Chow motives associated to certain algebraic Hecke characters

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.


Introduction
The Langlands philosophy predicts a correspondence between certain automorphic representations and Galois representations. Moreover, the Fontaine-Mazur conjecture and its underlying philosophy specify when these Galois representations are expected to arise from the ℓ-adic cohomology of a variety or, more generally, a motive. 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 M f yielding an equality of L-functions L(M f , s) = L(f, s) (or at least an equality of all but finitely many Euler factors)?
In this paper, we explore this question in a very special case, namely that of algebraic Hecke characters and CM motives. In 1961, Shimura and Taniyama proved that if A is a abelian variety over a number field F with CM by a field K linearly disjoint from F , then there is an algebraic Hecke character λ A of F K such that L(A/F, s) = L(λ A , s) [ST61]. Using more explicit methods, Weil had proved the same result in 1952 for factors of Jacobians of curves of the form C : y e = γx f + δ, for 2 ≤ e ≤ f and γ, δ ∈ Q × [Wei52].
In this paper we are concerned with a converse question. Fix C as above with (e, f ) = 1 as well as a primitive e-th (respectively, f -th) root of unity ζ e (respectively, ζ f ). Assume F = Q(γ, δ) is linearly disjoint from Q(ζ e , ζ f ) and that C has an F -rational point. Let λ be the algebraic Hecke character associated to an isotypic CM factor of the Jacobian of C. Fix a positive integer n and another integer n/2 < a ≤ n. We explicitly construct a Chow motive over F for the Hecke character λ n λ n−a , where the bar denotes complex conjugation.
More precisely, we define a group action of G = (Z/f Z × Z/eZ) n−1 on the product C n that depends on the integer a. There is a Chow motiveM such that any classical realization ofM is given by G-invariants of the corresponding realization of C n . We then decomposeM using idempotents coming from the motive h 1 (C). To describe the decomposition, recall that the Jacobian of C is F -isogeneous to a product (1) where each A i is an isotypic abelian variety defined over F that obtains CM by a certain cyclotomic field K i upon base change. The idempotent e i that cuts out A i can be viewed as an idempotent for h 1 (C), and thus E i = e ⊗n i is an idempotent for h(C n ). We show that the Chow motive M i := E i (M ) behaves very similarly to A i .
For a finite set of primes S, an (incomplete) L-function L (S) ( * , s) is the Euler product of the local L-factors in L( * , s) outside of the set S. Let S i denote the set of primes where the abelian variety A i has bad reduction. Let λ i : A × F K i /F K × i → C × be the algebraic Hecke character associated to A i by Weil and Shimura-Taniyama; that is, L(A i /F, s) = L(λ i , s). Then our main theorem is the following. There are a few important notes to make about the history of this problem. First, the case when C is given by y 2 = x 3 + δ (δ ∈ Q) was treated by Cynk and Hulek in [CH07], and our work is very much inspired by their approach. Secondly, 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 [Sch88, §I.4]. There are two main advantages to our construction when λ comes from one of Weil's curves, which are discussed more precisely in Section 3.4.1. First, our construction yields a Chow motive, which carries more information than a numerical motive. Second, our construction shows that the standard motive descends to a smaller field than that given by the standard construction, in the sense that our motive M i is defined over F and, when base-changed to F K i , coincides with the standard motives. Indeed, our theorem gives a positive answer to the following question in the case of algebraic Hecke characters arising from Weil curves.
Question 1.2. Let λ : A × k /k × → C × be an algebraic Hecke character. Assume there is a subfield k ′ ⊂ k such that the standard motive M (λ) descends to k ′ . For positive integers n, a as above, does M (λ a λ n−a ) also descend to k ′ ?
The structure of the paper is as follows. In Section 2 we establish some facts about the curve C and its Jacobian. The main point is to find a nice basis with respect to which we can compute the Galois action on theétale cohomology of C, which is done in Proposition 2.4. Then 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. Section 3 is devoted to constructing the relevant Chow motives and calculating their L-functions. We start with a brief introduction to the language and notation of motives in Section 3.1. The group action of G on C n is defined at the start of Section 3.2 and was inspired by similar constructions in [CH07] and [Sch15]. The main technical result in the paper is Proposition 3.2, where we compute the Betti realization of the motiveM by computing the G-invariants of the Betti realization of C n . This was due to Schreieder [Sch15] in the case when C is of the form y 2 = x 2g+1 + 1 and our calculation is a straightforward generalization of his result. Theorem 1.1 is proved in Proposition 3.4 and Corollary 3.5.
