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Symmetric square \(L\)-functions and Shafarevich-Tate groups. (English) Zbl 1039.11029

Summary: 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.

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F33 Congruences for modular and \(p\)-adic modular forms
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