This is the submitted version of the article: Bars Cortina, Francesc; González Rovira, Josep. «Bielliptic modular curves X0* (N)». Journal of Algebra, Vol. 559 (October 2020), p. 726-759. DOI 10.1016/j.jalgebra.2020.02.028 This version is available at https://ddd.uab.cat/record/240651 under the terms of the license Bielliptic modular curves X0∗(N ) Francesc Bars∗and Josep González † Abstract Let N ≥ 1 be a integer such that the modular curve X0∗ (N ) has genus ≥ 2. We prove that X0∗ (N ) is bielliptic exactly for 69 values of N . In particular, we obtain that X0∗ (N ) is bielliptic over the base field√for all these values of N , except X0∗ (160) that is not bielliptic over Q but it does over Q( −1). Moreover, we prove that the set of all quadratic points over Q for the modular curve X0∗ (N ) is infinite exactly for 100 values of N . 1 Introduction Let X be a smooth projective curve defined over a number field K of genus gX at least two. For a finite extension L/K, the set of points of X defined over L, X(L), is finite. When we consider the set Γ2 (X, K) := ∪[L:K]≤2 X(L) , we know that this set is infinite if, and only if, X is hyperelliptic over K, i.e. there is an involution w defined over K such that the quotient curve has genus zero and K-rational points, or X is bielliptic over K, i.e. there exist an involution u (called bielliptic) whose quotient curve E has genus one, such that E is an elliptic curve over K and its K-rank is at least one (cf. [HS91] and [Bar18, Theorem 2.14]). We focus our attention on the modular curves X0∗ (N ) defined as the quotient of the modular curve X0 (N ) by the group of all Atkin-Lehner involutions, which is defined over Q. In this case, we know that X0∗ (N ) is hyperelliptic for sixty-four values of N (cf. [Has97] ), and in [BG19] the following result is presented. Theorem 1.1. Let N > 1 be a square-free integer and assume that the genus of X0∗ (N ) is at least two. Then (i) The modular curve X0∗ (N ) is bielliptic if, and only if, N is in the following table genus N 2 106, 122, 129, 158, 166, 215, 390 3 178, 183, 246, 249, 258, 290, 303, 318, 430, 455, 510 4 370 In all these cases X0∗ (N ) is bielliptic over Q. ii) The set Γ2 (X0∗ (N ), Q) is infinite if, and only if, N is in the set {67, 73, 85, 93, 103, 106, 107, 115, 122, 129, 133, 134, 146, 154, 158, 161, 165, 166, 167, 170, 177, 178, 183, 186, 191, 205, 206, 209, 213, 215, 221, 230, 246, 249, 255, 258, 266, 285, 286, 287, 290, 299, 303, 318, 330, 357, 370, 390, 430, 455, 510} . ∗ The first author is supported by MTM2016-75980-P and MDM-2014-0445. † The second author is partially supported by DGI grant MTM2015-66180-R. 1 The goal of this paper is to complete the list of all bielliptic curves X0∗ (N ). The strategy and some tools used when N is square-free in [BG19] are based on the two following properties. On the one hand, all bielliptic involutions of the curve are defined over Q. So, reductions at a prime p - N of bielliptic quotients can be treated in the finite field Fp . On the other hand, all endomorphisms of the jacobian are defined over Q and, moreover, the endomorphism algebra EndQ (Jac(X0∗ (N ))) ⊗ Q is isomorphic to the product of totally real number fields. So, the action of the involutions on these number fields can only be the product by ±1. When N is non square-free and X0∗ (N ) is bielliptic, both properties can fail. It may be that a bielliptic involution of X0∗ (N ) is not defined over Q when the genus of the curve is ≤ 5. Also, the algebra EndQ (Jac(X0∗ (N )))⊗Q can be non commutative and it can exist endomorphisms of the jacobian not defined over Q. For these reasons, it is important to know the decomposition of the jacobian of X0 (N ) over Q and over Q, the number field where the involutions are defined and to determine the action of these involutions on the jacobian of the curve. Accordingly, the procedure used in [BG19] have to be adapted and expanded to this panorama. The paper is organized as follows. In section 2, we present preliminary results. Some of these, Propositions 2.2, 2.4 and 2.6, will be used to decide when a non hyperelliptic curve X0∗ (N ), with N non square-free, is bielliptic or not. In section 3, we determine a finite set of positive integers that contains all values N such that X0∗ (N ) is bielliptic. Differently from the square-free case, a key point to delimit this finite set is based on the fact that if X0∗ (N ) is bielliptic and M is the biggest square-free integer such that M |N and for each prime p|M the p-valuation at N is odd, then X0∗ (M ) is bielliptic or its genus is ≤ 2. In sections 4, 5, 6 and 7 we determine the values N such that X0∗ (N ) is bielliptic with genus 2, 3, 4 and 5 respectively. Section 8 is devoted to determine the bielliptic curves X0∗ (N ) with genus ≥ 6. In fact, we prove that the curve X0∗ (558) of genus 7 is the unique bielliptic curve X0∗ (N ) with genus ≥ 6. Finally, in section 9 we determine the curves such that the set Γ2 (X0∗ (N ), Q) is infinite. The main result of this article is the following. Theorem 1.2. Let N > 1 be a non square-free integer. Assume that the genus of the modular curve X0∗ (N ) is at least 2. Then, (i) The modular curve X0∗ (N ) is bielliptic if, and only if, N is in the following table genus N 2 88, 112, 116, 121, 153, 180, 184, 198, 204, 276, 284, 380 3 128, 144, 152, 164, 189, 196, 207, 234, 236, 240, 245, 248, 252, 294, 312, 315, 348, 420, 476 4 148, 160, 172, 200, 224, 225, 228, 242, 260, 264, 275, 280, 300, 306, 342 5 364, 444, 495 7 558 √ The curve X0∗ (160) is bielliptic over Q( −1), but not over Q. For the remaining values of N , the curve X0∗ (N ) is bielliptic over Q. (ii) The set Γ2 (X0∗ (N ), Q) is infinite if, and only if, N is in the set {88, 104, 112, 116, 117, 121, 125, 128, 135, 136, 147, 148, 152, 153, 164, 168, 171, 172, 176, 180, 184, 198, 204, 207, 224, 225, 228, 234, 236, 240, 248, 252, 260, 264, 276, 279, 280, 284, 312, 315, 342, 348, 364, 380, 420, 444, 476, 495, 558}. 2 2 Preliminary results Let N > 1 be an integer. We fix, once and for all, the following notation. (i) We denote by B(N ) the group of the Atkin-Lehner involutions of X0 (N ). If N 0 |N and gcd(N 0 , N/N 0 ) = 1, B(N 0 ) denotes the subgroup of B(N ) formed by the Atkin-Lehner involutions wd such that d|N 0 . (ii) The integer ω(N ) is the number of primes dividing N . So, |B(N )| = 2ω(N ) . ∗ (iii) The integers gN and gN are the genus of X0 (N ) and X0∗ (N ) respectively. (iv) We denote by NewN the set of normalized newforms in S2 (Γ0 (N )) and New∗N is the subset of NewN formed by the newforms invariant under the action of the group of the Atkin-Lehner involutions B(N ). (v) S2 (N )∗ is the vector space S2 (Γ0 (N ))B(N ) , J0 (N ) = Jac(X0 (N )) and J0∗ (N ) = Jac(X0∗ (N )). (vi) For a normalized eigenform g ∈ S2 (Γ0 (N )), Ag denotes the abelian variety defined over Q attached by Shimura to g. This abelian variety can be viewed as the optimal quotient of J0 (N ) such that the pullback of Ω1Ag is the vector space generated by the Galois conjugates of g(q) dq/q. K (vii) Given two abelian varieties A and B defined over the number field K, the notation A ∼ B stands for A and B are isogenous