Finally, we briefly discuss the relationship between the motives constructed here and other motives and varieties in the literature in Section 3.4. In particular, we discuss the constructions of motives in [Sch88] and [Sch90]. Moreover, we note in Corollary 3.8 that our theorem yields modularity results for a class of smooth projective varieties constructed in [Sch15], which generalizes the modularity results of [CH07].
Notation: Throughout the paper we will use ϕ to denote Euler's totient function, ϕ(n) = #(Z/nZ) × . For a positive integer n, let ζ n denote a primitive n-th root of unity. If X is a variety defined over a field F and K is an extension of F , we will write X K for the base change X × F K of X to K. For a field k, we fix an algebraic closure k and write G k for the absolute Galois group Gal(k/k).

Weil's curves
In this section we introduce the curves C studied by Weil in [Wei52] and give explicit descriptions of the deRham, Betti, and ℓ-adic cohomology of C. In particular, the computation of the ℓ-adic cohomology will be given in terms of some algebraic Hecke characters, and it is the powers of these Hecke characters to which we will attach motives in Section 3.
2.1. The curve C. Fix integers 2 ≤ e < f such that (e, f ) = 1. Let γ, δ ∈ Q × such that the field F = Q(γ, δ) is linearly disjoint from the field K = Q(ζ e , ζ f ). From a notational point of view, it is easier to assume F = Q, e = 2, and f is prime. We suggest the reader make these assumptions upon a first reading of the paper. Let Y be the smooth affine curve over F given by Let X be the projective closure of Y , which is usually singular, and C the normalization of X. The curve C then has genus g = (e−1)(f −1) 2 . After base-change to F K, the curve X F K is equipped with two automorphisms ψ e , ψ f of orders e, f , given for projective coordinates [x ′ : y ′ : z ′ ] by: The universal property of normalization ensures that these automorphisms extend uniquely to automorphisms of C F K , which we shall also denote by ψ f and ψ e .
2.2. The de Rham cohomology of C. We shall often want to consider the complex points of C. Doing this requires that we base change C to C, which depends on a choice of embedding F ֒→ C. We fix such an embedding once and for all, with the understanding that our computations depend on this choice, and write C C for the base change of C to C with respect to our fixed embedding.
Let Ω 1 C C be the sheaf of holomorphic differential 1-forms on C C . There are explicit differential 1-forms on Y given by Y ֒→ X is the natural inclusion and N : C → X is the normalization map, then write . Note that the ω i,j are defined over F and thus can be viewed in the algebraic de Rham cohomology of the curve C. The forms ω i,j are eigenvectors for the automorphisms ψ e and ψ f . Indeed, directly from the definitions we calculate Lemma 2.1. A C-basis for H 0 dR (C C , Ω 1 C C ) is given by the set of forms Proof.
Since (e, f ) = 1, the forms ω i,j are eigenvectors with distinct eigenvalues for the automor- , it follows that the ω i,j indeed form a basis for H 0 dR (C C , Ω 1 C C ) To simplify notation, let us set Thus IJ = (e−1)(f −1) 2 = g regardless of the parity of f .
2.3. The Betti cohomology and Jacobian of C. In order to understand the Betti cohomology of the curve C as well as the Jacobian Jac C, we will make use of some Hodge theoretic terminology, which we introduce here.
2.3.1. Preliminaries on Hodge Theory. A Q-Hodge structure V of weight w is a finite dimensional Q-vector space together with a decomposition into linear subspaces V C = p+q=w V p,q , such that V p,q = V q,p , where the bar denotes the action of complex conjugation. If V is a Q-Hodge structure, then an endomorphism α of V is a Q-vector space endomorphism of V that preserves the linear subspaces V p,q when base changed to C.
A polarization of a Q-Hodge structure V of weight w is a bilinear form , : V × V → Q that is alternating if w is odd, symmetric if w is even, and whose extension to V C satisfies: For X a smooth complex projective variety, a choice of ample line bundle on X determines a polarization on the Hodge structure given by the rational cohomology H w (X, Q). The category of polarizable Q-Hodge structures is a semisimple abelian category, which we will denote by Q-HS.

The Betti Cohomology of C.
