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.

Constructing Abelian Varieties with Small Isogeny Classes

Speaker: 

Travis Scholl

Institution: 

UC Irvine

Time: 

Thursday, October 11, 2018 - 3:00pm to 4:00pm

Location: 

RH 306

In this talk we will focus on constructing "super-isolated abelian varieties". These are abelian varieties that have isogeny class which contains a single isomorphism class. Their motivation comes from security concerns in elliptic and hyperelliptic curve cryptography. Using a theorem of Honda and Tate, we transfer the problem of finding such varieties to a problem in algebraic number theory. Finding these varieties turns out to be related to finding primes of the form n2 + 1 and to solving Pell's equation.

Ramification of $p$-adic etale sheaves coming from overconvergent $F$-isocrystals on curves

Speaker: 

Joe Kramer-Miller

Institution: 

UCI

Time: 

Thursday, December 6, 2018 - 3:00pm to 4:00pm

Wan conjectured that the variation of zeta functions along towers of curves associated to the $p$-adic etale cohomology of a fibration of smooth proper ordinary varieties should satisfy several stabilizing properties. The most basic of these conjectures state that the genera of the curves in these towers grow in a regular way. We state and prove a generalization of this conjecture, which applies to the graded pieces of the slope filtration of an overconvergent $F$-isocrystal. Along the way, we develop a theory of $F$-isocrystals with logarithmic decay and provide a new proof of the Drinfeld-Kedlaya theorem for curves.

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