over K. (viii) For an integer m ≥ 1 and f ∈ NewN , am (f ) is the m-th Fourier coefficient of f . (ix) As usual, ψ denotes the Dedekind psi function. When we deal with the bielliptic curves X0∗ (N ), there are two important differences depen- ding on whether N is square-free or not. One of them concerns to the pullback of the regular differentials of the elliptic quotients and the other to the field of definition of the bielliptic involutions. Indeed, on the one hand, we know that s Y J0∗ (N ) ∼ Q Afi , i=1 for some fi ∈ NewMi with Mi |N . When N is square-free, these abelian varieties Afi are pairwise non-isogenous over Q and, in particular, EndQ (J0∗ (N ))⊗Q is a commutative algebra isomorphic the product of totally real number fields. Every newform fi provides an only normalized eigenform gi ∈ S2 (N )∗ lying in the vector space generated by {fi (q d ) : 1 ≤ d|N/Mi }. More precisely, X X gi = wd (fi ) = d fi (q d ) , 1≤d|N/Mi 1≤d|N/Mi where wd is the Atkin-Lehner involution attached to d. Hence, if dim Afi = 1, then the pullback of Ω1Af in Ω1X ∗ (N ) under any morphism X0∗ (N )  Afi is always the vector space generated by i 0 the differential gi (q) dq/q. This property can fail for N non square-free. The next lemma and Proposition 2.2 allow us to find a basis of normalized eigenforms of the old part in S2 (N )∗ coming from an eigenform whose level is a strict divisor of N . By Bd : S2 (Γ0 (D)) → S2 (Γ0 (d · D)) we mean the linear map defined as Bd (g(q)) = g(q d ). 3 Lemma 2.1. Let M and N be positive integers such that M |N . Let M1 be a positive di- visor of M such that gcd(M, M/M1 ) = 1 and let d be a positive divisor of N/M such that gcd(M1 d, N/(M1 d)) = 1. If f ∈ S2 (Γ0 (M )) is an eigenvector of the Atkin-Lehner involution wM1 with eigenvalue ε(f ) and ε ∈ {−1, 1}, then f + ε d Bd (f ) ∈ S2 (Γ0 (N )) is an eigenvector of the Atkin-Lehner involution wM1 d with eigenvalue ε(f ) ε. Proof. An  automorphism  u on X0 (N ) whose action on the upper half-plane is given by a A B matrix γ = ∈ GL2 (R) with positive determinant, sends a weight two cusp form C D det γ h to h(γ(z)) (Cz+D) . The action on the Atkin-Lehner involution wM1 d of X0 (N ) is given for 2  M1 d A B any matrix γ = ∈ GL2 (Z) with determinant M1 d. The following matrix   N C M1 d D M1 A B gives the action of the Atkin-Lehner involution wM1 of X0 (M ). It is easy N/d C M1 d D to check that wM1 d sends f to ε(f ) d Bd (f ). Hence, f + ε(f ) · ε wM1 d (f ) is an eigenvector of the Atkin-Lehner involution wM1 d with eigenvalue ε(f ) ε.  Proposition 2.2. Assume that N = pk · M , where k ≥ 1, p is a prime and M is an integer coprime to p. For 0 ≤ i < k, let f ∈ S2 (Γ0 (pi · M ))B(M ) be such that wpi (f ) = ε · f (clearly ε = 1 when i = 0). Let S be the vector subspace of S2 (Γ0 (pk · M )) generated by the k − i + 1 linearly independent eigenforms {f, Bp (f ), · · · , Bpk−i (f )}. Then, (i) The following normalized eigenforms g0 = (1 + pBp )k−i f, · · · , gj = (1 + pBp )k−i−j (1 − pBp )j f, · · · , gk−i = (1 − pBp )k−i f , form a basis of S. (ii) Every gj is an eigenvector of wpk with eigenvalue (−1)j ε. (iii) The dimension s of the vector space S2 (pk · M )∗ ∩ S is   k−i+1  if k − i is odd, s= 2 k−i+1+ε   if k − i is even. 2 Proof. To prove the statement, we fix i and proceed by induction over k. For k = i + 1, it is clear that the statement is true. Indeed, by Lemma 2.1, the eigenforms g0 = (1 + pBp )f and g1 = (1 − pBp )f are eigenvectors of wi+1 with eigenvalues ε and −ε respectively and, moreover, s = 1. Assume that the statement is right for k − 1 > i. The linear map 1 + pBp : hf, · · · , f (q k−i−1 )i −→ S = hf, · · · , f (q k−i )i is injective and, moreover, f is not in its image. By induction, {g0 , · · · , gk−i−1 } is a basis of the image of 1 + pBp and, thus, {g0 , · · · , gk−i−1 , f } is a basis of S. From the equality k−i   k−i k−i X k−i 2 f = (1 + pBp + 1 − pBp ) f= gj , j=0 j it follows part (i) of the statement. By Lemma 2.1, each gj is an eigenvector of the Atkin-Lehner involution wpk with eigenvalue (−1)j ε. Part (iii) can be easily checked.  4 For instance, consider a prime p not dividing M for which there is f ∈ Newp·M such that f ∈ S2 (Γ0 (p · M ))B(M ) and wp (f ) = ε f . In this case, f provides a vector subspace in the old part of S2 (p3 M )∗ of dimension 1 or 2 depending on whether  = −1 or not. Indeed, for ε = −1, we get the only normalized eigenform 2 g = (1 + p Bp )(1 − p Bp )(f ) = f (q) − p2 f (q p ) . For ε = 1, we get the two following eigenforms 2 2 g1 = (1+p Bp )2 (f ) = f (q)+2 p f (q p )+p2 f (q p ) , g2 = (1−p Bp )2 (f ) = f (q)−2 p f (q p )+p2 f (q p ) . In this particular case, if dim Af = 1, any direction in the vector space generated by g1 (q) dq/q and g2 (q) dq/q can be the pullback of Ω1Af in Ω1X ∗ (p3 M ) under a morphism X0∗ (p3 M )  Af . 0 On the other hand, when N is square-free, all automorphisms of X0∗ (N ) are defined over Q, but this property can fail for N non square-free. The following results will be useful to know the number field where bielliptic involutions are defined. Lemma 2.3 (Silverman-Harris). [HS91, Prop.2.1] Let X be a curve defined over a number field K with genus ≥ 6. If X is bielliptic, then there is an only bielliptic involution and, moreover, it is defined over K. Note that if X0∗ (N ) has a bielliptic involution u defined over a number field K, the bielliptic quotient is an elliptic curve defined over K because X0∗ (N ) is non empty. Proposition 2.4. Let A be a modular abelian variety defined over Q such that A ∼ m ni Q Q i=1 Afi for some fi ∈ NewNi , where Afi are pairwise non-isogenous over Q. All endomorphisms of A are defined over Q if, and only if, for every non trivial quadratic Dirichlet character χ, the newform fi ⊗ χ is different from any Galois conjugate of fj for all i and j. Q Proof. For two newforms f ∈ NewN and f 0 ∈ NewN 0 , we have that Af ∼ Af 0 if, and only if, N = N 0 and f 0 is a Galois conjugate of f (see Proposition3.2 in [BGGP05]). It is well-known that the abelian variety Af is simple over Q if, and only if, f does not have any inner twist, i.e. there is not any quadratic Dirichlet character χ such that f ⊗ χ is a Galois conjugated of f . This condition amounts to saying that EndQ (Af ) = EndQ (Af ). Assume that f and f 0 does not have any inner twist. If Af and Af 0 are isogenous over Q but not over Q, then there exists a Dirichlet character χ such that Af 0 = Af ⊗χ (cf. Proposition 4.2 in [GJU12]).  Remark√ 2.5. If χ is the quadratic Dirichlet character attached to the quadratic number field K = Q( D), then there is an isogeny between the abelian varieties Af and Af ⊗χ defined over K. By using Petri’s Theorem, in [BG19, Lemma 13], it is characterized the existence of a bielliptic involution of X0∗ (N ), when gN ∗ > 2, X0∗ (N ) is not hyperelliptic and N is square-free. Next, we present Proposition 2.6 that will be useful to determine the existence of a bielliptic involution of X0∗ (N ) with bielliptic quotient E, when E appears in the splitting of J0∗ (N ) with exponent > 1 and X0∗ (N ) is not hyperelliptic. This result generalizes the mentioned lemma and can be applied to any curve X (modular or not). Let X be a non-hyperelliptic curve of genus g ≥ 3 defined over a number field K. For a fixed basis ω1 , · · · , ωg of Ω1X/K and an integer i ≥ 2, we denote by Li the K-vector space formed by the homogenous polynomials Q ∈ K[x1 , · · · , xg ] of degree i such that Q(ω1 , · · · , ωg ) = 0. 