We now return to our discussion of the rational Betti cohomology H 1 B (C, Q) := H 1 (C C (C), Q) of the curve C, which is a Q-Hodge structure of type {(1, 0)+(0, 1)} by the Betti-de Rham comparison isomorphism for complex varieties. By abuse of notation, we will consider the differential forms ω i,j ∈ H 0 dR (C C , Ω 1 C C ) as elements in H 1 B (C, Q) ⊗ Q C ∼ = H 1 B (C, C). Furthermore, let ω f −i,e−j be the image of ω i,j in H 1 B (C, C). Namely the involution of complex conjugation acts on H 1 Since the Betti-de Rham comparison isomorphism is equivariant with respect to the action of End C, and in particular with respect to ψ e and ψ f , it follows that . Now the Abel-Jacobi map yields an isomorphism of Q-Hodge structures H 1 B (C, Q) ∼ = H 1 B (Jac C, Q). We will frequently make use of this isomorphism together with the equivalence of categories discussed above to go back and forth between the language of cohomology and abelian varieties.
For a field k ⊇ F , let End k (Jac C) denote the algebra of endomorphisms of (Jac C) k defined over k, and let End 0 k (Jac C) = End k (Jac C) ⊗ Z Q. We adopt the following conventions for the rest of the paper to simplify notation. For a proper divisor d|f (respectively, d ′ |e), write f d (respectively, e d ′ ) for the quotient f /d (respectively, e/d ′ ). The notation (d, d ′ ) will always mean f = d|f and e = d ′ |e. If we take a product or sum over d, d ′ , we mean let d and d ′ run over all proper divisors of f and e. Furthermore, let In particular, there is an embedding of Q-algebras d,d ′ Q ֒→ End 0 F (Jac C). Proof. Since ψ e and ψ f are endomorphisms of C defined over F K, they induce endomorphisms ψ * e , ψ * f ∈ End F K (Jac C). We want to calculate the subalgebra of End 0 F K (Jac C) that they generate. In order to do this, we observe that there is an injection (depending on our fixed embedding F ֒→ C, or rather, an extension of that to F K ֒→ C) . So it suffices to calculate the subalgebra of End 0 C (Jac C) generated by ψ * e and ψ * f . Now, (Jac C) C is a complex abelian variety. Since the category of complex abelian varieties up to isogeny is equivalent to the category of polarizable Q-Hodge structures of type {(1, 0) + (0, 1)}, computing End 0 C (Jac C) is the same as computing End Q−HS (H 1 B (Jac C, Q)). Since the Abel-Jacobi map induces an isomorphism H 1 , respectively. Thus the characteristic polynomials of ψ * e and ψ * f acting on H 1 (C, Q) are respectively. Hence the minimal polynomial of ψ * e acting on H 1 B (C, Q) is y e −1 y−1 , and the minimal polynomial of ψ * f acting on H 1 It follows that the subalgebra of End Q−HS (H 1 B (C, Q)) generated by ψ * e is isomorphic to Q[y]/( y e −1 y−1 ) and the subalgebra generated by ψ * f is isomorphic Since (e, f ) = 1, the polynomials T f −1 T −1 and T e −1 T −1 are relatively prime. Thus the subalgebra of End Q−HS (H 1 B (C, Q)) generated by ψ * e and ψ * f is isomorphic to the Q-algebra Q[x, y]/( x f −1 x−1 , y e −1 y−1 ). The factorization of cyclotomic polynomials (see, for instance, [Was97, Chapter 2]) yields the isomorphism The last sentence in the statement of the lemma follows from the first since End 0 , which is an abelian variety defined over F . By definition, (2) The Betti cohomology of C decomposes as For the last statement, recall that the Abel-Jacobi map induces Let Φ n denote the n-th cyclotomic polynomial. We can identify by sending x to ζ f d and y to ζ e d ′ . But under the embedding we sent x to ψ * f and y to ψ * e . Therefore we have Therefore ω i,j ∈ V d,d ′ if and only if (i, f ) = d and (j, e) = d ′ , which proves the last statement.
