p-converse to a theorem of Gross-Zagier, Kolyvagin and Rubin

Speaker: 

Ashay Burungale

Institution: 

Caltech

Time: 

Thursday, November 1, 2018 - 3:00pm to 4:00pm

Location: 

RH 306

Let E be a CM elliptic curve over the rationals with conductor N and p a prime coprime to 6N. If the p^{infty}-Selmer group of E has Z_{p}-corank one, we show that the analytic rank of E is also one (joint with Chris Skinner and Ye Tian). We plan to discuss the setup and strategy in the ordinary case.

Teichmuller curves mod p

Speaker: 

Ronen Mukamel

Institution: 

Rice University

Time: 

Monday, November 26, 2018 - 4:00pm to 5:00pm

Location: 

RH 340P

A Teichmuller curve is a totally geodesic curve in the moduli space of Riemann surfaces. These curves are defined by polynomials with integer coefficients that are irreducible over C.  We will show that these polynomials have surprising factorizations mod p.  This is joint work with Keerthi Madapusi Pera.

Cohen-Lenstra in the Presence of Roots of Unity

Speaker: 

Jacob Tsimerman

Institution: 

University of Toronto

Time: 

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

Location: 

RH 306

The class group is a natural abelian group one can associate to a number field, and it is natural to ask how it varies in families. Cohen and Lenstra famously proposed a model for families of quadratic fields based on random matrices of large rank, and this was later generalized by Cohen-Martinet. However, their model was observed by Malle to have issues when the base field contains roots of unity. We study this in detail in the case of function fields using methods of Ellenberg-Venkatesh-Westerland, and based on this we propose a model in the number field setting. Our conjecture is based on keeping track not only of the underlying group structure, but also certain natural pairings one can define in the presence of roots of unity (joint with Lipnowski, Sawin).

Selmer groups, Tate-Shafarevich groups, and ranks of abelian varieties in quadratic twist families

Speaker: 

Zev Klagsbrun

Institution: 

CCR-La Jolla

Time: 

Thursday, May 31, 2018 - 3:00pm to 4:00pm

Location: 

RH 306

We determine the average size of the Φ-Selmer group in any quadratic twist family of abelian varieties having an isogeny Φ of degree 3 over any number field. This has several applications towards the rank statistics in such families of quadratic twists. For example, it yields the first known quadratic twist families of absolutely simple abelian varieties over Q, of dimension greater than one, for which the average rank is bounded; in fact, we obtain such twist families in arbitrarily large dimension. In the case that E/F is an elliptic curve admitting a 3-isogeny, we prove that the average rank of its quadratic twists is bounded; if F is totally real, we moreover show that a positive proportion of these twists have rank 0 and a positive proportion have 3-Selmer rank 1. We also obtain consequences for Tate-Shafarevich groups of quadratic twists of a given elliptic curve. This is joint work with Manjul Bhargava, Robert Lemke Oliver, and Ari Shnidman.

Arithmetic stability in p-adic towers of global function fields.

Speaker: 

Daqing Wan

Institution: 

UC Irvine

Time: 

Thursday, May 17, 2018 - 3:00pm to 4:00pm

Location: 

RH 306

Given a global function field K of characteristic p>0, the fundamental arithmetic invariants include the genus, the class number, the p-rank and more generally the slope sequence of the zeta function of K. In this expository lecture, we explore possible stability of these invariants in a p-adic Lie tower of K. Strong stability is expected when the tower comes from algebraic geometry, but this is already sufficiently interesting and difficult in the case of Zp towers.

A Motivic Deuring-Shafarevich Formula

Speaker: 

Bryden Cais

Institution: 

University of Arizona

Time: 

Thursday, April 12, 2018 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

Let Y --> X be a branched G-covering of curves over a field k.  The genus of X and the genus of Y are related by the famous Hurwitz genus formula.  When k is perfect of characteristic p and G is a p-group, one also has the Deuring-Shafarevich formula which relates the p-rank of X to that of Y.  In this talk, we will discuss our attempts to find a "motivic" generalization of the Deuring-Shafarevich formula by studying how the p-torsion group schemes of the Jacobians of X and Y are related.  In particular, we will explain how to promote the numerical Deuring-Shafarevich formula to an isomorphism of (etale) group schemes. This is ongoing joint work with Rachel Pries.

Adventures in Binary Quadratic Forms

Speaker: 

Vadim Ponomarenko

Institution: 

San Diego State University

Time: 

Thursday, May 24, 2018 - 3:00pm to 4:00pm

Location: 

RH 306

Since Fermat characterized (without proof) those integers represented by the quadratic form x^2+y^2, number theorists have been extending these results.  Recently a paper appeared in Journal of Number Theory answering the question for x^2 ± xy ± y^2.  It turns out that this was not news (although JNT refuses to correct or retract).  Naively, today's speaker extended these results, through elementary means.  This talk will outline these methods, and contrast them with more traditional techniques.

Choosing distinct points on cubic curves

Speaker: 

Weiyan Chen

Institution: 

University of Minnesota

Time: 

Tuesday, April 17, 2018 - 3:00pm to 4:00pm

Host: 

Location: 

RH 340P

It is a classical topic dating back to Maclaurin (1698–1746) to study certain special points on smooth cubic plane curves, such as the 9 inflection points (Maclaurin and Hesse), the 27 sextatic points (Cayley), and the 72 points "of type 9" (Gattazzo). Motivated by these algebro-geometric constructions, we ask the following topological question: is it possible to choose n distinct points on a smooth cubic plane curve as the curve varies continuously in family, for any integer n other than 9, 27 and 72? We will present both constructions and obstructions to such continuous choices of points, state a classification theorem for them, and discuss conjectures and open questions.

Uniform Bounds of Families of Twists

Speaker: 

Bianca Thompson

Institution: 

Harvey Mudd

Time: 

Thursday, January 25, 2018 - 3:00pm to 4:00pm

Location: 

RH 340P

The study of discrete dynamical systems boomed in the age of computing. The Mandelbrot set, created by iterating 0 in the function z^2+c  and allowing c to vary, gives us a wealth of questions to explore. We can ask about the number of rational preperiodic points (points whose iterates end in a cycle) for z^2+c. Can this number be uniform as we allow c to vary? It turns out this is a hard question to answer. Instead we will explore places where this question can be answered; twists of rational functions. 

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