Abstracts
Daniel Le, University of Toronto
Title: Diagrams in the mod p cohomology of Shimura curves
Shiang Tang, UIUC
Title: Full-image p-adic Galois Representations and Galois Deformation Theory
Abstract: The classical inverse Galois problem asks whether or not every finite group appears as the Galois group of an extension of the rationals. It is still wide open for finite groups of Lie type, for example, SL_2(F_p). On the other hand, number theorists are interested in reductive algebraic groups over p-adic fields for various reasons. We then ask an analogue of the inverse Galois problem in the p-adic world, for example, whether or not there is a Galois extension of the rationals whose Galois group is an open subgroup of SL_2(Z_p). This kind of question fits into the study of continuous p-adic Galois representations valued in a connected reductive group. Representations into general linear groups, general symplectic groups or general orthogonal groups may be constructed by studying the action of the Galois group over Q on the cohomology of an algebraic variety over Q. Representations into exceptional algebraic groups appear in the work of Nick Katz, Stefan Patrikis and Zhiwei Yun. In this work, we construct p-adic Galois representations using Galois deformation theory, which first appeared in Barry Mazur's work and became one of the central tools in modern algebraic number theory after Andrew Wiles published his proof of Fermat's Last Theorem. We will explain a strategy (due to Ravi Ramakrishna and others) for “deforming" a continuous Galois representation in characteristic p to characteristic zero, together with its application to the p-adic analogue of the inverse Galois problem.
Vlad Matei, UC Irvine
Title: Higher Moments of Arithmetic Functions in Function Fields
Abstract: In joint work with Daniel Hast, we have recast the paper of John Keating and Zeev Rudnick "The variance of the number of prime polynomials in short intervals and in residue classes" by studying the geometry of these short intervals through an associated highly singular variety. We manage to recover their results for a general class of arithmetic functions up to a constant and also obtain information about the higher moments. Recent work by Will Sawin give more insight into the geometry of the variety and new geometric approaches to other arithmetic questions.
Jen Berg, Rice
Title: Odd order obstructions to rational points on general K3 surfaces
Abstract: K3 surfaces are 2-dimensional analogues of elliptic curves, but lack a group structure. They need not have rational points. However, in 2009 Skorobogatov conjectured that the Brauer group should account for all failures of the local-to-global principle for rational points on K3 surfaces. In this talk I will briefly describe the geometric origin of certain 3-torsion classes in the Brauer group of a K3 surface. We utilize this geometric description to show that these classes can in fact obstruct the existence of rational points. This is joint work with Anthony Varilly-Alvarado.
Elija Allen
Title: Advancements on Schinzel's hypothesis H
Kyle Pratt, UIUC
Title: Primes from sums of two squares and missing digits
Abstract: In recent decades there have been significant advances made in finding primes in "thin" sequences. One such advance was the work of Friedlander and Iwaniec, in which they proved there are infinitely many primes that can be represented as the sum of a square and a biquadrate. A more recent advance is due to Maynard, who showed the existence of infinitely many primes in the thin sequence of integers missing a fixed digit in their decimal expansion. In this talk I discuss a marriage of some of the ideas of Friedlander-Iwaniec and Maynard which allows one to find primes in other interesting thin sequences.
Simon Marshall, Wisconsin
Title: Counting nontempered forms.
Abstract: It was once believed that nontempered cusp forms on reductive groups shouldn't exist. Today we know that they do, but we expect there to be few of them. I will discuss the problem of bounding the multiplicity of nontempered forms, and present a theorem (joint with Sug Woo Shin) which applies endoscopy to solve this for cohomological forms on unitary groups.
Yong Suk Moon, Purdue
Title: Relative crystalline representations and p-divisible groups when e < p-1
Abstract: Let k be a perfect field of characteristic p > 2, and let R be a relative base ring over W(k) with ramification index e < p-1. If R has Krull dimension 2, then we show that every crystalline representation of the etale fundamental group of Spec(R[1/p]) with Hodge-Tate weights in [0, 1] arises from a p-divisible group over R. If R has higher dimension, then we show that the analogous statement holds after base change to the p-adic completion of R[1/f], for some element f not divisible by p and depending on the representation.
Rachel Davis, Wisconsin
Title: How often is the order of a point on an elliptic curve coprime to $\ell$?
