NON-CRITICAL EQUIVARIANT L-VALUES OF MODULAR ABELIAN VARIETIES

Abstract. We prove an equivariant version of Beilinson’s conjecture on non-critical L-values of strongly modular abelian varieties over number fields. 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 Q-curves. The purpose of this article is to use the full strength of Beilinson’s theorem on modular curves to prove the following result. Theorem 1. Let A be an abelian variety defined over a Galois number field K whose Hasse-Weil L-function L(A/K,s) is a product of L-functions of newforms of weight 2 without complex multiplication. Then for every integer n ⩾ 2, the weak form of Beilinson’s conjecture on L(A/K,n) holds. We in fact prove a slightly stronger result, namely an equivariant version of Beilinson’s conjecture for the Chow motive H1(A/K) with coefficients in the endomorphism algebra of A, at every non-critical integer (see Corollary 34). The abelian varieties satisfying the hypotheses of Theorem 1 are called strongly modular in [12]. Thanks to the work of Ribet and the proof of Serre’s conjecture, such abelian varieties are known to be modular in the sense that they arise as a quotient of the Jacobian J1(N) of the modular curve X1(N)