5 K Proposition 2.6. With the above notation, assume that Jac(X) ∼ E m × A, where E is an elliptic curve and A an abelian variety that does not have E as a quotient defined over K. Denote by Ig−m ∈ Mg−m (K) the identity matrix. Take the basis {ωi } such that {ω1 , · · · , ωm } and {ωm+1 , · · · , ωg } are bases of the pullbacks of Ω1E m /K and Ω1A/K respectively. Then, E is K-isogenous to the Jacobian of a bielliptic quotient of X over K if, and only if, there exists a matrix A ∈ GLm (K) that satisfies Q((−x1 , x2 , · · · , xg ) · B) ∈ L0i for all Q ∈ Li and for all i ≥ 2 , (2.1)   A 0 where B is the matrix ∈ GLg (K) and L0i = {Q((x1 , x2 , · · · , xg ) · B)) : Q ∈ Li }. 0 Ig−m Proof. Let u be a bielliptic involution of X over K such that the curve E is K-isogenous to the jacobian of the quotient curve X/hui. The linear map u∗ acting on the vector space Ω1X/K is diagonalizable and has an only eigenvalue equals to 1, because X/hui has genus one. Since E is not a quotient of A defined over K, the linear map u∗ leaves the vector spaces Ω1E m /K and Ω1A/K stable. Hence, u(ωi ) = −ωi for i > m and there exists a matrix A ∈ GLm (K) such that the regular differentials (ω10 , ω20 , · · · , ωm 0 ) = (ω1 , ω2 , . . . , ωm ) · A−1 are eigenvectors of u with eigenvalues 1, −1, · · · , −1 respectively. The condition 2.1 follows from the fact that the vector spaces L0i are those attached to the basis ω10 , · · · , ωm 0 , ωm+1 , · · · , ωg . For the converse, see the proof of [BG19, Lemma 13].  Remark 2.7. We recall that if g = 3, then the condition (2.1) can be restricted to i = 4 and dim L4 = 1. When g > 3, dim L2 = (g − 3)(g − 2)/2. In this case, it suffices to check (2.1) only for i = 2, 3 and, in the particular case that X is neither a smooth quintic plane curve (g = 6) nor a trigonal curve, we can restrict the condition to i = 2. As in [BG19], for 1 ≤ j ≤ g, we introduce the K-vector space L2,j = {Q ∈ L2 : Q(x1 , · · · , xj−1 , −xj , xj+1 , · · · , xg ) ∈ L2 } . By using that the polynomials in L2 are irreducible, in [BG19] it is proved that L2,j = {Q ∈ L2 : Q(x1 , · · · , xj−1 , xj , xj+1 , · · · , xg ) = Q(x1 , · · · , xj−1 , −xj , xj+1 , · · · , xg )} . 0 Since the spaces L0i in Proposition 2.6 are the spaces Li corresponding to the basis ω10 , · · · , ωm , ωm+1 , · · · , ωg of Ω1X/K , the space L02,1 is formed by the polynomials in L02 that are even in the first variable x1 . A similar result is obtained when g = 3 and we replace L2 with L4 . Remark 2.8. We have J0∗ (N ) ∼ si=1 Anfii , for some fi ∈ NewMi with Mi |N and the abelian Q Q varieties Afi are pairwise non isogenous over Q. Any fi determines ni normalized eigenforms gj in S2 (N )∗ such that J0∗ (N ) ∼ rj=1 Agj , where r = si=1 ni and g1 , · · · , gr are all of these Q Q P eigenforms. The basis of the Galois conjugates of the newforms fi together with the exponents ni allow us to compute |X0∗ (Fp )| for all primes p - N , thanks to the Eichler-Shimura congruence. The basis of the regular differentials formed by all Galois conjugates of gj (q) dq/q allows us to compute equations for X0∗ (N ). When dim Afi = 1, we take as basis of Ω1Ani /Q the regular differentials fi attached to the normalized eigenforms obtained from fi via Proposition 2.2. 6 Now, we assume that X0∗ (N ) has a bielliptic involution u defined over Q. Let us denote by E the elliptic quotient X0∗ (N )/hui and by π the non constant morphism X0 (N ) → X0∗ (N ) → E of degree 2ω(N )+1 which is defined over Q. Let M be the conductor of E. It is well-known, that M |N and there exists a morphism πM : X0∗ (M )  E and a normalized newform fE ∈ New∗M ∗ such that πM (Ω1E/Q ) = Q(fE (q)d q/q). Moreover, π ∗ (Ω1E/Q ) = Q(g(q)d q/q), where g ∈ S2 (N )∗ is an eigenform in the vector space generated by {fE (q i ) : 1 ≤ i|N/M }. Note that for a prime p - M , |E(Fp2 )| ≤ (p + 1)2 . More precisely, due to the congruence of Eichler-Shimura, |E(Fp2 )| = (p + 1)2 − a2p (fE ). From [BG19, Lemmas 5 and 7], we have the following two lemmas when E is a bielliptic quotient of X0∗ (N ) over Q. Lemma 2.9. The following inequalities hold: (i) For a prime p - N , ψ(N ) 2|E(Fp2 )| − 1 ∗ |E(Fp2 )| |E(Fp2 )| (a) ω(N ) ≤ 12 · , (b) gN ≤2 , (c) gN ≤ 2ω(N )+1 . 2 p−1 p−1 p−1 ∗ (ii) For a prime p such that the p-adic valuation vp (N ) = 1, then gN/p ≤ 1 or ψ(N/p) 2 |E(Fp2 )| − 1 |E(Fp2 )| |E(Fp2 )| ω(N )−1 ≤ 12 ∗ , gN/p ≤2 , gN/p ≤ 2ω(N ) , if p - M , 2 p−1 p−1 p−1 ψ(N/p) 2p2 + 1 ∗ p2 + 1 2 ω(N ) p + 1 ≤ 12 , g N/p ≤ 2 , gN/p ≤ 2 , if p|M . 2ω(N )−1 p−1 p−1 p−1 We recall that the condition vp (N ) = 1 implies that the normalization of the curve X0∗ (N )/Fp is X0∗ (N/p)/Fp . Lemma 2.10. Let E 0 be the elliptic curve in the Q-isogeny class of E that is an optimal quotient of the jacobian of X0∗ (M ). If M = N , then the degree of the modular parametrization πN : X0 (N ) → E 0 divides 2ω(N )+1 . 3 Selecting candidates for bielliptic curves X0∗(N ) The starting point of our selection is based on the following result. Lemma 3.1. Let ` be a prime. If for an integer k ≥ 2, the curve X0∗ (`k · M ) is bielliptic, then X0∗ (`k−2 · M ) is hyperelliptic, bielliptic or has genus at most one. Proof. We can assume without loss of generality that gcd(M, `) = 1. We claim that the morphism π : X0 (`k · M ) → X0 (`k−2 · M ) induced by the map z 7→ `z on the upper half- plane, provides a morphism from X0∗ (`k · M ) to X0∗ (`k−2 · M ). Indeed, for wd ∈ B(M ) one has π ◦ wd = wd ◦ π and for w`k ∈ B(`k · M ) one has π ◦ w`k = w`k−2 ◦ π. The statement follows from [HS91, Proposition 1].  Corollary 3.2. Let N > 1 be an integer such that X0∗ (N ) has genus ≥ 2. Let M be the biggest square-free integer such that M |N and each prime p|M its p-valuation at N is odd. If X0∗ (N ) is bielliptic, then X0∗ (M ) is bielliptic, hyperelliptic (equivalently gM ∗ ∗ = 2) or gM ≤ 1. When N is a power of a prime, we have that X0∗ (N ) = X0+ (N ) and from [Jeo18] we know all bielliptic curves X0∗ (N ). 