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. |
Feb 8
Yunqing Tang |
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 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. |