Rank-2 attractors and Deligne's conjecture

In this paper, we will study the arithmetic geometry of rank-2 attractors, which are Calabi-Yau threefolds with the Hodge structure splits into the direct sum of two sub-Hodge structures. We will introduce methods to analyze the algebraic de Rham cohomology of rank-2 attractors, while we will illustrate our methods by focusing on an example in a recent paper by Candelas, de la Ossa, Elmi and van Straten. But the methods in this paper certainly work for general cases. We will show there exist interesting connections between rank-2 attractors and Deligne's conjecture on the special values of $L$-functions.


Introduction
The attractor mechanism appears in the study of supersymmetric black holes in IIB string theory. At low energies, the ten dimensional IIB string theory is well-described by supergravity, which is a classical theory with extended stringy objects approximated by point particles. The attractor mechanism is a very important tool to construct BPS black holes by compactifying the supergravity theory on Calabi-Yau threefolds. The interested readers are referred to the paper [11] for a more rigorous treatment; while here we will only introduce some properties of the attractor mechanism that will be needed in this paper.
The ten dimensional spacetime of this physics theory is taken to be R 3,1 × X, where R 3,1 is the four dimensional Minkowski spacetime and X is a Calabi-Yau threefold. We will focus on the case where the Hodge number h 2,1 of X is one; moreover we will assume X has a deformation of the form π : X → P 1 . (1.1) A coordinate for the base P 1 is denoted by ϕ, and the fiber over ϕ will be denoted by X ϕ . Given a smooth fiber X ϕ , we have a Hodge decomposition H 3 (X, Q) ⊗ C = H 3,0 (X ϕ ) ⊕ H 2,1 (X ϕ ) ⊕ H 1,2 (X ϕ ) ⊕ H 0,3 (X ϕ ), (1.2) 1 which defines a pure Hodge structure on H 3 (X, Q) (H 3 (X, Q), F p ϕ ), F p ϕ := ⊕ q≥p H q,3−q (X ϕ ). (1.3) Notice that here X is the manifold structure of X ϕ . The Hodge filtration F p ϕ varies holomorphically with respect to ϕ [10]. The charge lattice of the black holes is H 3 (X, Z) (modulo torsion). A point ϕ ∈ P 1 is called an attractor point if there exists a nonzero charge vector γ ∈ H 3 (X, Z) such that the Hodge decomposition of γ (in the formula 1.2) only has (3,0) and (0, 3) components, i.e. γ = γ (3,0) + γ (0, 3) .
(1.4) The Calabi-Yau threefold X ϕ is called an attractor if ϕ is an attractor point. A point ϕ is called a rank-2 attractor point if there exist two linearly independent charge vectors γ 1 and γ 2 such that their Hodge decompositions satisfy the condition in the formula 1.4. The Calabi-Yau threefold X ϕ is called a rank-2 attractor if ϕ is a rank-2 attractor point. If X ϕ is a rank-2 attractor, then the pure Hodge structure (H 3 (X, Q), F p ϕ ) splits into the direct sum (1.5) Here M A B (resp. M E B ) is a pure Hodge structure whose Hodge decomposition only has (3, 0) and (0, 3) (resp. (2,1) and (1,2)) components [9]. We will say that the Hodge type of M A B (resp. M E B ) is (3, 0) + (0, 3) (resp. (2, 1) + (1, 2)). The purpose of this paper is to study the arithmetic geometry of rank-2 attractors and their connections to Deligne's conjecture on the special values of L-functions at critical integer points.
