Algebra, Geometry, and Number Theory Seminar

Abstracts

Karl Schwede - Perfectoid signature and an application to étale fundamental groups

In characteristic $p > 0$ commutative algebra, the $F$-signature measures how close a strongly $F$-regular ring is from being non-singular. Here $F$-regular singularities are a characteristic $p > 0$ analog of klt singularities. In this talk, using the perfectoidization of Bhatt-Scholze, we will introduce a mixed characteristic analog of $F$-signature. As an application, we show it can be used to provide an explicit upper bound on the size of the étale fundamental group of the regular locus of BCM-regular singularities (related to results of Xu, Braun, Carvajal-Rojas, Tucker and others in characteristic zero and characteristic p). This is joint work with Hanlin Cai, Seungsu Lee, Linquan Ma and Kevin Tucker. Pre-talk at 2:30.

Brandon Alberts - A Random Group with Local Data

The Cohen--Lenstra heuristics describe the distribution of $\ell$-torsion in class groups of quadratic fields as corresponding to the distribution of certain random p-adic matrices. These ideas have been extended to using random groups to predict the distributions of more general unramified extensions in families of number fields (see work by Boston--Bush--Hajir, Liu--Wood, Liu--Wood--Zureick-Brown). Via the Galois correspondence, the distribution of unramified extensions is a specific example of counting number fields with prescribed ramification and bounded discriminant. As of yet, no constructions of random groups have been given in the literature to predict the answers to famous number field counting conjectures such as Malle's conjecture. We construct a "random group with local data" bridging this gap, and use it to describe new heuristic justifications for number field counting questions. Graduate student-oriented pretalk at 2:30 in the same room.

Andrew Kobin - Categorifying zeta and L-functions

Zeta and L-functions are ubiquitous in modern number theory. While some work in the past has brought homotopical methods into the theory of zeta functions, there is in fact a lesser-known zeta function that is native to homotopy theory. Namely, every suitably finite decomposition space (aka 2-Segal space) admits an abstract zeta function as an element of its incidence algebra. In this talk, I will show how many 'classical' zeta functions in number theory and algebraic geometry can be realized in this homotopical framework. I will also discuss work in progress towards a categorification of motivic zeta and L-functions. There will be a 2:30 pretalk, aimed at grad students, on "Objective linear algebra: an introduction to linear algebra in a category".

- Spring Break

Robert Lemke Oliver - (Colloquium) Simple ways of encoding roots of polynomials and bounds on number fields

From an algebraic perspective, the square root of 2 can be represented quite simply by saying that it's a root of the polynomial x^2-2. This is also essentially the simplest way to represent this number. Similarly, the polynomial x^3-5 is the simplest way to represent the irrational number that's a cube root of 5. But what about the polynomial x^5 - 7810*x^3 - 121055*x^2 + 2116510*x + 18532349? It can't be factored and its roots are all irrational numbers, but is this complicated polynomial really the "simplest" way to encode those roots? It shouldn't be obvious either way! In this talk, I'll tell you how with a little bit of extra information about the polynomial (its Galois group) it's often possible to encode the roots much more efficiently. Pulling back the curtain, this talk is really about studying number fields of bounded discriminant, and is based on joint work with Frank Thorne and others.

Kirsten Wickelgren - (Colloquium) The Weil Conjectures and A1-homotopy theory

In a celebrated paper from 1948, André Weil proposed a beautiful connection between algebraic topology and the number of solutions to equations over finite fields: the zeta function of a variety over a finite field is simultaneously a generating function for the number of solutions to its defining equations and a product of characteristic polynomials of endomorphisms of cohomology groups. The ranks of these cohomology groups are the number of holes of each dimension of the associated complex manifold. This talk will describe the Weil conjectures, some A1-homotopy theory, and then combine them to enrich the zeta function to have coefficients in a group of bilinear forms. The enrichment provides a connection between the solutions over finite fields and the associated real manifold. No knowledge of A1-homotopy theory is necessary. The new work in this talk (https://arxiv.org/abs/2210.03035) is joint with Margaret Bilu, Wei Ho, Padma Srinivasan, and Isabel Vogt.

Robert Lemke Oliver - An approximate form of Artin's holomorphy conjecture and non-vanishing of Artin L-functions

The Chebotarev density theorem is a powerful tool in algebraic number theory used to guarantee the existence of primes with particular behavior (e.g., to find primes that split completely in a number field). For some applications, it is enough simply to know that these primes exist, while for others, it is necessary also to control the size of the primes involved. This is the content of an "effective" Chebotarev density theorem, which is intimately connected to two major unsolved problems in number theory: the generalized Riemann hypothesis and the Artin holomorphy conjecture. In this talk, we discuss joint work with Jesse Thorner and Asif Zaman in which we show that almost all number fields in many natural families unconditionally satisfy a very strong effective Chebotarev density theorem, and we discuss some arithmetic applications of this result to class groups of number fields. The pretalk will cover background on Artin L-functions, the Chebotarev density theorems, and the use of group character theory in analytic number theory.

Santiago Arango-Piñeros - q-Frobenius distributions of abelian varieties

Fix a $g$-dimensional abelian variety $A$ defined over a finite field $\mathbf{F}_q$. For every integer $r \geq 1$, consider the extension of scalars $A_{(r)} = A\times_{\mathbf{F}_q} \mathbf{F}_{q^r}$. We study the distribution of the normalized trace of the Frobenius endomorphism of $A_{(r)}$ in the compact interval $[-2g,2g]$, as $r$ varies. We show that these distributions are controlled by a certain compact abelian Lie group, and classify the possible groups when $g \leq 3$. Pre-talk: I will give a brief introduction to the theory of abelian varieties over finite fields. The main goal will be to sketch a proof of the Honda--Tate theorem. A good reference to read ahead is the section about abelian varieties over finite fields in Poonen's notes: https://math.mit.edu/~poonen/papers/curves.pdf

Jack Petok - A survey of cubic fourfolds over $\mathbb{F}_2$

A cubic fourfold is the zero locus of a homogeneous polynomial of degree 3 within a projective space of dimension 5. Many fascinating questions about cubic fourfolds remain open about cubic fourfolds over the complex numbers. For example, how can one classify the rational cubic fourfolds? Additionally, special cubic fourfolds are related to K3 surfaces via their Hodge structures. To investigate these questions, we turn to computation and enumerate all isomorphism classes of cubic fourfolds over the field of two elements. We attempt to glean insights into longstanding theoretical questions via this dataset. The talk is intended to be accessible to junior graduate students. Joint work with Asher Auel, Avinash Kulkarni, and Jonah Weinbaum.

Last semester's seminar.