7 Proposition 3.3 (Jeon). Assume N is a non-square free integer which is a power of a prime, ∗ ∗ with gN ≥ 2. Then, N is bielliptic if, and only if, N = 121 = 112 , or N = 128 = 27 (g121 =2 ∗ and g128 = 3). For a non-square free integer N > 1, let M be the biggest square-free divisor of N such that for each prime p|M its p-valuation at N is odd. Therefore, if X0∗ (N ) is bielliptic, then M is a value appearing in part (i) of Theorem 1.1 or in table 4 of the Appendix corresponding to ∗ gN ≤ 2. Moreover, we can assume that N is divisible by at least two primes. Let M be the set of such square-free integers M and let N be the set of integers N that are not a power of a prime, such that, its biggest square-free divisor such that for each prime p|M its p-valuation at N is odd is in the set M. To discard curves X0∗ (N ) with N ∈ N that are not bielliptic over Q, we will apply the same strategy as the used in [BG19], that is summarized in Proposition 3.4. Indeed, for a prime p - N , X0∗ (N )(Fpk ) denotes the set of the Fpk -points of the reduction modulo p of X0∗ (N ). Set X Pp (k) := mod [( µ(k/d)|X0∗ (N )(Fpk )|)/k, 2] d|k where mod [m, 2] denotes 0 Por 1 depending on whether m is even or not, and µ is the Moebius function. Set Qp (2j + 1) = ji≥0 (2i + 1)Pp (2i + 1). By [Gon17, Theorem 2.1], if X0∗ (N ) has an involution defined over Q, then ∗ Qp (2j + 1) ≤ 2gN + 2 for all j ≥ 0 . Moreover, if E is a bielliptic quotient over Q of X0∗ (N ), then n(N, E, pE kE ) := |X0∗ (N )(Fpk )| − 2|E(Fpk )| ≤ 0 for all k > 0. From previous arguments and Lemmas 2.9(i)(a) and 2.10, it follows the next result. Proposition 3.4. Suppose that the curve X0∗ (N ) with gN ∗ ≥ 2 satisfies one of the three following conditions: 1 p−1 (i) There exists a prime p - N such that 2ω(N ) · 12 ψ(N ) + 1 > 2 · (p + 1)2 , ∗ (ii) There exist a prime p - N and an integer j ≥ 0 such that Qp (2j + 1) > 2gN + 2, (iii) Every elliptic curve E defined over Q of conductor NE which appear in the Q-decomposition of J0∗ (N ) satisfies: • If NE is a strict divisor or N , then there exist a prime pE - N and a positive natural kE such that n(N, E, pE kE ) > 0. • If NE = N , then the above condition on the existence of a prime pE - N and a positive natural kE is satisfied or the degree of the Weil strong parametrization of E does not divide 2ω(N )+1 . Then, the curve X0∗ (N ) is not bielliptic over Q. All objects appeared in Proposition 3.4 can be computed by Magma (and Cremona tables). We refer to the web page http://mat.uab.cat/∼francesc/Magmaprogrammesxostar.html for the programmes in Magma computing the different objects involved in Proposition 3.4, and in particular the Q-decomposition of the Jacobian of X0∗ (N ) for any N . To find all bielliptic curves X0∗ (N ) over Q, we proceed by determining a finite set of pairs (N, E) such that E is the Q-isogeny class of an elliptic curve defined over Q and E can be a bielliptic quotient of X0∗ (N ) for N ∈ N . 8 ∗ 3.1 Case gN >5 ∗ If gN > 5 and X0∗ (N ) is bielliptic, then X0∗ (N ) is bielliptic over Q (cf. Lemma 2.3). We present the following example to show how Proposition 3.4 and Lemma 3.1 are applied. Take M = 178 = 2 · 89 for which X0∗ (178) is bielliptic and gM ∗ = 3. We claim that for all N = 2 · 89 · d with a, b, d ≥ 1 and a + b > 2 with gcd(2 · 89, d) = 1, the curves X0∗ (N ) are a b 2 not bielliptic curves. Indeed, by Lemma 3.1, it is enough to prove the assertion for N = 22 · 89, N = 892 , N = 2 · 892 , N = 23 · 89 and N = 2 · 893 . For N = 892 , 2 · 892 ,2 · 893 and 23 · 89 one has ∗ gN > 6 and we can discard these cases by applying part (i) of Proposition 3.4 because ψ(N ) ∗ satisfy the inequality with p = 3. For N = 4 · 89 = 356, we have gN = 8 and the Q-isogeny ∗ classes of the elliptic curves over Q in J0 (356) are E89a and E178a with Cremona notation. For p = 3 - N , we obtain n(356, E89a, 9) = n(356, E178a, 9) = 32 − 30 > 0. Hence, by part (iii) of Proposition 3.4, the curve X0∗ (356) is not bielliptic. ∗ Applying part (i) of Lemma 2.9, Proposition 3.4 and Lemma 3.1, for gN ≥ 6 we can reduce the study to the following pairs (N, E) (see the detailed results for all levels and elliptic curves involved in http://mat.uab.cat/∼francesc/Tablestudy.pdf). ∗ gN (N, E) 6 (244, 61a), (272, 34a), (332, 83a), (332, 166a), (336, 42a), (336, 112a), (564, 94a), (620, 62a), (780, 65a), (780, 130c) 7 (320, 32a), (324, 27a), (360, 20a), (360, 30a), (450, 15a) (450, 75b), (456, 57a), (456, 76a), (456, 152a), (492, 123b), (504, 21a), (504, 36a), (504, 42a), (550, 55a), (550, 275a), (550, 550a), (558, 558a), (636, 53a), (660, 110b) (924, 77a), (924, 462a) 8 (408, 102a), (468, 26b), (468, 234b), (468, 234c), (480, 20a), (480, 24a), (480, 80b), (480, 160a), (540, 45a), (540, 54b), (990, 66a), (990, 99a), (1020, 102a) 9 (560, 56a), (560, 70a), (560, 280a)(1140, 190b), (1140, 285b) 10 (840, 20a), (840, 140b), (840, 210d), (1050, 175b) 11 (672, 112c), (672, 224a) 13 (1260, 21a), (1260, 70a), (1260, 90b), (1260, 210d) Table 1 where the Q-isogeny class of E is described with Cremona notation. Lemma 3.5. The curve X0∗ (1020) of genus 8 is not bielliptic. Proof. If X0∗ (1020) is bielliptic, then it is bielliptic over Q because g1020 ∗ ≥ 6. Computing the n values n(1020, E, p ) for all possible bielliptic quotients, we get that E should be E102a (see http://mat.uab.cat/∼francesc/Tablestudy.pdf ). Suppose that there is a degree two morphism defined over Q from X0∗ (1020) to an elliptic curve in the Q-isogeny class of E102a. Since 9 - 1020, the normalization of X0∗ (1020)/F3 is the curve X0∗ (340)/F3 . Hence, this curve has an involution defined over F3 whose quotient is a curve of genus zero defined over F3 because 3 | 102. But this is not possible because |X0∗ (340)(F3 )| = 13 > 8.  ∗ 3.2 Case gN ≤5 ∗ In Table 4 of the Appendix, there are all the values of N such that gN ≤ 5. We point out that the abelian variety Af has an elliptic quotient over Q if, and only if, Af is isogenous to the 9 power of an elliptic curve over Q. When dim Af > 1 and f does not have complex multiplication (CM), i.e. f 6= f ⊗ χ for all quadratic Dirichlet characters, a necessary condition to have Af an elliptic quotient over Q is ap (f )2 ∈ Z for all primes p (cf. [Pyl04]). ∗ When the genus gN ≤ 5 and N ∈ N , we compute the splitting of J0∗ (N ) over Q. We can discard the values N such that none of the abelian varieties Af involved in the splitting of J0∗ (N ) over Q is a power of an elliptic curve over Q. This occurs for N = 117, 147, 261 and 279. Q For instance, for N = 261, one has J0∗ (261) ∼ Af1 × Af2 with dim Afi = 2, where f1 (q) = q + a q 2 + q 3 + (a − 1) q 4 + (−2a + 2) q 5 + a q 6 + (−2a − 1)q 7 + . . . , f2 (q) = q − a q 2 + (a − 1) q 4 − 2 q 5 + (2a − 1) q 7 + · · · , and a2 − a − 1 = 0. Since a2 ∈ / Z, the curve X0∗ (261) does not have any elliptic quotient and, thus, it is not bielliptic. For N = 136, 164, 175, 196, 234, 245, 250, 294, 376 and 495, all involutions are defined over Q because in all these cases in the splitting of J0∗ (N ) over Q, there is only one abelian variety Af of dimension 1 and the abelian varieties Af with dimension > 1 do not have any elliptic quotient over Q. When all abelian varieties Af with dimension > 1 involved in the splitting of J0∗ (N ) are not isogenous to a power of an elliptic curve over Q, if X0∗ (N ) is bielliptic over Q but not over Q, then there exist two elliptic curves over Q in the splitting of J0∗ (N ) such that they are not isogenous over Q but their attached newforms are twisted by a quadratic Dirichlet character. ∗ By the list of the values of N such that 2 ≤ gN ≤ 5 (see the appendix) and the above paragraph, we obtain that the only possible values of N for which X0∗ (N ) is a bielliptic curve ∗ when gN ≤ 5 are the following, ∗ gN N 2 88, 104, 112, 116, 135, 153, 168, 180, 184, 198, 204, 276, 284, 380 3 136, 144, 152, 162, 164, 171, 189, 196, 207, 234, 236, 240, 245, 248, 252, 270, 294, 312, 315, 348, 420, 476. 4 148, 160, 172, 176, 200, 224, 225, 228, 242, 260, 264, 275, 280, 300, 306, 308, 342, 350. 