In this paper, we find that it is more convenient to use the more abstract language of pure motives. This language certainly will sound very difficult at the beginning, but even physicists will appreciate its simplicity and beauty after becoming familiar with it. We will use pure motive as a black box, while focus on its classical realizations. The interested readers are referred to the papers [9,10,14] for more formal treatments. Given a smooth algebraic variety Y defined over Q, the pure motive h n (Y ), n ∈ Z has three important classical realizations: (1) Betti realization. The variety Y has a complex manifold structure, and the Betti realization is the singular cohomology group H n (Y, Q). The Betti realization admits a pure Hodge structure that is determined by the Hodge decomposition Together with this Hodge structure, the Betti realization is also called the Hodge realization. (2) de Rham realization, which is the algebraic de Rham cohomology group H n dR (Y ) defined by algebraic forms on Y . It also has a Hodge filtration F p (H n dR (Y )). There exists a canonical comparison isomorphism between the Betti realization and the de Rham realization, which is induced by the integration of global algebraic forms on homology cycles. Under the comparison isomorphism, the filtration F p (H n dR (Y )) corresponds to the filtration ⊕ q≥p H q,n−q (Y ).
(3)étale realization, which is theétale cohomology group H ń et (Y, Q ℓ ). It is also a continuous representation of the absolute Galois group Gal(Q/Q) (a Galois representation). Theétale realization is crucial for the study of zeta functions of Y . 2 It is certainly very safe (and also very helpful) to purely understand pure motives through these three classical realizations, which is also the point of view in this paper. Let us now look at an example [9,10,14]. The Tate motive Q(1) is by definition the dual of the Lefschetz motive h 2 (P 1 Q ), whose classical realizations are: (1) Q(1) B = 2πi Q, which has a pure Hodge structure of type (−1, −1).
Here µ ℓ n (Q) consists of the ℓ nth roots of unity which admits a natural action by Z/ℓZ. Hence Q ℓ (1) is a Galois representation. The Tate motive Q(n) is the n-fold tensor product Q(1) ⊗n .
In the paper [11], it is conjectured that rank-2 attractors are algebraically defined over number fields, while in this paper, we will focus on the rank-2 attractors that are defined over Q. More precisely, we will focus on the case where the deformation of X in the formula 1.1 is algebraically defined over Q, and we will only consider those rank-2 attractor points ϕ in this family which are rational, i.e. ϕ ∈ Q. Under these assumptions, the Hodge conjecture tells us that the pure motive h 3 (X ϕ ) of a rank-2 attractor X ϕ , ϕ ∈ Q splits over a number field K [9] h in the formula 1.5. If further K is the rational field Q, then theétale realization of this split tells us that the four dimensional Galois representation . The Tate twist M E ⊗Q(1) has Hodge type (1, 0) + (0, 1) [9,10,14]. From the modularity theorem of elliptic curves, we deduce that the Galois representation M É et ⊗ Q ℓ (1) is modular [5]. More precisely, at a good prime number p, the characteristic polynomial of the geometric Frobenius is of the form where a p is the p-th coefficient of the q-expansion of a weight-2 newform [5]. While for M É et , the characteristic polynomial of the geometric Frobenius at p will be 1 − a p (pT ) + p(pT ) 2 . (1.10) From the paper [6], M Á et is also modular, i.e. at a good prime number p, the characteristic polynomial of the geometric Frobenius is of the form where b p is the p-th coefficient of the q-expansion of a weight-4 Hecke eigenform. On the other hand, the pure motives M A ⊗ Q(2), M A ⊗ Q(1) and M E ⊗ Q(2) are all critical, therefore a very interesting question is to show whether they satisfy Deligne's conjecture or not [4,14]. In the paper [2], the authors have found an example of a rank-2 attractor defined over Q, which is a smooth fiber over the rational point ϕ = −1/7 in a one-parameter family of Calabi-Yau threefolds. So we will denote this rank-2 attractor by X −1/7 . The authors have numerically computed the zeta functions of X −1/7 for small prime numbers, which are of the form [2] (1 − a p (pT ) + p(pT Here a p is the p-the coefficient of the q-expansion of a weight-2 modular form f 2 for the modular group Γ 0 (14), which is designated as 14.2.a.a in LMFDB. While b p is the p-the coefficient of the q-expansion of a weight-4 modular form f 4 also for the modular group Γ 0 (14), which is designated as 14.4.a.a in LMFDB [2]. The authors have also numerically computed the special values of the L-functions, i.e. L(f 2 , 1), L(f 4 , 1) and L(f 4 , 2). They find a special number v ⊥ such that the j-value of 1 2 + i v ⊥ is 215 3 /28 3 , which corresponds to the modular curve X 0 (14) for the modular group Γ 0 (14). The weight-2 eigenform associated to X 0 (14) under the modularity theorem of elliptic curves is just f 2 .
