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 Vf denote the p-adic Tate module of f. Beilinson constructed a class f H¹ (, 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) (p) and L'' (f, 0) (p) with circular units. The proofs make use of delicate Galois properties satisfied by various integral lattices within Vf 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.