5 192, 208, 212, 216, 316, 364, 376, 378, 396, 414, 440, 444, 495, 572, 630. Table 2 ∗ When X0∗ (N ) is bielliptic over Q and gN ≤ 5, after applying Proposition 3.4, for each N we can restrict our study to the following possible bielliptic quotients E. ∗ gN N:E 2 88 : 44a, 88a; 104 : 26b, 52a; 112 : 56a, 112a; 116 : 58a, 58b; 135 : 45a, 135a; 153 : 51a, 153a; 168 : 42a, 84b; 180 : 30a, 90b; 184 : 92a, 184b; 198 : 66a, 99a; 204 : 34a, 102a; 276 : 138a, 138c; 284 : 142b, 142d; 380 : 38b, 190b. 3 136 : 34a; 144 : 24a, 36a, 48a; 152 : 38b, 152a; 162 : 27a, 54a; 164 : 82a; 171 : 57a, 57c; 189 : 21a; 196 : 14a; 207 : 69a; 234 : 234c; 236 : 118a, 118b, 118c; 240 : 20a, 24a, 240c; 245 : 35a; 248; 62a, 124b; 252 : 21a, 42a; 270 : 30a, 45a, 135a; 294 : 14a; 312 : 26b, 52a, 312b; 315 : 21a; 348; 58a, 58b, 174c; 420 : 42a, 70a, 210d; 476 : 34a, 238b, 238d. 4 148 : 37a; 160 : 20a, 160a; 172 : 43a; 176 : 44a; 200; 40a, 50b; 224 : 56a, 224a; 225 : 15a, 225a; 228 : 57a; 242 : 11a, 121b; 260 : 65a; 264; 44a, 66b, 88a, 132b; 275 : 55a; 280 : 20a, 70a, 280a; 300 : 15a, 50b, 150c; 306 : 102a, 102c; 342 : 342e; 350 : 175b, 350c. 5 192 : 24a, 32a, 192a; 216 : 36a, 54b; 364 : 91a; 376; 94a; 378 : 63a, 189a; 396; 66a, 99a, 198c; 414 : 69a, 138a; 444 : 37a, 222b; 495 : 99a; 630; 21a, 210d. 10 Table 3 By using results due to Hasegawa, we can prove that some values of N in table 2 corresponds to bielliptic curves X0∗ (N ) over Q. First, we recall that if 4|N (resp. 9|N ) the map z 7→ z + 1/2 (resp. z 7→ z + 1/3) on the upper-half plane provides an automorphism S2 (resp. S3 ) of X√ 0 (N ). The automorphism S2 is an involution defined over Q and√S3 has order 3, defined over Q( −3) and σ(S3 ) = S3−1 for the non trivial conjugation σ of Q( −3). ∗ Proposition 3.6 (Proposition 7, Has97). Assume that gN ≥ 2 and N = 4N 0 with N 0 odd. Then, the three following quotients curves of X0 (N ) are isomorphic over Q: X0∗ (N ) ∼ = X0 (N )/hS2 and B(N 0 )i ∼ = X0 (2N 0 )/hB(N 0 )i . ∗ Corollary 3.7. Assume that gN ≥ 2 and N = 4N 0 with N 0 odd. If g2N ∗ ∗ 0 = 1, then X0 (N ) is a bielliptic curve over Q. Proof. Consider the degree two morphism of X0 (2N 0 )/hwd ∈ B(N 0 )i → X0∗ (2N 0 ) given by the quotient of the Atkin-Lehner involution w2 .  Corollary 3.8. The curve X0∗ (N ) is a bielliptic curve over Q for the following values of N : ∗ ∗ gN/2 gN N 1 2 116, 180, 204, 276, 284, 380. 1 3 164, 196, 236, 252, 348, 420, 476. 1 4 148, 172, 228, 260, 300. 1 5 364, 444. In order to study involutions of X0∗ (N ), Hasegawa obtained the following result. Proposition 3.9 (Corollary, p.371, Has97). Suppose 8|N (resp. 9||N ). Let 2ν be the maximal power of 2 dividing N . Then, V2 := S2 w2ν S2 (resp. V3 := S3 w9 S32 ) defines an involution on X0∗ (N ). It is clear that V2 is √ defined over Q. Also, V3 is defined over Q, because if σ is the non trivial conjugation of Q( −3), then σ(V3 ) = V3−1 = V3 . Hasegawa determined the genus of the quotients curves by these involutions and, in partic- ular, those which are 1. Corollary 3.10 (Table 3, Has97). For the following N , X0∗ (N ) is bielliptic over Q and some bielliptic involutions are also listed, ∗ gN N Bielliptic involution 3 144 V2 , V3 , V2 V3 152 V2 234 V3 240 V3 312 V2 4 264 V2 280 V2 306 V3 342 V3 Corollary 3.11. The hyperelliptic curves X0∗ (207) and X0∗ (315) of genus 3 are bielliptic over Q. 11 Proof. By Table 3 in [Has97], we know that the curves X0∗ (207)/hV3 i and X0∗ (315)/hV3 i have genus two. By [Acc94], if X is a genus 3 curve which has a map to a genus 2 curve then X is hyperelliptic and bielliptic. Since the hyperelliptic involution w is defined over Q, the involution w · V3 is bielliptic and defined over Q.  Remark 3.12. The levels N for which we know that X0∗ (N ) is bielliptic, by the work of ∗ Hasegawa, and gN > 2 will be studied again when we compute equations for these curves. In this way, we will be able to precise the bielliptic quotients and their ranks and, as a consequence, to determine the infinitude of the set Γ2 (X0∗ (N ), Q). Remark 3.13. Given a curve X over a number field of genus gX , its gonality γ, i.e. the lowest degree map to the projective line, satisfies the inequality   gX + 3 2≤γ≤ . 2 Therefore if X0∗ (N ) is bielliptic and non-hyperelliptic with gN ∗ = 3 or 4, then it has gonality three and is a trigonal curve. Hasegawa and Shimura listed all trigonal modular curves X0∗ (N ) in [HS00]. This result (jointly with [Has97]) allows us to present the complete list of the values ∗ N such that 3 ≤ gN ≤ 4 in Appendix A. Also, it is useful when we find equations for the candidate non hyperelliptic curves X0∗ (N ) such that gN ∗ > 4 (see Remark 2.7). ∗ 4 Case gN =2 Proposition 4.1. Let N be a non square-free integer. The curve X0∗ (N ) has genus two and is bielliptic if, and only if, N lies in the set {88, 112, 116, 121, 153, 180, 184, 198, 204, 276, 284, 380} . In all theses cases, the curves are bielliptic over Q. Proof. In all possible cases in Table 2, the splitting of the Jacobian of these curves over Q is the product of two elliptic curves. For N = 121, the curve X0∗ (N ) is bielliptic over Q (see [Gon16] or [Jeo18]). For N 6= 121, the Jacobian has all its endomorphisms defined over Q and, in particular, the bielliptic involutions are defined over Q. When N = 116, 180, 204, 276, 284, 380, we know that X0∗ (N ) is bielliptic (cf. Corollary 3.8). For the remaining cases to be considered, Q J0∗ (N ) ∼ Af1 × Af2 , where N f1 f2 B(11) 88 f1 ∈ New∗88 f2 ∈ New44 , w4 (f2 ) = −f2 B(13) B(13) 104 f1 ∈ New52 , w4 (f1 ) = −f1 f2 ∈ New26 , w2 (f2 ) = −f2 B(7) 112 f1 ∈ New∗112 f2 ∈ New56 , w8 (f2 ) = −f2 B(5) 135 f1 ∈ New∗135 f2 ∈ New45 , w9 (f2 ) = −f2 B(17) 153 f1 ∈ New∗153 f2 ∈ New51 , w3 (f2 ) = −f2 B(21) B(21) 168 f1 ∈ New84 , w4 (f1 ) = −f1 f2 ∈ New42 , w2 (f2 ) = −f2 B(23) 184 f1 ∈ New∗184 f2 ∈ New92 , w4 (f2 ) = −f2 B(22) 198 f1 ∈ New∗99 f2 ∈ New66 , w3 (f2 ) = −f2 12 Taking g1 and g2 as the following eigenforms N g1 g2 88 f1 (q) f2 (q) − 2f2 (q 2 ) 104 f1 (q) − 2f1 (q 2 ) f2 (q) − 4f2 (q 4 ) 112 f1 (q) f2 (q) − 2f2 (q 2 ) 135 f1 (q) f2 (q) − 3f2 (q 3 ) 153 f1 (q) f2 (q) − 3f2 (q 3 ) 168 f1 (q) − 2f1 (q 2 ) f2 (q) − 4f2 (q 4 ) 184 f1 (q) f2 (q) − 2f2 (q 3 ) 198 f1 (q) + 2f1 (q 2 ) f2 (q) − 3f2 (q 3 ) , g1 (q) d q/q and g2 (q) d q/q is a basis of Ω1X ∗ (N )/Q formed by the pullbacks of regular differentials 0 of the elliptic quotients. The functions g1 dX X= , Y = q, g2 g2 satisfy the following equations N = 88 : Y2 = 1 + X 2 + 3X 4 − X 6 , N = 104 : 81 Y 2 = 40 + 48X + 105X 2 + 244X 3 − 12X 4 + 336X 5 − 32X 6 , N = 112 : Y2 = 1 + X 4 + 2X 6 , N = 135 : 9Y 2 = −4 + 24X − 3X 2 + 32X 3 + 12X 4 + 12X 5 + 8X 6 , N = 153 : Y2 = 3 + 7X 2 + 9X 4 − 3X 6 , N = 168 : 81Y 2 = 112 + 192X + 105X 2 + 280X 3 − 48X 4 + 120X 5 − 32X 6 , N = 184 : 2Y 2 = 1 − X 2 + 7X 4 + X 6 , N = 198 : Y2 = (3 + X 2 )(−2 + 3X 2 + 3X 4 ) . Only the curves whose equations are of the form Y 2 = P (X 2 ) for a polynomial P of degree 3 are bielliptic over Q (cf. [BG19, Remark 10]). Since for N = 104, 135, 168 the newforms involved in the splitting of J0∗ (N ) are not twisted and without complex multiplication, all endomorphisms of J0∗ (N ) are defined over Q and X0∗ (N ) is not bielliptic over Q.  ∗ 5 Case gN =3 Proposition 5.1. Let N be a non square-free integer. The curve X0∗ (N ) has genus 3 and is bielliptic if, and only if, N lies in the set {128, 144, 152, 164, 189, 196, 207, 234, 236, 240, 245, 248, 252, 294, 312, 315, 348, 420, 476} . In all these cases, the curves are bielliptic over Q. Proof. In Table 2, we only have hyperelliptic curves for N = 136, 171, 207, 252, 315 and, in all cases, the jacobians have all their endomorphisms defined over Q. Their equations can be 13 found in [Has97]: N = 136 : Y2 = X(1 + X)(−2 + 3X + X 2 )(−4 + 2X + 5X 2 + 4X 3 + X 4 ) , N = 171 : Y2 = (3 + 3X + X 2 )(67 + 241X + 324X 2 + 209X 3 + 72X 4 + 13X 5 + X 6 ) , N = 207 : Y2 = 1 − 6X + 11X 2 − 12X 3 + 9X 4 − 12X 5 + 11X 6 − 6X 7 + X 8 , N = 252 : Y2 = 21 − 42X + 73X 2 − 74X 3 + 64X 4 − 38X 5 + 17X 6 − 6X 7 + X 8 , N = 315 : Y2 = (1 + X + 3X 2 + X 3 + X 4 )(1 + 5X + 3X 2 + 5X 3 + X 4 ) . It can be checked that the automorphisms group for N = 136, 171 has order 2 and, thus, the curves are not bielliptic. For N = 207, 252, 315, the automorphism group is always isomorphic to (Z/2Z)2 , generated by the hyperelliptic involution and the involutions (X, Y ) 7→ (1/X, Y /X 4 ), (X, Y ) 7→ ((X + 1)/(X − 1), 4Y /(X − 1)4 ) and (X, Y ) 7→ (1/X, Y /X 4 ) respectively . Hence, X0∗ (207), X0∗ (252) and X0∗ (315) are bielliptic. More precisely, their elliptic quotients for such bielliptic involutions are respectively N equation E X 2 +1 Y 207 V = U − 6U + 7U 2 + 6U − 11 , 2 4 3 with U = X , V = X2 E69a , X 2 +1 Y 252 V 2 = U 4 − 6U 3 + 7U 2 − 6U + 1 , with U = X−1 , V = (X−1) 2 E21a , X 2 +1 Y 315 V 2 = U 4 + 6U 3 + 7U 2 + 6U + 1 , with U = X , V = X2 E21a . We know that X0∗ (128) is bielliptic over Q (cf. [Jeo18]). For the remaining 18 cases to be considered, the splitting of the jacobian over Q is as follows: Q Q3 Q Q Q J0∗ (144) ∼ A fi , Af1 ∼ E24a , Af2 ∼ E36a , Af3 ∼ E48a , Q Q3i=1 Q Q Q J0∗ (152) ∼ A fi , Af1 ∼ E38b , Af2 ∼ E76a , Af3 ∼ E152a , Q Qi=1 3 Q Q Q J0∗ (162) ∼ Afi , Af1 ∼ E27a , Af2 ∼ E54a , Af3 ∼ E162a , Q Q2i=1 Q B(41) J0∗ (164) ∼ Afi , Af1 ∼ E82a , f2 ∈ New82 , dim Af2 = 2 , Q Q3i=1 Q Q Q J0∗ (189) ∼ Afi , Af1 ∼ E21a , Af2 ∼ E63a , Af3 ∼ E189a , Q Q2i=1 Q B(49) J0∗ (196) ∼ Afi , Af1 ∼ E14a , f2 ∈ New98 , dim Af2 = 2 , Q Qi=1 2 Q B(13) J0∗ (234) ∼ Afi , Af1 ∼ E234c , f2 ∈ New39 , dim Af2 = 2 , Q Q3i=1 Q Q Q J0∗ (236) ∼ A fi , Af1 ∼ E118a , Af2 ∼ E118b , Af3 ∼ E118c , Q Qi=1 3 Q Q Q J0∗ (240) ∼ A fi , Af1 ∼ E20a , Af2 ∼ E24a , Af3 ∼ E240c , Qi=1 2 Af2 ∈ New∗245 , Q Q J0∗ (245) ∼ Afi , Af1 ∼ E35a , dim Af2 = 2 , Q Q3i=1 Q Q Q J0∗ (248) ∼ A fi , Af1 ∼ E62a , Af2 ∼ E124b , Af3 ∼ E248a , Q Qi=1 3 Q Q Q J0∗ (270) ∼ Afi , Af1 ∼ E30a , Af2 ∼ E45a , Af3 ∼ E135a . Qi=1 2 f2 ∈ New∗147 , Q Q J0∗ (294) ∼ Afi , Af1 ∼ E14a , dim Af2 = 2 , Q Q3i=1 Q Q Q J0∗ (312) ∼ Afi , Af1 ∼ E26b , Af2 ∼ E52a , Af3 ∼ E312b . Q Q3i=1 Q Q Q J0∗ (348) ∼ Afi , Af1 ∼ E58a , Af2 ∼ E58b , Af3 ∼ E174c . Q Q3i=1 Q Q Q J0∗ (420) ∼ Afi , Af1 ∼ E42a , Af2 ∼ E70a , Af3 ∼ E210d . Q Q3i=1 Q Q Q J0∗ (476) ∼ i=1 Afi , Af1 ∼ E34a , Af2 ∼ E238b , Af3 ∼ E238d . The existence of twists only occurs for N = 144: f3 = f1 ⊗ χ−1 , f2 = f2 ⊗ χ−3 , √ where χD denotes the quadratic Dirichlet character √ attached to the quadratic field Q( D). The splitting of J0∗ (144) over Q and over Q( −3) agree. This fact implies that if Af2 is a 14 bielliptic quotient of X0∗ (144), √ then the bielliptic involution is defined over Q. The splitting √ ∗ Q( −1) over Q( −1) is J0 (144) ∼ A2f1 × Af2 . Hence, it may be that Af1 is the quotient of the √ curve X0∗ (144) by an involution defined over Q( −1) but not over Q. We take a basis ω1 , ω2 , ω3 of Ω1X ∗ (N )/Q obtained as the union of the bases corresponding to 0 each Afi and following the order exhibited in the splitting of its Jacobian. A basis of Ω1Af i formed by eigenforms is obtained by applying Proposition 2.2 to all Galois conjugate of fi . If dim Afi > 1, we replace this with a basis of the Z-module Ω1Af /Q ∩Z[[q]]. We compute a nonzero i homogenous polynomial Q ∈ Q[x, y, z] of degree 4 such that Q(ω1 , ω2 , ω3 ) = 0. We obtain N Q 144 x4 − 4x2 y 2 + 3y 4 + 2x2 z 2 − 6y 2 z 2 + 4z 4 152 8x + 16x y + 3x y − 32xy + 5y 4 − 27x2 z 2 − 108xyz 2 + 54y 2 z 2 + 81z 4 4 3 2 2 3 162 x4 + 2x3 y − 3x2 y 2 + 2xy 3 − 2y 4 + 9xy 2 z − 9xyz 2 + 9y 2 z 2 − 6xz 3 + 6yz 3 − 9z 4 164 x4 − 4x2 y 2 − 4x2 yz + 32y 3 z − 2x2 z 2 + 4y 2 z 2 − 12yz 3 + z 4 189 9x4 − 18x2 y 2 + 9y 4 − 12x2 yz − 4y 3 z − 24x2 z 2 + 24y 2 z 2 + 16z 4 196 x4 − 12x2 y 2 + 32y 4 − 4x2 yz + 32y 3 z − 2x2 z 2 + 12y 2 z 2 − 12yz 3 + z 4 234 3x4 − 13x2 y 2 + 12y 4 + 19x2 yz − 12y 3 z − x2 z 2 + y 2 z 2 + yz 3 − 2z 4 236 81x4 + 54x2 y 2 + 73y 4 − 108x2 yz + 92y 3 z − 108x2 z 2 − 84y 2 z 2 + 32yz 3 − 32z 4 240 x4 − y 4 + 2x2 z 2 + 2y 2 z 2 − 4z 4 245 x4 + 2x2 y 2 − 3y 4 − 8x2 yz + 8y 3 z − 8x2 z 2 − 8y 2 z 2 + 16yz 3 + 16z 4 248 16x4 + 32x3 y + 6x2 y 2 + 44xy 3 − 17y 4 − 54x2 z 2 + 108xyz 2 − 54y 2 z 2 − 81z 4 270 9x + 36x3 y − 42x2 y 2 + 4xy 3 − 7y 4 + 72x3 z + 72xy 2 z − 16y 3 z + 24x2 z 2 − 48xyz 2 4 +24y 2 z 2 − 64xz 3 + 64yz 3 − 128z 4 294 x4 − y 4 + 6x2 yz + 2y 3 z − 10x2 z 2 + 2y 2 z 2 − 24yz 3 + 24z 4 312 32x + 64x y + 30x y − 56xy 3 + 11y 4 − 126x2 z 2 − 288xyz 2 + 90y 2 z 2 + 243z 4 4 3 2 2 348 27x4 − 30x2 y 2 − 13y 4 − 84x2 yz + 4y 3 z + 60x2 z 2 + 36y 2 z 2 − 32yz 3 + 32z 4 420 5x4 + 12x3 y − 4x2 y 2 − 8xy 3 + 3y 4 + 2x2 z 2 − 12xyz 2 − 6y 2 z 2 + 8z 4 476 3x4 + 6x2 y 2 + 8y 4 + 4x3 z − 4xy 2 z − 4x2 z 2 − 18y 2 z 2 + 8xz 3 − 3z 4 Let ω be the regular differential attached to an elliptic quotient E of one of these curves X0∗ (N ). An equivalent condition for E to be a bielliptic quotient is that the polynomial Q must be even in the variable corresponding to ω. Only the curves corresponding to N = 162, 175, 270 are not bielliptic. The Q-isogeny classes of all bielliptic involutions for the remaining cases are N E N E N E 144 24a, 36a, 48a 234 234a 294 14a 152 152a 236 118a 312 312b 164 82a 240 20a, 24a, 240c 348 58a 189 21a 245 35a 420 210d 196 14a 248 248a 476 238b The curve X0∗ (144) is bielliptic over Q and has 3 bielliptic involutions. Applying Proposition 2.6, it can be checked that this curve only has these 3 bielliptic involutions.  ∗ 6 Case gN =4 Proposition 6.1. Let N be a non square-free integer. The curve X0∗ (N ) has genus 4 and is biel- liptic if, and only if, N ∈ {148, 160, 172, 200, 224, 225, 228, 242, 260, 264, 275, 280, 300, 306, 342}. 