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- W3123158548 abstract "Let $f$ be a cuspidal eigenform of weight two and level $N$, let $pnmid N$ 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 $kappa_fin H^1(mathbb Q,V_f(1))$ 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." @default.
- W3123158548 created "2021-02-01" @default.
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- W3123158548 date "2021-01-25" @default.
- W3123158548 modified "2023-09-27" @default.
- W3123158548 title "Motivic congruences and Sharifi's conjecture" @default.
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