Motivic congruences and Sharifi's conjecture

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 κf∈ 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.