Chow motives associated to certain algebraic Hecke characters

TL;DR

This paper explores a converse to a classical result, constructing Chow motives linked to algebraic Hecke characters of certain abelian varieties, and compares their L-functions to establish deep connections in number theory.

Contribution

It constructs Chow motives associated to specific algebraic Hecke characters for abelian varieties arising from particular curves, extending the understanding of their L-functions.

Findings

Constructed Chow motives over number fields with matching L-functions outside finitely many primes.

Established a correspondence between motives and algebraic Hecke characters for certain abelian varieties.

Provided conditions under which the motives' L-functions align with those of 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 $n/2 < a \leq n$. 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.