2.4. TheÉtale Cohomology of C. Let us begin by fixing, once and for all, an algebraic closure F of F as well as embeddings K ֒→ F and F ֒→ C. Let Z /F be a smooth projective variety. Recall that for a given rational prime ℓ, the fixed embedding F ֒→ C yields the identity Moreover, for a fixed embedding ι ℓ : Q ℓ ֒→ C, the Betti-étale comparison isomorphism for complex varieties yields an isomorphism of complex vector spaces [Har77, Appendix C, 3.7]: . In particular, it follows from Proposition 2.3 that we have isomorphisms that respect the decomposition given by the pairs (d, d ′ ). By abuse of notation, write ω i,j for the Since the isomorphisms in (4) are equivariant with respect to the action of End C, it follows that the Note that since A d,d ′ is defined over F , itsétale cohomology inherits an action of the Galois group G F = Gal(F /F ). This action is unramified away from the (finitely many) primes of bad reduction for A d,d ′ . Fix a prime p of F where A d,d ′ has good reduction. In particular, it follows that p ∤ ef . Let q denote the size of the residue field at p. That is, if O F is the ring of integers of F , then q = #O F /p. Now fix a rational prime ℓ such that p ∤ ℓ and an embedding ι ℓ : Q ℓ ֒→ C. Consider the action of a Frobenius element Frob * is a generalized permutation matrix, that is, a matrix with exactly one non-zero entry in each row and column. Given for some a s,t ∈ C.
A direct calculation shows that we have the following equality on C(F q ) for ν ∈ {e, f }: We compute both sides of this equality with respect to the basis B d,d ′ : Since s, t ∈ I d,d ′ , the above congruences are equivalent to the conditions For the claim about the structure of ρ, we identify I d, In the above calculation, q −1 denotes the (multiplicative) inverse of q in Therefore ρ indeed has the desired structure.
The characteristic polynomial of a generalized permutation matrix can be calculated using basic linear algebra (see [NN13, Section 1.2] for details).
Lemma 2.5. Let M ∈ GL N (C) be a generalized permutation matrix given by the data of a permutation ρ and z 1 , . . . , z N ∈ C × . Write C 1 , . . . , C n for the supports of the disjoint cycles of ρ, and let c i = #C i . For 1 ≤ m ≤ n, define the complex number Then the characteristic polynomial of M is the polynomial n m=1 (T cm − Z m ).
As in the proof of Proposition 2.4, let us write the matrix of Frob p with respect to B d,d ′ as (a s,ρ(s) ) s∈I d,d ′ for the permutation ρ. Let q −1 denote the (multiplicative) inverse of q in ( Proof. This follows directly from Proposition 2.4 and Lemma 2.5 since the characteristic polynomial of Frob * p can be computed after any base change. Remark 2.7. Note that while the a s,t may be complex numbers, Corollary 2.6 implies that the Z b are algebraic since the characteristic polynomial of Frob * p vanishes at all of the (ord f d ·e d ′ q)-th roots of Z b .
We shall also want to understand the action of complex conjugation on Z b . Unfortunately, we can only do this when F is totally real. Letting the − symbol denote complex conjugation, we record the following proposition for later use.
Proposition 2.8. Suppose that F is totally real, and let p be a prime of F where A d,d ′ has good reduction. Then for any s, t ∈ I d,d ′ we have a s,t = a −s,−t . In particular, for Proof. Since F is totally real, there is a well-defined complex conjugation c ∈ G F , and for p as in the statement of the proposition, we have the relation c•Frob p = Frob p •c. Hence Frob * p •c * = c * •Frob * p . We compute both sides of this relation, making use of the fact that theétale-Betti comparison isomorphism under which we have identified the various incarnations of ω t is equivariant with respect to complex conjugation. Thus for any t ∈ I d,d ′ , we know that c * (ω t ) = ω t = ω −t . Hence we have This gives a s,−t = a −s,t , as desired. The fact that Z b = Z −b now follows directly from the definition of Z b .
2.5. The L-function of C. Let k and E be number fields. Write O k for the ring of integers of k.
The language of compatible systems of Galois representations is useful for defining the L-functions of interest. Recall that a d-dimensional compatible system of G k -representations V = {V λ } λ is a collection of representations {ρ λ : G k → GL d (E λ )} λ prime of E such that there is a finite set S of primes of k such that: (1) ρ λ is unramified outside S and the primes of k lying over ℓ = λ ∩ Z (2) for all primes p of k outside S, there is a polynomial F p (X) ∈ E[X] such that, for all primes λ of E such that λ ∩ Z = p ∩ Z, the characteristic polynomial ρ λ (Frob p ) is equal to F p (X), independent of λ. For example, if Z/k is a smooth projective variety, then {H n et (Z k , Q ℓ )} ℓ is a compatible system of G k -representations with E = Q for any n. If M is a Chow motive defined over k (see Section 3.1 for the definition), then {H n et (M k , Q ℓ )} ℓ is a system of G k -representations. It is not known in general whether the system is compatible. However, for the motives we construct, we prove that this system of Galois representations is compatible in Proposition 3.4.