In the paper [14], a new method to compute Deligne's periods using mirror symmetry has been developed. Based on this method, the author has verified that the critical motive h 3 (X −1/7 ⊗ Q(2)) does satisfy Deligne's conjecture. This paper can be considered as a further development of the paper [14]. In this paper, we will study the algebraic de Rham cohomology of rank-2 attractors, which sheds further light on the nature of the split 1.7. Then we will compute Deligne's periods of the critical motives M A ⊗ Q(2), M A ⊗ Q(1) and M E ⊗ Q(2), based on which we will show they satisfy Deligne's conjecture. Moreover, we will show that the pure motive M E essentially comes from the modular curve X 0 (14), i.e. (1.13) The outline of this paper is as follows. In Section 2, we will give an overview of some results of mirror symmetry and introduce the attractor equation. In Section 3, we will review some results about the rank-2 attractor found in the paper [2]. All results in this section are from that paper. In Section 4, we will study the algebraic de Rham cohomology of rank-2 attractors. In Section 5, we will compute Deligne's periods for the critical motives M A ⊗ Q(2), M A ⊗ Q(1) and M E ⊗ Q(2), and we will numerically verify that they satisfy Deligne's conjecture. Moreover, we will show that the pure motive M E essentially comes from the modular curve X 0 (14), i.e. formula 1.13. Section 6 is about the conclusion of this paper.

Mirror symmetry and the attractor equation
In this section, we will briefly review some results of the mirror symmetry of Calabi-Yau threefolds that will be needed in this paper [14]. The interested readers are referred to [1,3,7,10] for more details. We will only focus on one-parameter mirror pairs of Calabi-Yau threefolds. Given a mirror pair (X ∨ , X) of Calabi-Yau threefolds, one-parameter means that their Hodge numbers satisfy For simplicity, we will assume the mirror threefold X has an algebraic deformation defined over Q of the form From now on, X will also mean the differential manifold structure of a smooth fiber in this family. The coordinate of the base variety P 1 Q has been chosen to be ϕ. Following mirror symmetry, we will also assume that for each smooth fiber X ϕ , there exists a nowherevanishing algebraic threeform Ω ϕ that varies algebraically with respect to ϕ. Moreover, as a threeform on a smooth open subvariety of X , Ω is defined over Q. In particular, for a rational point ϕ, Ω ϕ is defined over Q [3,7,10]. From Griffiths transversality, Ω ϕ satisfies a fourth-order Picard-Fuchs equation The point ϕ = 0 is called the large complex structure limit if the monodromy is maximally unipotent. More precisely, there exists a small disc ∆ of ϕ = 0 such that the Picard-Fuchs operator L 2.4 has four canonical solutions of the form ] that converge on ∆. We will further impose the condition f 0 (0) = 1, f 1 (0) = f 2 (0) = f 3 (0) = 0, (2.6) under which the four canonical solutions in the formula 2.5 are unique. From now on, we will assume ϕ = 0 is the large complex structure limit. The canonical period vector ̟ is the column vector defined by Remark 2.1. In this paper, the multi-valued homomorphic function log ϕ satisfy The algebraic de Rham cohomology H dR (X ϕ ) of a smooth rational fiber X ϕ , ϕ ∈ Q is completely determined by the threeform Ω ϕ and its derivatives: where we have used Griffiths transversality.
Here Ω ϕ means the rational vector space spanned by Ω ϕ , etc. The notation Ω ′ ϕ means the first derivative of Ω ϕ with respect to ϕ, etc. The readers are referred to the papers [10,13,14] for more details.

2.2.
