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