Symmetric Square $L$-Functions and Shafarevich--Tate Groups

Abstract

We use Zagier's method to compute the critical values of the symmetric square $L$-functions of six cuspidal eigenforms of level one with rational coefficients. According to the Bloch--Kato conjecture, certain large primes dividing these critical values must be the orders of elements in generalised Shafarevich--Tate groups. We give some conditional constructions of these elements. One uses Heegner cycles and Ramanujan-style congruences. The other uses Kurokawa's congruences for Siegel modular forms of degree two. The first construction also applies to the tensor product $L$-function attached to a pair of eigenforms of level one. Here the critical values can be both calculated and analysed theoretically using a formula of Shimura.