Mirror symmetry. From Poincaré duality, there exists a unimodular skew symmetric pairing on H 3 (X, Z) (modulo torsion), which allows us to choose a symplectic basis {A 0 , A 1 , B 0 , B 1 } that satisfy the following intersection pairing [1,3,7] (2.10) Suppose the dual of this basis is Remark 2.2. The torsion of homology or cohomology groups will be ignored in this paper.
The integral periods come from the integration of the threeform Ω ϕ over the symplectic which are multi-valued holomorphic functions [1,3,7]. Now we define the integral period vector ∐(ϕ) by (2.14) For later convenience, let us also define the row vector β by which is the basis vector of H 3 (X, Z). Under the comparison isomorphism between Betti and de Rham cohomology, Ω ϕ has an expansion of the form Since the integral period vector ∐ forms another basis of the solution space of the Picard-Fuchs equation 2.3, there exists a matrix S ∈ GL(4, C) such that The transformation matrix S is crucial in this paper, and it will be determined by mirror symmetry [10,14].
In all examples of one-parameter mirror pairs, there exists an integral symplectic basis where λ is a nonzero rational number whose exact value is not important. The mirror map is defined by the quotient Near the large complex structure limit, formula 2.6 implies 20) 6 therefore the large complex structure limit ϕ = 0 corresponds to t = ∞i. In mirror symmetry, the prepotential F admits an expansion near the large complex structure limit of the form [1, 3, 10] where F np is the non-perturbative instanton correction that admits a series expansion of the form The coefficient Y 111 in the formula 2.21 is the topological intersection number [1,3,7] where e is a basis of H 2 (X ∨ , Z) that lies in the Kähler cone of X ∨ ; hence Y 111 is a positive integer. The coefficients Y 011 and Y 001 are rational numbers [10].
In all examples of mirror pairs, Y 000 is always of the form [1] where χ(X ∨ ) is the Euler characteristic of X ∨ . A detailed study of the appearance of ζ(3) from the motivic point of view is presented in the paper [10]. Using mirror symmetry, it can be shown that the matrix S is given by [10,14] (2.25) 2.3. The attractor equation. Now we are ready to write down the attractor equation for a nonzero charge γ ∈ H 3 (X, Z). Given a point ϕ, as the dimension of H 3,0 (X ϕ ) is 1, we immediately deduce that the component γ 3,0 satisfies where C ∈ C is a nonzero constant. Similarly we also have Suppose the expansion of γ with respect to the basis β is If γ only has (3, 0) and (0, 3) parts, then we will have P = C S · ̟(ϕ) + C S · ̟(ϕ). Similarly, ϕ is a rank-2 attractor point if and only if there exist two different nonzero constants C 1 and C 2 such that C 1 S · ̟(ϕ) + C 1 S · ̟(ϕ) and C 2 S · ̟(ϕ) + C 2 S · ̟(ϕ) are two linearly independent vectors of Z 4 .

Remark 2.3.
At an arbitrary point ϕ, the numerical value of ̟ i (ϕ) can be evaluated by numerically solving the Picard-Fuchs equation 2.3. For more details, the readers can consult the paper [13]. However, it is still very challenging to search for rank-2 attractor points numerically using software like Mathematica.

