On the rationality of theta elements for a Rankin-Selberg setting

Abstract

In 1987 Mazur and Tate used the theory of modular symbols to define theta elements that interpolate special values of the Hasse-Weil L-function of an elliptic curve. These elements are essentially finite layer analogues of the L-function, and were conjectured to satisfy many interesting properties. In this thesis, we aim to follow their steps and try to study theta elements for a different L-function. Instead of the Hasse-Weil L-function, we will consider a certain Rankin-Selberg L-function of an elliptic curves twisted by an Artin representation (under some technical conditions). We show that, despite the lack of a theory of modular symbols for our setting, it is still possible to use a theorem of Rankin-Selberg-Shimura to construct similar rational group ring elements that interpolate special values of the L-function. For the original Mazur-Tate setting, the theta elements can be quickly computed due to the many algorithms and libraries that exist for modular symbols. For the Rankin-Selberg setting however, not much is implemented. Still, we show that our theta elements can be explicitly computed and we implement them in SageMath.