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2001 Symmetric Square $L$-Functions and Shafarevich--Tate Groups
Neil Dummigan
Experiment. Math. 10(3): 383-400 (2001).


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.


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Neil Dummigan. "Symmetric Square $L$-Functions and Shafarevich--Tate Groups." Experiment. Math. 10 (3) 383 - 400, 2001.


Published: 2001
First available in Project Euclid: 25 November 2003

zbMATH: 1039.11029
MathSciNet: MR1917426

Keywords: Bloch--Kato conjecture , modular form , Shafarevich--Tate group

Rights: Copyright © 2001 A K Peters, Ltd.


Vol.10 • No. 3 • 2001
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