Skip to main content
eScholarship
Open Access Publications from the University of California

ANDRE-OORT CONJECTURE AND NONVANISHING OF CENTRAL L-VALUES OVER HILBERT CLASS FIELDS

  • Author(s): Burungale, Ashay A
  • Hida, Haruzo
  • et al.

Published Web Location

https://msp.org/ant/2017/11-8/ant-v11-n8-p05-s.pdf
No data is associated with this publication.
Abstract

Let $F/\mathbf{Q}$ be a totally real field and $K/F$ a complex multiplication (CM) quadratic extension. Let $f$ be a cuspidal Hilbert modular new form over $F$. Let ${\it\lambda}$ be a Hecke character over $K$ such that the Rankin–Selberg convolution $f$ with the ${\it\theta}$-series associated with ${\it\lambda}$ is self-dual with root number 1. We consider the nonvanishing of the family of central-critical Rankin–Selberg $L$-values $L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})$, as ${\it\chi}$ varies over the class group characters of $K$. Our approach is geometric, relying on the Zariski density of CM points in self-products of a Hilbert modular Shimura variety. We show that the number of class group characters ${\it\chi}$ such that $L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})\neq 0$ increases with the absolute value of the discriminant of $K$. We crucially rely on the André–Oort conjecture for arbitrary self-product of the Hilbert modular Shimura variety. In view of the recent results of Tsimerman, Yuan–Zhang and Andreatta–Goren–Howard–Pera, the results are now unconditional. We also consider a quaternionic version. Our approach is geometric, relying on the general theory of Shimura varieties and the geometric definition of nearly holomorphic modular forms. In particular, the approach avoids any use of a subconvex bound for the Rankin–Selberg $L$-values. The Waldspurger formula plays an underlying role.

Many UC-authored scholarly publications are freely available on this site because of the UC's open access policies. Let us know how this access is important for you.

Main Content

This item is under embargo until December 31, 2999.

You may have access to the publisher's version here:

https://msp.org/ant/2017/11-8/ant-v11-n8-p05-s.pdfNotify me by email when this item becomes available