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Non-critical equivariant \(L\)-values of modular abelian varieties. (English) Zbl 1454.11121

Summary: We prove an equivariant version of Beilinson’s conjecture on non-critical \(L\)-values of strongly modular abelian varieties over number fields. The proof builds on Beilinson’s theorem on modular curves as well as a modularity result for endomorphism algebras. As an application, we prove a weak version of Zagier’s conjecture on \(L(E, 2)\) and Deninger’s conjecture on \(L(E, 3)\) for non-CM strongly modular \(\mathbb{Q}\)-curves.

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G55 Polylogarithms and relations with \(K\)-theory
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
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