Motivic congruences and Sharifi's conjecture

Abstract

Let f be a cuspidal eigenform of weight two and level N , let p N be a prime at which f is congruent to an Eisenstein series and let V(f )denote the p-adic Tate module off. Beilinson constructed a class kappa f is an element of H-1(Q,Vf(1)) arising from the cup product of two Siegel units and proved a striking relationship with the first derivative L '(f, 0) at the near central point s = 0 of the L-series of f , which led him to formulate his celebrated conjecture. In this note we prove two congruence formulae relating the motivic part of L '(f, 0) ( mod p) and L ''(f, 0) ( mod p) with circular units. The proofs make use of delicate Galois properties satisfied by various integral lattices within V(f )and exploits Perrin-Riou's, Coleman's and Kato's work on the Euler systems of circular units and Beilinson-Kato elements and, most crucially, the work of Sharifi, Fukaya-Kato, and Ohta.