The L-function of a compatible system V of d-dimensional G k -representations is a complex analytic (or meromorphic) function that encodes the data of how the local Galois groups G kp ֒→ G k act on the Galois representations {V λ } λ . (Here p is any prime of k and λ is chosen such that p ∩ Z = λ ∩ Z.) It is defined as an Euler product with one local factor for each prime of k. It is easiest to define these local factors when the representation is unramified at p. As these are the only factors that will concern us, we restrict our definition to that case.
Definition 2.9. Let p be a prime of k at which a d-dimensional compatible system of Galois representations V is unramified. Choose a prime λ of E such that p ∩ Z = λ ∩ Z. Let P p (T ) be the characteristic polynomial of Frob p on V (which is independent of the choice of λ). Let then we write L p (Z/k, s) and call it the local L-factor of Z at p. This is the (incomplete) Hasse-Weil zeta function of Z. If Z = A is an abelian variety, then H r et (A k , Q ℓ ) = Λ r H 1 et (A k , Q ℓ ). Therefore we abuse notation slightly and write L(A/k, s) for the L-function of {H 1 et (A k , Q ℓ )} ℓ in the case of abelian varieties. This should not cause any confusion. Note that given a local L-factor L p (V /k, s), it is possible to recover P p (T ) by replacing q −s p in L p (V /k, s) with a variable T and multiplying the resulting Laurent polynomial by T d . We shall often switch between local L-factors and characteristic polynomials in what follows. In particular, we can restate Corollary 2.6 as follows.
Corollary 2.10. If p is a prime of F where A d,d ′ has good reduction, then the local L-factor at p of A d,d ′ over F is Proof. Recall that the compatible system of Galois representations associated to an abelian variety is ramified at exactly the primes of bad reduction. The result now follows from Definition 2.9 and Corollary 2.6.
As we shall only be concerned with the local factors where our representations are unramified, we shall define the incomplete L-function of a compatible system V as follows. Let S be the finite set of primes of k at which V is ramified. The incomplete L-function is It is important to note that the L-function defined above depends on the field k; it will change if we replace k by a larger field k ′ .
Weil proved [Wei52] that the varieties A d,d ′ have algebraic Hecke characters associated to them, a fact that was later generalized to all CM abelian varieties by Shimura and Taniyama [ST61]. In order to state this correspondence, let us recall some definitions. For a number field k, write A k for the ring of adeles of k and O v for the completion of the ring of integers of k at a finite place v. For a place v of k, write ι v : k × v ֒→ A × k for the embedding sending x ∈ k × v to the idele with x in the v-th component and 1 elsewhere.
Definition 2.11. A Hecke character is a continuous homomorphism λ : A × k /k × → C × . Such a character is said to be algebraic if for every archimedean place σ of k, there exists n σ ∈ Z (respectively, n σ , m σ ∈ Z) if σ is real (respectively, complex), such that λ(ι σ (x)) = x nσ (respectively, λ(ι σ (x)) = x nσxmσ ) for all x ∈ k × σ . The conductor of λ is the largest ideal m of k such that λ is trivial on v 1 + mO v .
Fix a uniformizer ̟ v of O v for each finite place v of k, and let p v be the corresponding prime ideal of F . Write Definition 2.12. With notation as in Definition 2.11, let p ∤ m be a prime of k with residue degree q p . The local L-factor of λ at p is Recall that for a positive integer n, if p is a prime not dividing n, then p splits into ϕ(n)/ ord n p distinct primes in Q(ζ n ), each of residue class degree ord n p [Was97, Theorem 2.13]. In particular, Let f p be the residue degree of p over p; that is, q = p fp using the notation introduced just before Proposition 2.4. Then the local L-factor of L(λ d,d ′ , s) at p is where the product runs over all primes p ′ of F d,d ′ lying over p.
Note that since q = p fp , it follows that ord For each prime p ′ of F d,d ′ lying over p, fix an m-th root of λ d,d ′ (p ′ ), and let µ m denote the group of m-th roots of unity. Then we have the following corollary.
Corollary 2.13. For primes p of F as above, we have an equality of sets of ord f d ·e d ′ q elements: where in the second set p ′ runs over primes of F d,d ′ lying over p. Furthermore, if F is totally real Proof. The first statement follows directly from Corollary 2.10 and equations (6) and (7). For the second equality, we have by Proposition 2.8.

