive algebraic integer. The Schur-Siegel-Smyth trace problem, as posed by Borwein in 2002, is to show that no sequence of totally positive algebraic integers could best this bound.

In this talk, we will resolve this problem in an unexpected way by constructing infinitely many totally positive algebraic integers whose conjugates have an average of at most 1.899. To do this, we will apply a new method for constructing algebraic integers to an example first considered by Serre. We also will explain how our method can be used to find simple abelian varieties with extreme point counts.

The distribution of the cokernel of a random p-adic matrix

Speaker: 

Myungjun Yu

Institution: 

Yonsei University

Time: 

Thursday, January 23, 2025 - 3:00pm to 4:00pm

Location: 

RH 306

The cokernel of a random p-adic matrix can be used to study the distribution of objects that arise naturally in number theory. For example, Cohen and Lenstra suggested a conjectural distribution of the p-parts of the ideal class groups of imaginary quadratic fields. Friedman and Washington proved that the distribution of the cokernel of a random p-adic matrix is the same as the Cohen–Lenstra distribution. Recently, Wood generalized the result of Friedman–Washington by considering a far more general class of measure on p-adic matrices. In this talk, we explain a further generalization of Wood’s work. This is joint work with Dong Yeap Kang and Jungin Lee.

Erdos-Kac type central limit theorem for randomly selected ideals in a Dedekind domain

Speaker: 

Michael Cranston

Institution: 

UCI

Time: 

Tuesday, January 21, 2025 - 3:00pm to 4:00pm

Location: 

RH 306

Using the Dedekind zeta function, one can randomly select an ideal in a Dedekind domain. Then the factorization of the randomly selected ideal into a product of prime ideals has very nice statistical properties. Using these properties one can examine the number of distinct prime ideals there are in the factorization and prove a central limit theorem as a certain parameter tends to one. This talk is based on joint work with E. Hsu.

Periods and p-adic Hodge structures

Speaker: 

Christian Klevdal

Institution: 

UCSD

Time: 

Thursday, January 30, 2025 - 3:00pm to 4:00pm

Location: 

RH 306

Complex Hodge theory equips the rational singular cohomology of smooth projective variety with a Hodge filtration, and period maps measure how these filtrations vary in families. Period maps are highly transcendental in nature, and the transcendence properties of these maps reflect interesting aspects of the geometry; for example a theorem of Cohen and Shiga-Wolfhart show that a complex abelian variety A has CM if and only both A and the Hodge filtration are defined over a number field.

In this talk, we describe a similar situation in p-adic Hodge theory, and discuss joint work with Sean Howe that proves an analogue of the theorem of Cohen and Shiga-Wolfhart. The first part of the talk will contain a brief introduction to Hodge theory, so no previous experience with Hodge theory (complex or p-adic) is required!

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