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

Let F/Q be a totally real field and K/F a complex multiplication (CM) quadratic extension. Let I he a cuspidal Hilbert modular new form over F. Let lambda he a Ilecke character over K such that the Rankin Selherg convolution f with the theta-series associated with lambda is self-dual with root number 1. We consider the nonvanishing of the family of central-critical Rankin Selberg L-values L(1/2, f circle times lambda(chi)), as 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 chi such that L(1/2, f circle times lambda(chi)) not equal 0 increases with the absolute value of the discriminant of K. We crucially rely on the Andre-Oort conjecture for arbitrary self-product of the Hilbert modular Shimura variety. In view of the recent results of Tsitnertnan, 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 suhconvex hound for the Rankin Selberg L-values. The Waklspurger formula plays an underlying role.