ABC Triples in Families

Speaker: 

Edray Goins

Institution: 

Pomona College

Time: 

Thursday, April 18, 2019 - 3:00pm to 4:00pm

Location: 

RH 306

Given three positive, relative prime integers A, B, and C such that the first two sum to the third i.e. A + B = C, it is rare to have the product of the primes p dividing them to be smaller than each of the three.  In 1985, David Masser and Joseph Osterlé made this precise by defining a "quality" q(P) for such a triple of integers P = (A,B,C); their celebrated "ABC Conjecture" asserts that it is rare for this quality q(P) to be greater than 1 -- even through there are infinitely many examples where this happens.  In 1987, Gerhard Frey offered an approach to understanding this conjecture by introducing elliptic curves.  In this talk, we introduce families of triples so that the Frey curve has nontrivial torsion subgroup, and explain how certain triples with large quality appear in these families.  We also discuss how these families contain infinitely many examples where the quality q(P) is greater than 1.  This joint work with Alex Barrios.

Graphs of Hecke operators and Hall algebras

Speaker: 

Roberto Alvarenga

Institution: 

USP-Brazil, visiting UCI

Time: 

Thursday, February 14, 2019 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

 

Abstract: Motivated by questions of Zagier, Lorscheid develops in his Ph. D thesis the theory of graphs of Hecke operators. These graphs encode the action of Hecke operators on automorphic forms over a global function field. In this talk, I intend to present this theory and connect it with the Hall algebras theory. This connection allow us to describe these graphs for an elliptic curve.  

 

Rank 2 local systems and abelian varieties

Speaker: 

Raju Krishnamoorthy

Institution: 

University of Georgia

Time: 

Wednesday, February 6, 2019 - 3:00pm

Location: 

RH 306

Let X/k be a smooth variety over a finite field. Motivated by work of Corlette-Simpson over the complex numbers, we formulate a conjecture that certain rank 2 local systems on X come from families of abelian varieties. After an introduction to l/p-adic companions, we explain how the existence of a complete set of p-adic companions can be used to approach the conjecture. We also prove Lefschetz theorems for families of abelian varieties over F_q, analogous to work of Simpson over C. This is joint work with A. Pál.

Higher Eisenstein Congruences

Speaker: 

Catherine Hsu

Institution: 

University of Bristol

Time: 

Thursday, May 2, 2019 - 3:00pm

Location: 

RH 306

In his seminal work on modular curves and the Eisenstein ideal, Mazur studied the existence of congruences between certain Eisenstein series and newforms, proving that Eisenstein ideals associated to weight 2 cusp forms of prime level are locally principal. In this talk, we re-examine Eisenstein congruences, incorporating a notion of "depth of congruence," in order to understand the local structure of Eisenstein ideals associated to weight 2 cusp forms of squarefree level.

 

Multiplicative functions over F_q[X] and the Erdos Discrepancy problem.

Speaker: 

Oleksiy Klurman

Institution: 

KTH

Time: 

Thursday, February 28, 2019 - 3:00pm to 4:00pm

We discuss analog of several classical results about mean values of multiplicative functions over F_q[X] explaining some features that are not present in the number field setting. In the first part of the talk, which is based on the joint work with C. Pohoata and K. Soundararajan, we describe spectrum of multiplicative functions over F_q[x]. In the second part of the talk (based on the joint work with A. Mangerel and J. Teravainen), we will focus on the "corrected" function field analog of the Erdos Discrepancy Problem.

Low degree points on curves

Speaker: 

Isabel Vogt

Institution: 

MIT

Time: 

Wednesday, February 27, 2019 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

In this talk we will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite.  By work of Faltings, Harris--Silverman and Abramovich--Harris, it is well-understood when this invariant is 1, 2, or 3; by work of Debarre--Fahlaoui these criteria do not generalize to e at least 4.  We will study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality . This is joint work with Geoffrey Smith.

Random matrices over finite fields follow the Cohen-Lenstra distribution

Speaker: 

GilYoung Cheong

Institution: 

University of Michigan

Time: 

Thursday, March 7, 2019 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

Given a random n x n matrix over the finite field of p elements for a fixed prime p, we will compute the probability that its Jordan normal form contains a specified 0-Jordan block. As n goes to infinity, we will see that our answer converges to the Cohen-Lenstra distribution, which is conjectured to compute the probability that the class group of a random imaginary quadratic field has a specified p-part (when p is odd). We will see why this happens by making connections between our statistics of random matrices and a heuristic distribution of finite abelian groups given by Cohen and Lenstra.

Much of the talk is from a joint work with Yifeng Huang and Zhan Jiang. We will not assume any background from the audience beyond basic graduate algebra classes.

The Monsky-Washnitzer site

Speaker: 

Dingxin Zhang

Institution: 

Harvard University

Time: 

Tuesday, October 23, 2018 - 3:00pm to 4:00pm

Location: 

RH 340P

Rigid cohomology defined by Berthelot agrees with the formal
cohomology defined Monsky and Washnitzer for smooth affine varieites.
Motivated by this, mimicking the convergent theory of Ogus, we define a
site using weakly completed algebras. We show a certain sheaf cohomology of
this site agrees with Berthelot's rigid cohomology

On higher direct images of convergent isocrystals

Speaker: 

Daxin Xu

Institution: 

Caltech

Time: 

Thursday, November 15, 2018 - 3:00pm

Let k be a perfect field of characteristic p > 0 and W the ring of Witt vectors of k. In this talk, we give a new proof of the Frobenius descent for convergent isocrystals on a variety over k relative to W. This proof allows us to deduce an analogue of the de Rham complexes comparison theorem of Berthelot without assuming a lifting of the Frobenius morphism. As an application, we prove a version of Berthelot's conjecture on the preservation of convergent isocrystals under the higher direct image by a smooth proper morphism of k-varieties in the context of Ogus' convergent topos.

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