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 motive M , defined over F = Q(γ, δ), such that L(M/F, s) and L(λ a A λ n−a A , s) have the same Euler factors outside finitely many primes.

On the Chow Ring of Cynk–Hulek Calabi–Yau Varieties and Schreieder Varieties

“…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].…”

Some explicit elliptic modular surfaces

“…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.…”

Geometry of Schreieder’s varieties and some elliptic and K3 moduli curves

“…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].…”