Constructing a motive attached to powers of λ
Let λ = λ d,d ′ be the Hecke character attached to A d,d ′ obtained as in the previous section. Fix a positive integer n and n/2 < a ≤ n. The goal of this section is to construct a Chow motive 3.1. Background on motives. Let k be a number field. We begin by briefly recalling the construction of the category of Chow motives over k, closely following [And04]. Let V k denote the category of smooth projective varieties over k. To any object Z in V k , let Z n (Z) denote the group of algebraic cycles in Z of codimension n and Z n (Z) Q = Z n (Z) ⊗ Z Q. If ∼ is an adequate equivalence relation on algebraic cycles [And04, Definition 3.1.1.1], write Z n ∼ (Z) Q = Z n (Z) Q / ∼. For a fixed adequate equivalence relation as above, define a category CV ∼ k whose objects are smooth projective varieties and See [And04, §3.1.3] for the definition of composition of morphisms. The category CV ∼ k is a tensor category. In particular, it is an additive category with a tensor product structure that is symmetric. The category of motives M ∼ k is defined to be the pseudoabelian closure of CV ∼ k . Objects in M ∼ k can be represented explicitly as triples (X, p, m), where X is a smooth projective variety, p ∈ End CV ∼ k (X) such that p 2 = p, and m ∈ Z. Morphisms of motives are given by Furthermore, there is a natural contravariant functor h : V k → M k given by h(X) = (X, ∆ X , 0), where ∆ X is the diagonal subvariety of X × k X. 3.1.1. The motive h 1 of a curve. Let C ∈ V k be a geometrically connected curve, and assume that C has a k-rational point P . Let p 0 be the cycle on C × C given by {P } × C and p 2 the cycle given by C × {P }. Then p 0 and p 2 are idempotent for any choice of adequate equivalence relation. Define p 1 = 1 − p 0 − p 2 , which is also idempotent. Let h 1 (C) = (C, p 1 , 0); that is, h 1 (C) is the image of p 1 . It is well defined up to unique isomorphism and we have [Sch94, Proposition 3.3]: End M ∼ k (h 1 (C)) = End 0 k (Jac C). In particular, End M ∼ k (h 1 (C)) is independent of the choice of adequate equivalence relation ∼. Note that since Jac C is isogenous to a product of isotypic abelian varieties, Jac C ∼ i∈I A i , it follows that there is an orthogonal system of idempotents {e i } i∈I ∈ End M ∼ k (h 1 (C)) corresponding to the decomposition of Jac C. For any positive integer n it follows that {e i 1 ⊗· · ·⊗e in : i j ∈ I} is an orthogonal system of idempotents in End M ∼ k (h 1 (C) ⊗n ). In particular, when C is one of the Weil curves introduced in Section 2 we have an orthogonal system of idempotents in End M ∼ F (h 1 (C)) corresponding to the e d,d ′ introduced prior to Proposition 2.3. We continue to write e d,d ′ for the corresponding element of End M ∼ k (h 1 (C)). Furthermore, we can view e d,d ′ as an element in . Write k-FVS for the category of finitedimensional k-vector spaces with a filtration. The de Rham realization is a functor where H * dR (X) denotes the algebraic de Rham cohomology of the k-scheme X. For any rational prime number ℓ, let Rep Q ℓ (G k ) denote the category of continuous representations of G k on finite dimensional Q ℓ -vector spaces. The ℓ-adic realization is a functor If this system of ℓ-adic representations is compatible, then we define the L-function of a Chow motive as the L-function of its ℓ-adic realizations.

Galois descent.
There is a theory of Galois descent for motives. Let k/k ′ be a finite Galois extension. There is a base change functor − × k ′ k : M ∼ k ′ → M ∼ k as well as a Weil restriction of scalars functor Res k/k ′ : 1.16, Lemma 1.17].

Group actions on varieties.
There is a way to construct a motive h(X) G ∈ M rat k from the action of a finite group G on a smooth projective variety X ∈ V k , assuming that the action of G on X is defined over k. Explicitly, this motive can be written as where Γ g ∈ Z dim X rat (X × k X) Q is the transpose of the graph of the automorphism g [dBnRNA98, proof of Proposition 1.2]. In coordinates, Thus Γ 1 G = ∆ X . Furthermore, this construction behaves well with respect to realizations in the following sense. Each realization of X inherits an action of G. Any realization of h(X) G is just the G-invariant vectors in the corresponding realization of X. Finally, the motive h(X) G descends to a subfield k ′ ⊆ k if both X and the cycle 1 |G| g∈G Γ g descend to k ′ .
