Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture

Details Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture

2005-12-13

arxiv.org/abs/math/0512279v1

Mathematics

Number Theory

36 Pages Submitted to Duke Mathematical Journal

Scientific paper

Let f be a newform of weight 2k-2 and level 1. In this paper we provide evidence for the Bloch-Kato conjecture for modular forms. We demonstrate an implication that under suitable hypothesis if a prime divides the algebraic part of L(k,f), then the prime divides the order of the Selmer group associated to f. We demonstrate this by establishing a congruence between the Saito-Kurokawa lift of f and a cuspidal Siegel eigenform that is not a Saito-Kurokawa lift. We then examine what this congruence says in terms of Galois representations to produce a non-trivial p-torsion element in the Selmer group.

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