Chow motives associated to certain algebraic Hecke characters

Abstract

Shimura and Taniyama proved that if $A$ is a potentially CM abelian variety over a number field $F$ with CM by a field $K$ linearly disjoint from F, then there is an algebraic Hecke character ${\lambda }_A$ of $K$ such that $L(A/F,s)=L({\lambda }_A,s)$. We consider a certain converse to their result. Namely, let $A$ be a potentially CM abelian variety appearing as a factor of the Jacobian of a curve of the form $y^e=\gamma x^f+\delta$. Fix positive integers $a$ and $n$ such that $n/2<a\le n$. Under mild conditions on $e,f,\gamma ,\delta$, we construct a Chow motive $M$, defined over $F=\mathbb{Q}(\gamma ,\delta )$, such that $L(M/F,s)$ and $L({\lambda }_A^a{\bar{\lambda }}_A^{n-a},s)$ have the same Euler factors outside finitely many primes.

20 pages

References in corpus (1)

• On the construction problem for Hodge numbers