15 √ For N = 160, the curve is bielliptic over Q( −1) but not over Q. For the remaining cases X0∗ (N ) is bielliptic over Q. Proof. For N 6= 160, 200, 225, 242, 275 in Table 2, the absence of twists between the newforms involved in the splitting of J0∗ (N ) over Q implies that all involutions of the curve are defined over Q. By using Table 3, we can discard N = 308. Only for N = 176, the curve X0∗ (N ) is hyperelliptic. In [Has97], it is determined the following equation: Y 2 = X(X 3 − 4 X + 4)(X 3 − 2 X 2 )(X 3 + 2 X − 2) . For instance, by using Magma, it can be checked that the automorphism group has order 2 and, thus, the curve is not bielliptic. For the remaining 16 cases, the splitting of the Jacobian over Q is as follows: Q Q B(37) J0∗ (148) ∼ A2f1 × Af2 , Af1 ∼ E37a , f2 ∈ New74 , dim Af2 = 2 , J0∗ (160) ∼ A2f1 3i=2 Afi , Q Q Q Q Q Af1 ∼ E20a , Af2 ∼ E80b , Af3 ∼ E160a , Q Q B(43) J0∗ (172) ∼ A2f1 × Af2 , Af1 ∼ E43a , f2 ∈ New86 , dim Af2 = 2 , Q Q4 Q Q Q Q J0∗ (200) ∼ i=1 Afi , Af1 ∼ E20a , Af2 ∼ E40a , Af3 ∼ E50a , Af4 ∼ E100a , Q Q4 Q Q Q Q J0∗ (224) ∼ i=1 Afi , Af1 ∼ E56a , Af2 ∼ E112a , Af3 ∼ E112c , Af4 ∼ E224a , Q Q4 Q Q Q Q J0∗ (225) ∼ i=1 Afi , Af1 ∼ E15a , Af2 ∼ E75b , Af3 ∼ E75c , Af4 ∼ E225a , A2f1 3i=2 Afi , Af1 ∼ E57a , Q Q Q Q J0∗ (228) Q ∼ Af2 ∼ E38b , Af3 ∼ E114c , Q3 Af3 ∈ New∗242 , Q Q Q J0∗ (242) ∼ i=1 Afi , Af1 ∼ E11a , Af2 ∼ E121b , dim Af3 = 2 , A2f1 3i=2 Afi , Af1 ∼ E65a , Q Q Q Q J0∗ (260) Q ∼ Af2 ∼ E26a , Af3 ∼ E130c , Q Q4 Q Q Q Q J0∗ (264) ∼ i=1 Afi , Af1 ∼ E44a , Af2 ∼ E66b , Af3 ∼ E88a , Af4 ∼ E132b , Q3 Af3 ∈ New∗275 , Q Q Q J0∗ (275) ∼ i=1 Afi , Af1 ∼ E55a , Af2 ∼ E275a , dim Af3 = 2 , Q Q4 Q Q Q Q J0∗ (280) ∼ i=1 Afi , Af1 ∼ E20a , Af2 ∼ E70a , Af3 ∼ E140b , Af4 ∼ E280a , A2f1 3i=2 Afi , Af1 ∼ E15a , Q Q Q Q J0∗ (300) Q ∼ Af2 ∼ E50b , Af3 ∼ E150c , Q Q4 Q Q Q Q J0∗ (306) ∼ i=1 Afi , Af1 ∼ E51a , Af2 ∼ E102a , Af3 ∼ E102c , Af4 ∼ E153c , Q Q4 Q Q Q Q J0∗ (342) ∼ i=1 Afi , Af1 ∼ E57a , Af2 ∼ E57b , Af3 ∼ E57c , Af4 ∼ E342e , Q Q3 Q Q B(7) J0∗ (350) ∼ i=1 Afi , Af1 ∼ E175b , Af2 ∼ E350c , Af3 ∈ New35 , dim Af3 = 2 , For the values N ∈ {160, 200, 225, 242, 275}, we have the following twists: 160 200 225 242 275 f2 = f1 ⊗ χ−1 f4 = f1 ⊗ χ5 f3 = f1 ⊗ χ5 f2 = f2 ⊗ χ−11 f2 = f1 ⊗ χ5 . The splitting over Q of J0∗ (N ) is as follows: √ √ √ Q( −1) Q( 5) Q( 5) J0∗ (160) ∼ A3f1 × Af3 , J0∗ (200) ∼ A2f1 × Af2 × Af3 , J0∗ (225) ∼ A2f1 × Af2 × Af4 , √ √ ∗ Q( −11) ∗ Q( 5) J0 (242) ∼ Af1 × Af2 × Af3 , J0 (275) ∼ A2f1 × Af3 . √ For N = 242, √ A f 2 is an elliptic curve with CM by Q( −11) and End(J0∗ (242))⊗Q is isomorphic to Q × Q( −11) × Q. So, an involution of X0∗ (242) has √ to be defined over Q. For the remaining four values of N , all involutions are defined over Q( D). 16 We take a basis ω1 , ω2 , ω3 , ω4 of Ω1X ∗ (N )/Q following the order exhibited in the splitting of 0 its Jacobian. We know that dim L2 = 1. It turns out that all these curves are trigonal (cf. Remark 3.13). Hence, the curves are determined by L2 and L3 , whose dimension is 5. So, the curve is determined by a nonzero polynomial Q2 ∈ L2 and any polynomial Q3 ∈ L3 that is not a multiple of Q2 . Now, we focus our attention to bielliptic curves over Q. First, we only consider the val- ues of N 6= 148, 160, 172, 228, 260, 300. For each of these values, the endomorphism algebra EndQ (J0∗ (N )) ⊗ Q is isomorphic to the product of totally real number fields and we can apply the same procedure as in [BG19]. We compute a nonzero polynomial Q2 ∈ L2 : N Q2 200 −10t + 15tx + 9x − 18y 2 − 10tz − 6xz + 20z 2 2 2 224 −4t2 + 2x2 + xy − y 2 + 3xz + yz − 2z 2 225 −9t2 + 3x2 + 5y 2 + z 2 242 −36t2 + 9x2 + 24ty − 5y 2 + 12tz + 4yz − 8z 2 264 −112t2 − 51tx + 63x2 + 80ty − 30xy − 175y 2 + 225z 2 275 −8t2 + 3x2 + 5y 2 + 4tz − 20z 2 280 −75t2 + 12x2 + 20xy + 25y 2 + xz + 30yz − 13z 2 306 27t2 + x2 − 3xy − 9y 2 − 17xz − 15yz + 16z 2 342 −216t2 + 117x2 + 72xy − 56y 2 + 90xz − 104yz + 97z 2 350 −171t + 150tx + 85x2 − 204ty − 20xy + 160y 2 + 468tz − 420xz + 240yz − 432z 2 2 We can discard the value N = 350 because for every elliptic differential ωi , one has Q2 (· · · , xi , · · · ) 6= Q2 (· · · , −xi , · · · ) . For the remaining values, we compute a polynomial Q3 ∈ L3 that is not a multiple of Q2 : N Q3 200 −5t − 27tx + 54ty − 54xy + 99txz + 27x2 z − 162y 2 z − 30tz 2 − 18xz 2 + 116z 3 3 2 2 2 224 4x3 − 3xy 2 + 2y 3 + 4x2 z − 2xyz − 3y 2 z + xz 2 − 3z 3 225 −9x2 y + 5y 3 + 4z 3 242 144t3 − 36tx2 − 24x2 y − 44ty 2 + 12y 3 − 3x2 z − 32tyz − 9y 2 z + 40tz 2 + 24yz 2 264 20t2 x − 45tx2 − 72t2 y + 104txy + 18x2 y + 180ty 2 + 20xy 2 − 255tz 2 + 180xz 2 − 150yz 2 275 −12t3 + 12tx2 − 45x2 y + 20ty 2 + 25y 3 + 51x2 z + 85y 2 z + 48tz 2 280 351t2 x − 156xy 2 + 130y 3 − 276t2 z − 12x2 z + 58xyz − 64y 2 z + 77xz 2 − 108yz 2 306 −32t x − 96t2 y + 18xy 2 + 33y 3 + 50t2 z + 22x2 z + 74xyz + 33y 2 z − 52xz 2 − 83yz 2 + 33z 3 2 342 −11349t2 x − 5626t2 y − 1067xy 2 + 970y 3 + 18421t2 z − 468x2 z − 1064xyz + 2067y 2 z −4628xz 2 + 2744yz 2 Hence, for all these values N 6= 225, the curve X0∗ (N ) is bielliptic over Q with a unique bielliptic involution, because there is only an elliptic differential ωi such that Q2 (· · · , ωi , · · · ) − Q2 (· · · , −ωi , · · · ) = 0, Q3 (· · · , ωi , · · · ) − Q3 (· · · , −ωi , · · · ) (6.1) ∈ L2 . xi The Q-isogeny classes of their bielliptic quotients are E40a, E224a, E11a, E88a, E55a, E280a, E153c and E342e respectively. For N = 225 there are two bielliptic involutions corresponding to the elliptic differentials ω1 and ω4 and their bielliptic quotients are E15a and E225a. 17 Q For the cases N ∈ {148, 160, 172, 228, 260, 300, 308}, J0∗ (N ) ∼ A2f1 × A, where Af1 is an elliptic curve. The polynomials Q2 are N Q2 148 −8t2 − 30tx + 25x2 + 6ty + 5xy + 2y 2 − 72tz − 10xz + 2yz − 112z 2 160 −48t2 + 8tx + 3x2 − 8ty + 6xy − y 2 + 36xz + 12yz − 8z 2 172 −32t − 20tx + 25x2 + 4ty + 20xy + 3y 2 + 272tz − 40xz + 8yz − 128z 2 2 228 512t2 − 80tx + 75x2 + 16ty − 120xy − 3y 2 + 80tz − 100xz + 20yz − 400z 2 260 68t2 − 40tx + 25x2 + 50xy + 25y 2 − 136tz + 40xz − 32z 2 300 170t2 − 60tx + 45x2 − 18xy − 9y 2 + 80tz − 48xz − 160z 2 The splitting of A over Q does not contain any bielliptic quotient of X0∗ (N ) over Q. By Proposition 2.6, in the  class of Af1 there is a bielliptic quotient if, and only if, there  Q-isogeny a1 b 1 exists a matrix A = ∈ GL2 (Q) such that the polynomials a2 b 2 R2 (x, y, z, t) := Q2 (a1 x + a2 y, b1 x + b2 y, z, t), R3 (x, y, z, t) := Q3 (a1 x + a2 y, b1 x + b2 y, y, z, t) satisfy the conditions R2 (x, y, z, t) − R2 (−x, y, z, t) = 0, R3 (x, y, z, t) − R3 (−x, y, z, t) (6.2) = λ R2 (x, y, z, t) , x for some λ ∈ Q. For instance, we can assume a1 ∈ {0, 1}. It can be checked that the first condition in (6.2) does not happen for N = 160. For the remaining 5 cases, we obtain the following polynomial Q3 ∈ L3 : N Q3 148 344t x − 318tx + 35x − 72t y + 28x2 y − 18ty 2 + xy 2 + 2304t2 z + 216txz 2 2 3 2 −26x2 z − 72tyz − 80xyz − 6y 2 z + 768tz 2 + 16xz 2 − 48yz 2 172 −272t x + 125x + 112t2 y + 84txy − 4ty 2 − 37xy 2 − 8y 3 − 640t2 z + 712txz 2 3 −250x2 z − 632tyz − 32xyz − 38y 2 z − 480tz 2 + 112xz 2 + 208yz 2 228 −3248t2 x + 7725tx2 + 1350x3 − 4880t2 y + 4056txy + 555ty 2 + 1242xy 2 + 2352t2 z −4700txz − 3000x2 z + 76tyz − 600xyz − 528y 2 z − 4080tz 2 − 800xz 2 + 4480yz 2 260 −9916t2 x + 2245tx2 + 1700x3 + 3810txy + 1905ty 2 − 5440t2 z + 7672txz −2720x2 z − 1360xyz − 680y 2 z + 2240tz 2 + 544xz 2 300 4850t2 x − 1725tx2 + 1530x3 − 330txy − 165ty 2 − 1140t2 z + 1160txz −1020x2 z + 816xyz + 408y 2 z + 240tz 2 − 4624xz 2 . Taking the following matrix A 148,  172, 228   260, 300 1 5 0 1 A , 1 −3 1 −1 the reader can check that the conditions (6.2) are satisfied. The bielliptic Q-isogeny class of the bielliptic quotients are E37a, E43a, E57a, E65a and E15a respectively. Finally, we claim that X0∗ (160) is bielliptic over Q. For the basis chosen to compute Q2 , we obtain as Q3 (x, y, z, t) the following polynomial: −80t3 +13tx2 +2x3 +10txy+ty 2 −2xy 2 +60txz+6x2 z+20tyz−4xyz−2y 2 z−24tz 2 −4xz 2 +4yz 2 . 