3.2. The group action on C n and the Chow motives. We now return to the notation from Section 2. Recall that for proper divisors d|f , d ′ |e, the abelian variety A d,d ′ is defined over F and has CM by K d,d ′ after base change to F d,d ′ . Consider the n-fold product C n F 1,1 . Fix another positive integer a such that n/2 < a ≤ n. Everything that follows will depend on the choice of integers a and n, though this will not be reflected in the notation. Consider the subgroup G of Aut C n F 1,1 given by where the k-th factor ψ u k f ψ v k e in the product n k=1 ψ u k f ψ v k e acts on the k-th factor in the product C n F 1,1 . Note that we have an isomorphism of groups for the idempotent cutting out h(C n F 1,1 ) G . We now use Galois descent to produce a motive defined over the field F instead of F 1,1 .
Proof. Recall that the motive h(C n F 1,1 ) G consists of the data (C n F 1,1 , ε, 0). Since the curve C is defined over the field F , it suffices to check that Let [a : b : c] be projective coordinates of the curve C. Recall that C is the normalization of the singular projective curve X given by (y ′ ) e z ′f −e = γ( Since resolving the singular point [0 : 1 : 0] of X will result in new coordinates obtained as rational functions in the variables x ′ , y ′ , z ′ , both ψ f and ψ e will act by powers of ζ e and ζ f , respectively, on the coordinates a, b, and c of C F 1,1 .
Hence there are integers m i , n j for 1 ≤ i, j ≤ 3 such that for any point Then the class Γ g 0 ⊂ (C n × F C n ) × F F 1,1 is given by Let τ ∈ Gal(F 1,1 /F ) ∼ = Gal(Q(ζ f , ζ e )/Q). Say ζ τ f = ζ i f and ζ τ e = ζ j e . Then for an element Namely, we have α τ ∈ Γ g ′ , where g ′ = n k=1 ψ iu k f ψ jv k e ∈ G. Thus g∈G Γ g is Gal(F 1,1 /F )-stable, as desired.
We can now define the Chow motive associated to λ a d, ) as in Section 3.1.1. Then by Lemma 3.1 we may compose We reiterate that M d,d ′ depends on the choice of a and n even though this is not reflected in the notation.
3.3. Computing the cohomology of the motives M d,d ′ . We begin by studying the Betti realization of M . Once we understand this, we will then determine its decomposition with respect to the idempotents induced by the E d,d ′ . We begin by establishing some notation. Let π k : C n → C be the natural projection map onto the k-th copy of C in C n . Write Ω for the fundamental class of C and Ω k = π * k (Ω). In particular, Ω represents a nonzero class in H 1,1 (C) = H 2 B (C, C) under the Betti-de Rham comparison isomorphism. Let ω k,i,j = π * k (ω i,j ) for all 1 ≤ i ≤ I, 1 ≤ j ≤ J. For the same range of i, j, define Σ i,j = ω 1,i,j · · · ω a,i,jωa+1,i,j · · ·ω n,i,j .
Following the conventions established with ω i,j , we denote by Σ i,j the image of Σ i,j under the standard comparison isomorphisms between Betti, deRham, and ℓ-adic cohomology theories.
The following proposition is due to Schreieder in the case when the curve C is of the form y 2 = x 2g+1 + 1 [Sch15, Lemma 8]. The proof we give below is a straightforward generalization of his argument.
(2) The subspace V p,p consists of all G-invariant homogeneous polynomials of degree p in the classes Ω 1 , . . . , Ω n .
Observe that the product of any (1, 0) and (0, 1) class of the factor C k lies in H 1,1 (C k ) and thus is a multiple of the fundamental class Ω k . Hence we may assume that the intersection Since H 2,0 (C k ) = 0 = H 0,2 (C k ), we may assume that k 1 , . . . , k s are pairwise disjoint and the same for k s+1 , . . . , k t , so all of k 1 , . . . , k t are pairwise distinct. Moreover, since H 2,2 (C k ) = 0, the elements l 1 , . . . , l r are pairwise distinct. Since H 2,1 (C k ) = 0 = H 1,2 (C k ) we then have That is, in the expression for η, all of l 1 , . . . , l r , k 1 , . . . , k t are pairwise distinct. Namely we have l 1 < · · · < l r , k 1 < · · · < k s , and k s+1 < · · · < k t . Now note that the element ψ u 1 f ψ v 1 e × · · · × ψ un f ψ vn e of the group G acts on the monomial η by multiplication by the scalar ζ where µ u 1 ,...,ut is of the form i 1 u 1 ± · · · ± i t u t and ν v 1 ,...,vt is of the form j 1 v 1 ± · · · ± j t v t . First we show that t < n implies that t = 0. Indeed, begin by choosing values for u 1 , . . . , u t and v 1 , . . . , v t such that µ u 1 ,...,ut ≡ 0 mod f and ν v 1 ,...,vt ≡ 0 mod e. Since t < n, we can still choose u t+1 , . . . , u n and v t+1 , . . . , v n such that Namely, the automorphism of C n F 1,1 given by ψ u 1 f ψ v 1 e × · · · × ψ un f ψ vn e lies in the group G, but it acts non-trivially on the monomial η when t > 0. Since η is G-invariant, we must have t = 0 and η = Ω l 1 · · · Ω lr .
