Arithmetic of the Yoshida Lift.

Abstract

Co-Chairs: Stephen M. DeBacker and Christopher M. Skinner

This thesis concerns the arithmetic properties of the Yoshida lift, Y, which is a scalar- valued holomorphic Siegel modular form of degree 2 obtained as the theta lift of a pair of automorphic forms f1,f2 on D×, where D is a definite quaternion algebra over Q.

Specifically, we define a refined version of the Yoshida lift, Y, which has the special property that it preserves p-integral structures and is not identically zero under mild conditions. For p-integrality, we compute a formula for the Fourier coefficients aT of Y by exploiting an inherent freedom in the definition of Y. The formula for aT in turn allows us to compute the Bessel model of the Yoshida lift, and apply an argument of Cornut–Vatsal to conclude that Y is non-zero. Furthermore, if we assume Artin’s conjecture on primitive roots, then we show that Y is in fact not zero modulo p.