## University of Chicago Number Theory Seminar## Winter 2017: Tuesday 1:30-2:50pm, Eckhart 308 |

This is the homepage of the Number Theory Seminar at the University of Chicago. To get on or off the mailing list, you can either go to lists.uchicago.edu or contact Jack Shotton.

Click here to see the location of Eckhart Hall, and here for directions to the University of Chicago.

Click here to see the schedule of previous quarters: Spring 2013 / Fall 2013 / Spring 2014 / Fall 2014 / Winter 2015 / Spring 2015 / Fall 2015 / Winter 2016 / Spring 2016

/ Spring 2016/ Fall 2016-
Date

Speaker

Title

January 10

**Chao Li**

(*Columbia*)**Goldfeld's conjecture and congruences between Heegner points**

Given an elliptic curve E over Q, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (resp. 1). We show this conjecture holds whenever E has a rational 3-isogeny. We also prove the analogous result for the sextic twists of j-invariant 0 curves. For a more general elliptic curve E, we show that the number of quadratic twists of E up to twisting discriminant X of analytic rank 0 (resp. 1) is >>X/log^{5/6}X, improving the current best general bound towards Goldfeld's conjecture due to Ono--Skinner (resp. Perelli--Pomykala). We prove these results by establishing a congruence formula between p-adic logarithms of Heegner points based on Coleman's integration. This is joint work with Daniel Kriz. ( Hide Abstract)January 17

**Ananth Shankar**

(*Harvard*)**The p-curvature conjecture and monodromy about simple closed loops.**

The Grothendieck-Katz p-curvature conjecture is an analogue of the Hasse Principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its p-curvature vanishes modulo p, for almost all primes p. We prove that if the variety is a generic curve, then every simple closed loop has finite monodromy. (Hide Abstract)January 24

**Frank Calegari***(Chicago)***Ramanujan, the Bloch group, and modularity**

The Rogers-Ramanujan identity:

1 + \frac{q}{(1-q)} + \frac{q^4}{(1-q)(1-q^2)} + \frac{q^9}{(1-q)(1-q^2)(1-q^3)} + ... = \frac{1}{(1-q)(1-q^4)(1-q^6)(1 - q^9) ...}

says that a certain $q$-hypergeometric function (the left hand side) is equal to a modular form (the right hand side, up to a power of~$q$). To what extent can one classify all $q$-hypergeometric functions which are modular? We discuss this question and its relation to conjectures in knot theory and K-theory. This is joint work with Stavros Garoufalidis and Don Zagier. (Hide Abstract)January 31

**Rahul Krishna**

(*Northwestern*)**A new approach to Waldspurger's formula**

I present a new trace formula approach to Waldspurger's formula for toric periods of automorphic forms on PGL_2. The method is motivated by interpreting Waldspurger's result as a period relation on SO_2 \times SO_3, which leads to a strange comparison of relative trace formulas. I will explain the local results needed to carry out this comparison, and discuss some optimistic dreams for extending these results to high rank orthogonal groups. (Hide Abstract)February 7

**Rong Zhou**

(*Harvard*)**Mod p isogeny classes on Shimura varieties with parahoric level structure.**

The Langlands-Rapoport conjecture gives a description of the mod p points of suitable integral models for Shimura varieties. Such results are of use, for example, in computing the local factor of the (semi-simple) Hasse-Weil zeta function of the Shimura variety. In this talk, we show the mod p isogeny classes on the integral models of Shimura varieties with parahoric level constructed by Kisin and Pappas have the form predicted by the conjecture, when the group is residually split at p. Along the way, we verify some of the He-Rapoport axioms which allow us to deduce the non-emptiness of Newton strata for these models. (Hide Abstract)February 14

**Baiying Liu**

(*Purdue*)**On the local converse theorem for p-adic Gl_n**

In this talk, I will introduce a complete proof of a standard conjecture on the local converse theorem for generic representations of GLn(F), where F is a non-archimedean local field. This is a joint work with Prof. Herve Jacquet. (Hide Abstract)February 21

**No seminar**

February 28

**Ramin Takloo-Bighash**

*(UIC)***Rational points on zero loci of Brauer elements**

We consider the problem of counting the number of rational points of bounded height in the zero-loci of Brauer group elements on semi-simple algebraic groups over number fields. We obtain asymptotic formulae for the counting problem for wonderful compactifications using the spectral theory of automorphic forms. Applications include asymptotic formulae for the number of matrices over Q whose determinant is a sum of two squares. These results provide a positive answer to some cases of a question of Serre concerning such counting problems. This is joint work with Daniel Loughran and Sho Tanimoto. (Hide Abstract)March 7

**no seminar**

One may also want to check out:

This page is maintained by Jack Shotton; it was shamelessly copied from Brandon Levin's page, which in turn was shamelessly copied from Davide Reduzzi page, which was shamelessly copied from Liang Xiao's page, which was shamelessly copied from Kiran Kedlaya's page, which in turn was shamelessly copied from Jason Starr's page, which in turn was shamelessly copied from Ravi Vakil's page, which in turn was shamelessly copied from Pasha Belorousski's page at the University of Michigan. For more sites with a similar pedigree, see Michael Thaddeus's list or Jim Bryan's list.