Chow motives associated to certain algebraic Hecke characters

Laure Flapan,Jaclyn Lang

doi:10.1090/btran/27

Abstract

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

Highlights

The Langlands philosophy predicts a correspondence between certain automorphic representations and Galois representations
To each of these objects—automorphic representations, Galois representations, or motives—one can attach a natural invariant, called an L-function, that is a meromorphic function on some right-half complex plane
In light of these two general conjectures, one can ask: given an automorphic representation f, how can one construct a motive Mf yielding an equality of L-functions L(Mf, s) = L(f, s)?

Summary

Introduction

The Langlands philosophy predicts a correspondence between certain automorphic representations and Galois representations. We explore this question in a very special case, namely that of algebraic Hecke characters and CM motives. Given a number field k and any algebraic Hecke character λ : A×k /k× → C×, there is a standard way to construct a numerical motive M (λ) defined over k such that L(S)(M (λ), s) = L(S)(λ, s) for some finite set of primes S [6, §I.4]. We introduce L-functions and use them to relate the algebraic Hecke characters in question to the matrices describing the Galois action with respect to our chosen basis. This relationship is recorded in Corollary 2.13. We fix an algebraic closure k and write Gk for the absolute Galois group Gal(k/k)

Weil’s curves

Constructing a motive attached to powers of λ