, Let f be a newform of weight 2 without complex multiplication, and let F be a finite abelian extension of Q. Let X = End F (A f ) ? Q and G = Gal(F Q). For every integer n 2
By Theorem 30, we have an isomorphism of Chow motives e f (H 1 (J 1 (N ) F )complex multiplication, and let F, F ? be finite abelian extensions of Q such that, )) is a newform of level N. We use Theorem 31 with the subgroup K = K 1 (N ) F defined in 3.2, so that M K = X 1 (N ) F. Let J 1 (N ) F be the Jacobian of X 1 (N ) F. We have an isomorphism H 1 (X 1 (N ) F ) ? H 1 (J 1 (N ) F ) in CHM Q (T N,F ), vol.29 ,
, so that we have a canonical embedding R X ? {G ? }. By Theorem 32 and Proposition 15, Conjecture 10 holds for L( R op H 1 (B f,F F ? ) ?m , n). We conclude by projecting onto H 1 (B f,F F ? ) using Proposition 16, By definition of B f,F , we have an isogeny A f ? F B m f,F for some m 1, and thus an isomorphism of Chow motives H 1 (A f F ? ) ? H 1 (B f,F F ? ) ?m. Let R = M m (X{G}) ? M m (X){G}. Put X ? = End F ? (A f ) ? Q and G ? = Gal(F ? Q)
, Let A be an abelian variety over a number field K such that L(AK, s) is a product of L-functions of newforms of weight 2 without complex multiplication
, Let E be a Q-curve without complex multiplication over a number field K such that L(EK, s) is a product of L-functions of newforms of weight 2. Then for every integer n 2, Corollary, vol.35
, Let E be a Q-curve without complex multiplication over a number field K such that L(EK, s) is a product of L-functions of newforms of weight 2. Then the weak form of Zagier, Corollary, vol.36
, Deninger predicted that for an elliptic curve EQ, the L-value L(E, 3) can be expressed in terms of certain double Eisenstein-Kronecker series evaluated at algebraic points of E [10
, Let E be a Q-curve without complex multiplication over a number field K such that L(EK, s) is a product of L-functions of newforms of weight 2. Then the weak form of Deninger, Corollary, vol.37
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