Motivic congruences and Sharifi's conjecture

Let $f$ be a cuspidal eigenform of weight two and level $N$, let $p\nmid N$ be a prime at which $f$ is congruent to an Eisenstein series and let $V_f$ denote the $p$-adic Tate module of $f$. Beilinson constructed a class $κ_f\in H^1(\mathbb Q,V_f(1))$ 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 $L'(f,0) \,(\mathrm{mod} \, p)$ and $L''(f,0) \,(\mathrm{mod} \, p)$ 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.

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