NTS ABSTRACTSpring2018
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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. |