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

Jim Brown
2007 Compositio Mathematica  
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 hypotheses if where p is a suitably chosen prime and a uniformizer of a finite extension K/Q p . We demonstrate this by establishing a congruence between the Saito-Kurokawa lift F f of f and a cuspidal Siegel eigenform G that is not a Saito-Kurokawa lift. We then examine what this congruence says in terms of Galois
more » ... in terms of Galois representations to produce a non-trivial p-torsion element in H 1 f (Q, W f (1 − k)).
doi:10.1112/s0010437x06002466 fatcat:55kvzjerbvhfnl2evraho3xjiq