Abstract: Given a point $\alpha \in E(\mathbb{Q})$ of infinite order, what can we say about the order of $\alpha$ when it is reduced in $E(\mathbb{F}_p)$? For example, how often is the order of $\alpha$ odd? In 2010, Jones and Rouse answered 11/21 of the time for ``most" elliptic curves $E/\mathbb{Q}$. In this work, we give an explicit bound for the rate of convergence of their limit. This is joint work with Huynh, Keaton, and Rouse.
Karl Schaefer, Univeristy of Chicago
Title: Unramified p-Extensions of Kummer Fields via Cup Products in Galois Cohomology
Abstract: In their paper "On the Ramification of Hecke Algebras at Eisenstein Primes", Calegari and Emerton prove a relationship between Merel's number and the rank of the p-part of the class group of Q(N^1/p) for primes N = 1 mod p. We answer a question of Calegari-Emerton on this rank and give an effective method for computing this rank in the case p = 5. Our results are obtained by relating the existence of Galois representations satisfying certain local conditions to generalizations of Merel's number. This is joint work with Eric Stubley.
Dave Hansen, Notre Dame
Title: Some remarks on perfectoid Shimura varieties
Abstract: I'll discuss a (somewhat optimistic) conjecture in p-adic Hodge theory. This conjecture, if true, would imply that every Shimura variety becomes perfectoid after adding infinite level structure at p; I'll also explain this implication.
Frank Calegari, University of Chicago
Title: Modularity of abelian surfaces.
Abstract: I will discuss some joint work with Boxer, Gee, and Pilloni on the modularity of abelian surfaces.
Tong Liu, Purdue
Title: Fontaine-Messing theory for formal proper smooth scheme.
Abstract: Let X be a proper smooth scheme over O_K such that K is unramified over Q_p. Fontaine-Messing theory constructed a comparison between torsion crystalline cohomology and torsion etale cohomology (under certain restriction). In this talk, I report an ongoing project on extending Fontaine-Messing theory to allow formal proper smooth scheme. This collaborated work with Bryden Cais and Deepam Patel.
Andrei Jorza, Notre Dame
Title: Taylor expansions of two-variable p-adic L-functions
Abstract: p-adic L-functions attached to p-adic families of cuspidal representations we first used by Greenberg and Stevens in their proof of the Mazur-Tate-Teitelbaum conjecture. They have since been constructed in many instances, most generally for ordinary families on unitary groups by Eischen, Harris, Li and Skinner. I will describe recent work with Mladen Dimitrov on their Taylor expansions that relies on partial p-adic families that vary the local representations at some places above p.
Eun Hye Lee, UIC
Title: On Certain Multiple Dirichlet Series
Abstract: In this talk, I will be talking about the analytic properties of multiple Dirichlet series defined using the space of binary cubic forms. First I will construct the two variable multiple Dirichlet series associated with the two (out of four) semi-invariants of the binary cubic forms, and then as the time permits I will prove its meromorphic continuation to the whole $\mathbb{C}^2$ via sufficiently many “almost" functional equations.
Patrick Allen, UIUC
Title: Automorphy of mod 3 representations over CM fields
Abstract: Wiles's proof of the modularity of semistable elliptic curves over the rationals uses, as a starting point, the Langlands-Tunnell theorem, which implies that the mod 3 Galois representation attached to an elliptic curve over the rationals arises from a modular form of weight one. In order to feed this into modularity lifting theorems, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over CM fields, and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 representations over CM field arise from the "correct" automorphic forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch" that gives a criterion for when a given mod 6 representation arises from an elliptic curve. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.
Darko Trifunovski, UIC
Title: Rankin-Selberg Zeta Integrals for GU(1, 1)
Abstract: Jacquet (1972) analyzes L-functions attached to pairs of cuspidal automorphic representations of GL(2), which he obtains by looking at integrals of the product of a pair of cusp forms with an Eisenstein series. We discuss Jacquet's results and talk about adapting his ideas to the group GU(1, 1).
Dylon Chow, UIC
Title: Height functions on group compactifications
Abstract: In this talk I will discuss some recent results on the distribution of integral points on wonderful compactifications of semisimple groups.
James Freitag, UIC
Title: Ax-Lindemann-Weierstrass for Fuchsian functions
Abstract: Over the last decade, various functional transcendence results and special points conjectures have been proved using a model-theoretic tool called o-minimality and techniques pioneered by Pila in his proof of the André-Oort conjecture. In this talk, we will discuss how an entirely new model-theoretic approach recently lead to a results generalizing Pila’s ALW theorem. The results also imply new cases of the André-Pink conjecture, and resolve an old conjecture of Painlevé (1903).