NTS ABSTRACTSpring2018

Return to NTS Spring 2018

Jan 25

Asif Ali Zaman

A log-free zero density estimate for Rankin-Selberg $L$-functions and applications

Abstract:We discuss a log-free zero density estimate for Rankin-Selberg $L$-functions of the form $L(s,\pi\times\pi_0)$, where $\pi$ varies in a given set of cusp forms and $\pi_0$ is a fixed cusp form. This estimate is unconditional in many cases of interest, and holds in full generality assuming an average form of the generalized Ramanujan conjecture. There are several applications of this density estimate related to the rarity of Landau-Siegel zeros of Rankin-Selberg $L$-functions, the Chebotarev density theorem, and nontrivial bounds for torsion in class groups of number fields assuming the existence of a Siegel zero. We will highlight the latter two topics. This represents joint work with Jesse Thorner.

Feb 1

Yunqing Tang

Exceptional splitting of reductions of abelian surfaces with real multiplication

Abstract: Zywina showed that after passing to a suitable field extension, every abelian surface $A$ with real multiplication over some number field has geometrically simple reduction modulo $\frak{p}$ for a density one set of primes $\frak{p}$. One may ask whether its complement, the density zero set of primes $\frak{p}$ such that the reduction of $A$ modulo $\frak{p}$ is not geometrically simple, is infinite. Such question is analogous to the study of exceptional mod $\frak{p}$ isogeny between two elliptic curves in the recent work of Charles. In this talk, I will show that abelian surfaces over number fields with real multiplication have infinitely many non-geometrically-simple reductions. This is joint work with Ananth Shankar.

Feb 8

Roman Fedorov

A conjecture of Grothendieck and Serre on principal bundles in mixed characteristic

Abstract: Let G be a reductive group scheme over a regular local ring R. An old conjecture of Grothendieck and Serre predicts that such a principal bundle is trivial, if it is trivial over the fraction field of R. The conjecture has recently been proved in the "geometric" case, that is, when R contains a field. In the remaining case, the difficulty comes from the fact, that the situation is more rigid, so that a certain general position argument does not go through. I will discuss this difficulty and a way to circumvent it to obtain some partial results.

Feb 13

Frank Calegari

Recent Progress in Modularity

Abstract: We survey some recent work in modularity lifting, and also describe some applications of these results. This will be based partly on joint work with Allen, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne, and also on joint work with Boxer, Gee, and Pilloni.

Feb 15

Junho Peter Whang

Integral points and curves on moduli of local systems

Abstract: We consider the Diophantine geometry of moduli spaces for special linear rank two local systems on surfaces with fixed boundary traces. After motivating their Diophantine study, we establish a structure theorem for their integral points via mapping class group descent, generalizing classical work of Markoff (1880). We also obtain Diophantine results for algebraic curves in these moduli spaces, including effective finiteness of imaginary quadratic integral points for non-special curves.

Feb 15

Junho Peter Whang

Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus

Abstract: In this talk we consider the rational torsion subgroup of the generalized Jacobian of the modular curve X_0(N) with respect to a reduced divisor given by the sum of all cusps. When N=p is a prime, we find that the rational torsion subgroup is always cyclic of order 2 (while that of the usual Jacobian of X_0(p) grows linearly as p tends to infinity, according to a well-known result of Mazur). Subject to some unproven conjecture about the rational torsions of the Jacobian of X_0(p^n), we also determine the structure of the rational torsion subgroup of the generalized Jacobian of X_0(p^n). This is a joint work with Takao Yamazaki.