An example of rank-2 attractors
In the paper [2], the authors have found two rank-2 attractors defined over Q. In this section, we will review those results of the paper [2] that will be used in later sections. The two rank-2 attractors are very similar, e.g. they have the same zeta functions. Therefore we will focus on one example, while the analysis of the second one is exactly the same. We will only review the results that will be used in this paper, while the readers can check [2] for more details. Here X will also mean the differential manifold structure of a smooth fiber of this family. The details of the construction of X and its deformation will not be needed in this paper, hence they are left to [2]. There exists a family of threefroms Ω ϕ for the deformation 3.1 that satisfies all the assumptions in Section 2. The Picard-Fuchs equation of the threeform Ω ϕ is D Ω ϕ = 0 with D =θ 4 − ϕ(35θ 4 + 70θ 3 + 63θ 2 + 28θ + 5) + ϕ 2 (θ + 1) 2 (259θ 2 + 518θ + 285) − 225ϕ 3 (θ + 1) 2 (θ + 2) 2 , θ = ϕ d dϕ . (3. 2) The Picard-Fuchs operator D has five regular singularities at the points ϕ = 0, 1/25, 1/9, 1, ∞, (3.3) 8 while ϕ = 0 is the large complex structure limit. The canonical period ̟ 0 is given by ̟ 0 = 1 + ∞ n=1 a n ϕ n ; a n = i+j+k+l+m=n n! i!j!k!l!m! 2 . (3.4) The other three canonical periods in the formula 2.5 can be computed by Frobenius method [13]. The numbers that appear in the prepotential of X ∨ have also been computed in [2], and they are given by Therefore, the matrix S in the formula 2.25 is determined uniquely up to a non-zero rational multiple by λ ∈ Q × . The authors have found that the smooth fiber X −1/7 over ϕ = −1/7 is a rank-2 attractor. More precisely, they numerically have shown that there exists two non-zero constants C + and C − such that where the column vectors A + and A − are given by [2] A + = (16, −60, 0, 5) ⊤ , A − = (0, 0, 2, 1) ⊤ . (3.7) Therefore X −1/7 is a rank-2 attractor and the pure Hodge structure (H 3 (X, Q), Here the rational vector space of M A B is spanned by β · A + and β · A − , which is a two dimensional subspace of H 3 (X, Q). So the Hodge type of M A B is (3, 0) + (0, 3), and that of M E B is (2, 1) + (1, 2). Remark 3.1. In next section, we will construct a natural basis for M E B . 3.2. Zeta functions and L-functions. The zeta function of H 3 et (X −1/7 , Q ℓ ) at a good prime p is of the form Here, a p is the p-th coefficient of the q-expansion of a weight-2 modular form f 2 for the modular group Γ 0 (14), which is designated as 14.2.a.a in LMFDB. While b p is the p-th coefficient of the q-expansion of a weight-4 modular form f 4 also for the modular group Γ 0 (14), which is designated as 14.4.a.a in LMFDB. This property has been numerically checked by them for small prime numbers [2]. The Hodge conjecture combined with the factorization of zeta functions in the formula 3.9 suggest that the pure motive h 3 (X k,−1/7 ) splits over Q [9] where the Hodge realization of the pure motive M A (resp. M E ) is M A B (resp. M E B ). Thé etale realization of M A (resp. M E ) is a two dimensional Galois representation whose zeta function at a good prime p is 1 − b p T + p 3 T 2 (resp. 1 − a p (pT ) + p(pT ) 2 ).

(3.11)
There is also another number v ⊥ that is very interesting, and its numerical value is v ⊥ = 0.37369955695472976699767292752499463211766555651682 · · · . (3.12) The authors have found that the j-value of τ ⊥ : (3.13) They also find that LMFDB includes only one rationally defined elliptic curve with the above j-invariant, which also has f 2 (14.2.a.a) as its associated eigenform under modularity theorem of elliptic curves. In fact, this curve is defined by which is the modular curve X 0 (14).

The algebraic de Rham cohomology of rank-2 attractors
In this section, we will study the algebraic de Rham cohomology of rank-2 attractors that are defined over Q. We will illustrate our method by working on the rank-2 attractor X −1/7 of [2], which has been reviewed in Section 3; while our method works for general rank-2 attractors. The results obtained in this section shed further lights on the split in the formula 3.10.
First, complex conjugation defines an involution on the complex manifold X −1/7 . Intuitively, it is constructed from the complex conjugation of the complex coordinates of a point in the complex manifold X −1/7 [4,14]. This involution induces an involution which acts on the singular cohomology groups of X [12,14]. We will denote the action of this involution on H 3 (X, Q) by F ∞ : H 3 (X, Q) → H 3 (X, Q).
