2018
DOI: 10.1090/btran/27
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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 λA of F K such that L(A/F, s) = L(λA, 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 = γx f + δ. Fix positive integers a and n such that n/2 < a ≤ n. Under mild conditions on e, f, γ, δ, we construct a Chow mo… Show more

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Cited by 3 publications
(3 citation statements)
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“…His construction starts with the hyperelliptic curve C which is the smooth projectivization of the affine curve {y 2 = x 3 c − 1} equipped with the action of a primitive 3 c -th root of unity ζ acting as (x, y) → (ζ • x, y). The variety X is an explicit smooth projective birational model of C n /G, where G is a certain subgroup of (Z 3 c ) n isomorphic to (Z 3 c ) n−1 (see Proposition 4. From an arithmetic perspective, the construction of Schreieder has been used by Flapan and Lang [11] to construct motives associated to certain algebraic Hecke characters, thereby generalizing the modularity result of Cynk and Hulek [7].…”
mentioning
confidence: 99%
“…His construction starts with the hyperelliptic curve C which is the smooth projectivization of the affine curve {y 2 = x 3 c − 1} equipped with the action of a primitive 3 c -th root of unity ζ acting as (x, y) → (ζ • x, y). The variety X is an explicit smooth projective birational model of C n /G, where G is a certain subgroup of (Z 3 c ) n isomorphic to (Z 3 c ) n−1 (see Proposition 4. From an arithmetic perspective, the construction of Schreieder has been used by Flapan and Lang [11] to construct motives associated to certain algebraic Hecke characters, thereby generalizing the modularity result of Cynk and Hulek [7].…”
mentioning
confidence: 99%
“…For c = 1, Cynk and Hulek prove in [CH07] that the varieties Z 1,n are modular, in the sense that one may identify a corresponding L-function attached to their cohomology. This result was then extended to all c ≥ 1 in [FL17]. Recently, Laterveer and Vial show in [LV17] that the subring of the Chow ring of Z c,n generated by divisors, Chern classes, and intersections of two positive-dimensional cycles injects into cohomology via the cycle class map.…”
Section: Introductionmentioning
confidence: 92%
“…From Cynk and Hulek's inductive construction, it follows that the X 1,n may be realized as families of Calabi-Yau varieties over P 1 . Additionally, they proved that these Calabi-Yau varieties are modular, a result which was generalized to all of Schreieder's varieties in [FL17,Corollary 3.8].…”
Section: Introductionmentioning
confidence: 98%