The above calculation shows that the matrix [Frob orde d ′ q , the product of all the nonzero entries in The result now follows from Lemma 2.5. The second statement is a restatement of the first using the standard translation between characteristic polynomials and local L-factors.
Recall that S d,d ′ is the set of primes of F where A d,d ′ has bad reduction and S = ∪ d,d ′ S d,d ′ .
Corollary 3.5. Let n ≥ 1 and n/2 < a ≤ n, where we require a = n if F is not totally real. Then, we have an equality of (incomplete) L-functions: Proof. This follows from Corollary 2.13 and Proposition 3.4. Therefore using the decomposition ofM given in (9) and letting S := ∪ d,d ′ S d,d ′ , we have proven the following theorem.
Theorem 3.6. Let n ≥ 1 and n/2 < a ≤ n, where we require a = n if F is not totally real. Then we have an equality of (incomplete) L-functions: That is, our construction proves that the motives M (λ a d,d ′ λ n−a d,d ′ ) ∈ M num F d,d ′ descend to F and can be realized at the level of Chow motives.
3.4.2. Motives of CM modular forms. Suppose F = Q, K is an imaginary quadratic field, and χ is an algebraic Hecke character of K. Then one can form the theta series θ(χ), which is a modular eigenform. Scholl gave a construction attaching a homological (or Grothendieck ) motive over Q to any given eigenform [Sch90]. In the very limited circumstances when F = Q and [K d,d ′ : Q] = 2, our construction proves that Scholl's motives attached to θ(λ a d,d ′ λ n−a d,d ′ ) can be realized at the level of Chow motives.

Modularity of Schreieder's varieties.
The original motivation for this project was to prove that the varieties constructed by Schreieder in [Sch15] are modular in a sense similar to [CH07, Theorem 3.3]. Fix a positive integer k ≥ 1 and let e = 2, f = 3 k , γ = δ = 1 in the Weil curve C introduced in Section 2.1. That is, C has affine coordinate patches y 2 = x 3 k + 1, v 2 = u 3 k +1 + u and genus g = 3 k −1 2 . The particular case of the group G acting on C n was considered for k = 1 and a = n by Cynk-Hulek in [CH07, Section 3] and for general k and a by Schreieder in [Sch15, Section 8]. In these papers they construct, for each k ≥ 1, a smooth model X of the singular quotient variety C n /G with the property: Proposition 3.7. [Sch15, Theorem 17] For any k ≥ 1 and n/2 < a ≤ n, the Betti cohomology of the n-dimensional variety X is of the form where all classes in V p,p are algebraic.
In other words, the "transcendental part" of X, which we denote T (X), is equal to our motiveM in this case. Write A i for the isotypic 3 k−i−1 -dimensional abelian variety defined over Q appearing in the decomposition of (Jac C) K 0 that obtains CM by K i = Q(ζ 3 i 3 k ) when base changed to K i . Let S i denote the finite set of primes of Q where A i has bad reduction, and let λ i be the Hecke character of K i associated to A i by Weil. We obtain the following corollary.
Corollary 3.8. For any choices of k ≥ 1 and n ≥ 1 and n/2 < a ≤ n, the n-dimensional smooth projective variety X is modular. That is, we have an equality of (incomplete) L-functions Proof. The particular decomposition of the Jacobian of C is given by Corollary 2.3 and the result then follows from Theorem 3.6.
In particular, when n is odd we obtain Corollary 3.8 was proved by Cynk and Hulek when C is the elliptic curve y 2 = x 3 + δ (δ ∈ Q) and a = n [CH07, Theorem 3.3].