(4.1) The matrix of F ∞ with respect to the basis β = (β 0 , β 1 , α 0 , α 1 ) of H 3 (X, Q) in the formula 2.15 has been computed in the paper [14] The two linearly independent eigenvectors of F ∞ associated to the eigenvalue 1 are The two linearly independent eigenvectors of F ∞ associated to the eigenvalue −1 are v − 1 = (0, 0, 2, 1) ⊤ , v − 2 = (−1, 2, 0, 0) ⊤ , (4.5) hence the subspace H 3 − (X, Q) of H 3 (X, Q) that F ∞ acts as −1 is spanned by 2α 0 + α 1 and − β 0 + 2β 1 . (4.6) From Section 3.1, the two vectors A + and A − in the formula 3.7 satisfy the relation Notice that β · A + and β · A − form a basis of M A B . The action of F ∞ on the Betti realization we obtain the action of F ∞ on M A B . Recall that the Hodge filtration of the algebraic de Rham cohomology H 3 dR (X −1/7 ) of X −1/7 is determined by formula 2.9. The de Rham realization M A dR of M A is determined by the Hodge filtration of H 3 dR (X −1/7 ). More precisely, the Hodge filtration Here Ω −1/7 means the linear space spanned by Ω −1/7 over Q. Now let E ± be (4.10) Our numerical results show that This property determines a canonical choice of M E B as a subspace of H 3 (X, Q): To determine the Hodge filtration of the de Rham realization M A dR , we only need to determine F 0 (M A dR ), which is a two dimensional vector space over Q. We numerically find that The action of F ∞ on M E B is determined by the equations We now compute the Hodge filtration of the de Rham realization M E dR . Numerically, we have also found that (4.16) The Hodge filtration F p (M E dR ) is also a subspace of F p (H 3 dR (X −1/7 )). Since the Hodge type of M E is (2, 1) + (1, 2), so F 3 (M E dR ) is zero. From formula 4.16, the Hodge filtration of M E dR is canonically given by

The verification of Deligne's conjecture
In this section, we will numerically verify Deligne's conjecture for the critical motives associated to pure motives M A and M E in the formula 3.10. Moreover, we will also show that the motive M E is the Tate twist of the pure motive h 1 (X 0 (14)) that comes from the modular curve X 0 (14). 5.1. The attractive sub-motive. Since the Hodge type of the pure motive M A , which is a direct summand of h 3 (X −1/7 ) in the formula 3.10, is (3, 0) + (0, 3), M A ⊗ Q(n) is critical if and only if n = 1, 2 [4,14]. From Section 3.2, the L-function of the pure motive M A ⊗ Q(n) is given by [4] L(M A ⊗ Q(n), s) = L(f 4 , s + n). (5.1) From the results of Section 4, Deligne's periods of M A are given by [4,14] c While from [4,14], we have Deligne's conjecture predicts that c + (M A ⊗ Q(2)) (resp. c + (M A ⊗ Q(1))) is a rational multiple of L(f 4 , 2) (resp. L(f 4 , 1)) [4,14]. Now we will numerically verify these two predictions.
Under the comparison isomorphism between Betti and de Rham cohomology, the threeform Ω −1/7 has an expansion Ω −1/7 = β · Π(−1/7) = β · S · ̟(−1/7). (5.4) The numerical values of ̟ i (−1/7) can be evaluated to a high precision by numerically solving the Picard-Fuchs equation 3.2. The readers are referred to the paper [13] for more details about this method. Now plug the numerical values of ̟ i (−1/7) into formula 5.2, we obtain where we have used the formulas 2.12 and 5.4. Recall that λ is a nonzero rational number that appears in the formula 2.25, hence we have numerically shown that the critical motives M A ⊗ Q(2) and M A ⊗ Q(1) both satisfy Deligne's conjecture.

5.2.
The elliptic sub-motive. Since the Hodge type of the pure motive M E , which is a direct summand of h 3 (X −1/7 ) in the formula 3.10, is (2, 1) + (1, 2), M E ⊗ Q(n) is critical if and only if n = 2 [4,14]. From Section 3.2, the L-function of the pure motive M E ⊗ Q(n) is given by From the results of Section 4, Deligne's periods of M E are given by [4,14] c While from [4], we have Deligne's conjecture predicts that c + (M E ⊗ Q(2)) is a rational multiple of L(f 2 , 1) [4,14]. Numerically, we have found that which will be important in next section. Recall that λ is a nonzero rational number that appears in the formula 2.25 and the numerical value of v ⊥ is given in the formula 3.12.

5.3.
The construction of the elliptic curve. The Hodge type of the Tate twist M E ⊗Q (1) is (1, 0)+(0, 1), which is exactly the same as that of elliptic curves. The readers might wonder whether there exists an elliptic curve E defined over Q such that If so, the weight-2 newform associated to the elliptic curve E under the modularity theorem will be the modular form f 2 (14.2.a.a) in Section 3.2. In this section, we will show that one choice of such an elliptic curve is in fact the modular curve X 0 (14) in the formula 3.14.
First, let us look at the classical realizations of the motive M E ⊗ Q(1). From Section 4, the rational vector space of its Betti realization M E B ⊗ Q(1) has a basis given by [4,14] From formula 4.17, the Hodge filtration of its de Rham realization is given by To construct an elliptic curve E whose pure motive h 1 (E) is M E ⊗Q(1), we will need a lattice structure of M E B ⊗ Q(1). More concretely, we need four rational numbers a ij , i, j = 1, 2 such that a 11 (2πi)β · E + + a 12 (2πi)β · E − , a 21 (2πi)β · E + + a 22 (2πi)β · E − (5.14) generate a rank-2 lattice of M E B ⊗ Q(1). However, any rational numbers such that a 11 a 22 − a 12 a 21 = 0 (5.15) will always satisfy this property. To proceed, let us compute the j-invariant of the elliptic curve corresponding to such a lattice structure. From the Hodge filtration of M E dR ⊗ Q(1) in the formula 5.13, the nowhere-vanishing oneform of such an elliptic E is hence we can compute the period of E with respect to the lattice structure defined by formula 5.14. From formula 5.7, the period of this elliptic curve E is given by A further restriction is that we need this elliptic curve E to be defined over Q, i.e. we want j(τ ) to be a rational number. There are not many choices of a ij that satisfy this stringent restriction. From the result of [2], which is reviewed in Section 3.2, one such choice is a 11 = − 1 28 , a 12 = 1 2 , a 21 = 0, and a 22 = 1. (5.19) 14 Namely, the period τ is just τ ⊥ = 1 2 + v ⊥ i in the paper [2], and we have j(τ ) = 215 28 3 , (5.20) which corresponds to the modular curve X 0 (14). Moreover, since X 0 (14) is defined over Q, we can choose E to be E = X 0 (14). (5.21) The upshot is that M E ⊗ Q(1) = h 1 (X 0 (14)). (5.22)

Conclusion
In this paper, we have studied the arithmetic geometry of rank-2 attractors that are defined over Q, and their connections to Deligne's conjecture on the special values of L-functions. We illustrate our methods by focusing on the example X −1/7 from the paper [2], while our methods certainly work for general cases.
Hodge conjecture predicts that the pure motive h 3 (X −1/7 ) splits into the direct sum of two sub-motives M A and M E over a number field K, where the Hodge type of M A (resp. M E ) is (3, 0) + (0, 3) (resp. (2, 1) + (1, 2)). While from the numerical results about the zeta functions of h 3 (X −1/7 ) in [2], K should be Q. In this paper, we have developed a method to study the algebraic de Rham cohomology of rank-2 attractors. Our results have shown that the de Rham realization of h 3 (X −1/7 ) also splits accordingly over Q.
Furthermore, we have computed Deligne's periods of the critical motives M A ⊗ Q(2), M A ⊗ Q(1) and M E ⊗ Q(2), based on which we have numerically shown that they do satisfy Deligne's conjecture. We have also found that the pure motive M E in fact comes from the modular curve X 0 (14), i.e. M E ⊗ Q(1) = h 1 (X 0 (14)).