18   a1 b 1 c 1 √ Again by Proposition 2.6, it suffices to find a matrix A =  a2 b2 c2  ∈ GL3 (Q( −1)) a3 b 3 c 3 such that the polynomials R2 (x, y, z, t) = Q2 (a1 x + a2 y + a3 z, b1 x + b2 y + b3 z, c1 x + c2 y + c3 z, t) , R3 (x, y, z, t) = Q3 (a1 x + a2 y + a3 z, b1 x + b2 y + b3 z, c1 x + c2 y + c3 z, t) , satisfy R2 (x, y, z, t) = R2 (−x, y, z, t) , R3 (−x, y, z, t) ∈ L03 . (6.3) The second condition in (6.3) amounts to saying √ R3 (x, y, z, t) − R3 (−x, y, z, t) = λ x R2 (x, y, z, t) , λ ∈ Q( −1) . (6.4) √ As usual, set i = −1. Taking the matrix   i i 1 A =  1 −3 0  , 0 −4 i 1 we get R2 (x, y, z, t) = −48t2 − (16 − 48i)x2 + 32ty − 24y 2 + 32i tz − 48i yz + (8 − 48i)z 2 , R3 (x, y, z, t) = −80t3 − (48 − 80i)tx2 − (32 − 32i)x2 y − 8ty 2 − 16y 3 − (32 + 32i)x2 z −16ityz − 48iy 2 z − (40 + 80i)tz 2 + (16 − 32i)yz 2 + (32 − 16i)z 3 , which satisfy the conditions (6.4). More precisely, put (ω10 , ω20 , ω30 ) = (ω1 , ω2 , ω3 ) · A−1 . The linear map u : (ω10 , ω20 , ω30 , ω4 ) 7→ (ω10 , −ω20 , −ω30 , −ω4 ) acts on the basis ωi as follows 1 1 3 3 3 1 u : (ω1 , ω2 , ω3 , ω4 ) 7→ (− ω1 + ω2 + i ω3 , ω1 − ω2 + i ω3 , − i ω1 − i ω2 , −ω4 ) , 4 4 4 4 4 4 which defines a bielliptic involution on X0∗ (160).  Remark 6.2. When X0∗ (N ) is bielliptic and J0∗ (N ) has endomorphisms not defined over Q, we do not determine all bielliptic involutions in the group AutQ (X0∗ (N )). For instance, the curve X0∗ (275) has an only bielliptic involution defined over Q: (ω1 , ω2 , ω3 , ω4 ) 7→ (ω1 , −ω2 , −ω3 , −ω4 ) . √ Nevertheless, it has another involution defined over Q( 5): √ √ 1 5 5 1 (ω1 , ω2 , ω3 , ω4 ) 7→ (− ω1 − ω2 , 3 ω1 − ω2 , −ω3 , −ω4 ) . 2 2 10 2 ∗ 7 Case gN =5 Proposition 7.1. Let N be a non square-free integer. The curve X0∗ (N ) has genus 5 and it is bielliptic if, and only if, N = 364, 444, 495. In all these cases, the curve is bielliptic over Q. 19 Proof. For N 6= 192, 208, 216, 378 in Table 2, the absence of twists between the newforms involved in the splitting of J0∗ (N ) implies that all involutions of the curve are defined over Q. By Table 3, we can exclude N = 212, 316, 440, 572 because for these values none of the pairs (N, E) is bielliptic over Q. The splitting of the Jacobian over Q is as follows: J0∗ (192) ∼ A2f1 4i=2 Afi , Af1 ∼ E24a , Q Q Q Q Q Q Af2 ∼ E32a , Af3 ∼ E96b , Af4 ∼ E192a , J0∗ (208) ∼ A2f1 4i=2 Afi , Af1 ∼ E26b , Q Q Q Q Q Q Af2 ∼ E52a , Af3 ∼ E104a , Af4 ∼ E208b , Q Q5 Q Q Q Q J0∗ (216) ∼ i=1 Afi , Af1 ∼ E36a , Af2 ∼ E54b , Af3 ∼ E72a , Af4 ∼ E108a , Q Af5 ∼ E216a , Q Q4 Q Q Q Q J0∗ (364) ∼ A2f1 Afi , Af1 ∼ E91a , Af2 ∼ E26b , Af3 ∼ E182a , Af4 ∼ E182d , Qi=2 3 B(47) f3 ∈ New∗376 , Q Q J0∗ (376) ∼ i=1 Afi , Af1 ∼ E94a , f2 ∈ New188 , dim Af2 = 2 , dim Af3 = 2 , Q Q5 Q Q Q Q J0∗ (378) ∼ i=1 Afi , Af1 ∼ 21a , Af2 ∼ E63a , Af3 ∼ E126b , Af4 ∼ E189a , Q Af5 ∼ E378d , Q Q4 Q Q Q Q J0∗ (396) ∼ A2f1 Afi , Af1 ∼ E99a , Af2 ∼ E66a , Af3 ∼ E66b , Af4 ∼ E198c , Qi=2 4 Af4 ∈ New∗207 , Q Q Q Q J0∗ (414) ∼ i=1 Afi , Af1 ∼ E69a , Af2 ∼ E138a , Af3 ∼ E138b , dim Af4 = 2 , Q Af5 ∼ E440a , Q Q3 Q Q B(37) J0∗ (444) ∼ A2f1 Afi , Af1 ∼ E37a , Af2 ∼ 222b , f3 New74 , dim Af 3 = 2 , Qi=2 3 B(55) New∗165 Q Q J0∗ (495) ∼ i=1 Afi , Af1 ∼ E99a , f2 ∈ , dim Af2 = 2 , f3 ∈ New165 , dim Af3 = 2 , Q4 Af4 ∈ New∗315 , Q Q Q Q J0∗ (630) ∼ i=1 Afi , Af1 ∼ E21a , Af2 ∼ E30a , Af3 ∼ E210d , dim Af4 = 2 , For N ∈ {192, 208, 216, 378}, we have the following twists: 192 208 216 378 f4 = f3 ⊗ χ−2 f4 = f3 ⊗ χ−1 f1 = f1 ⊗ χ−3 f2 = f1 ⊗ χ−3 . The splitting over Q of the jacobians of these curves is as follows: √ √ ∗ Q( −2) ∗ Q( −1) J0 (192) ∼ A2f1 × Af2 × A2f3 , J0 (208) ∼ A2f1 × Af2 × A2f3 , √ 5 √ 5 ∗ Q( −3) Y ∗ Q( −3) Y J0 (216) ∼ A fi , J0 (378) ∼ A2f1 Afi . i=1 i=3 First, we find which curves X0∗ (N ) are bielliptic curves over Q. For the values of N different from 192, 208, 216, 364, 396 and 444, the endomorphism algebra of the Jacobian of X0∗ (N ) is isomorphic to the product of totally real number fields and we can apply the same procedure as in [BG19]. As before, we take a basis ω1 , ω2 , ω3 , ω4 , ω5 of Ω1X ∗ (N )/Q following the order exhibited 0 in the splitting of its Jacobian. We know that dim L2 = 3. For all these values, the curve X0∗ (N ) is neither trigonal (cf. [HS00]) nor a smooth plane quintic. Hence, if ωj is the regular differential of an elliptic curve E, the pair (X0∗ (N ), E) is bielliptic over Q if, and only if, dim L2,j = 3. N dim L2,j N dim L2,j 216 dim L2,1 = dim L2,2 = 1 414 dim L2,1 = dim L2,2 = 0 376 dim L2,1 = 0 495 dim L2,1 = 3 378 dim L2,2 = dim L2,4 = 0 630 dim L2,1 = dim L2,3 = 0 20 Hence, for N ∈ {216, 376, 378, 414, 630}, the curve X0∗ (N ) is not bielliptic over Q. The curve X0∗ (495) is bielliptic and the Q-isogeny class of its bielliptic quotient is E99a. By Table 3, we know that the curve X0∗ (208) is not bielliptic over Q. For the values of ∗ N ∈ {192, 364,  Af1 is a bielliptic quotient of X0 (N ) over Q if, and only if, there is a  396, 444}, a1 b 1 matrix A = ∈ GL2 (Q) such that for all Q2 ∈ L2 the polynomials a2 b 2 R2 (x1 , x2 , x3 , x4 , x5 ) := Q2 (a1 x1 + a2 x2 , b1 x1 + b2 x2 , x3 , x4 , x5 ) , satisfy the condition R2 (x1 , x2 , x3 , x4 , x5 ) = R2 (−x1 , x2 , x3 , x4 , x5 ) . (7.1) One can check that, for N = 192, 396, there is not any matrix A satisfying thecondition(7.1). 1 5 For N = 364, 444, the condition (7.1) is satisfied for the same matrix A = . By 1 −3 Table 3, there are no more bielliptic quotients over Q because one has N dim L2,j N dim L2,j N dim L2,j 192 dim L2,3 = dim L2,5 = 1 396 dim L2,3 = 2, dim L2,5 = 1 444 dim L2,3 = 0 To determine bielliptic curves over Q, we only have to consider the cases N = 192, 208, √ 216 and 378. The case N = 216 can be discarded. Indeed, the splitting of J0∗ (216) over Q( −3) and over Q are the same and we get dim L2,1 = 1 . For  the remaining cases, fi = fj ⊗ χD with √  a1 b 1 i < j. One can check that there is no matrix A = ∈ GL2 ( D) such that, for all a2 b 2 Q2 ∈ L2 , one has that Q2 (x1 , · · · , xi−1 , a1 xi + a2 xj , xi+1 , · · · , xj−1 , b1 xi + b2 xj , xj+1 , · · · , x5 ) is an even function in the variable xi .  ∗ 8 Case gN ≥6 For g ∗ ≥ 6, all bielliptic involutions are defined over Q. For the possible values of N to be considered, the curve X0∗ (N ) is not trigonal. Proposition 8.1. The curve X0∗ (558) of genus 7 is the only bielliptic curve X0∗ (N ) of genus ≥ 6. We split the proof according to the genus, following the values N in Table 1, except for the value appearing in Lemma 3.5. 21 ∗ 8.1 gN =6 Proof. In this case, dim L2 = 6. The splitting of the Jacobian over Q for the seven cases to be considered is as follows: B(61) J0∗ (244) ∼ A2f1 3i=2 Afi , Af1 ∼ E61a , Af2 ∼ E122a , f3 ∈ New122 , Q Q Q Q dim Af3 = 3, , Q 4 Q Q Q B(17) J0∗ (272) ∼ A2f1 i=2 Afi , Af1 ∼ E34a , Af2 ∼ E136b , Af3 ∼ E272a , f4 ∈ New68 , Q dim Af4 = 2 , ∗ Q 2 Q3 Q Q B(83) J0 (332) ∼ Af1 i=2 Afi , Af1 ∼ E83a , Af2 ∼ E166a , f3 ∈ New166 , dim Af3 = 3 , Q 5 Q Q Q Q J0∗ (336) ∼ A2f1 i=2 Afi , Af1 ∼ E42a , Q Af2 ∼ E24a , Af3 ∼ E36a , Af4 ∼ E84b , Q Af5 ∼ E112a , Q Q4 Q Q Q B(282) J0∗ (564) ∼ A2f1 i=2 Afi , Af1 ∼ E141d , Af2 ∼ E94a , Af3 ∼ E282a , f4 ∈ New282 , dim Af4 = 2 , Q Q4 Q Q Q B(310) J0∗ (620) ∼ A2f1 i=2 Afi , Af1 ∼ E155c , Af2 ∼ E62a , Af3 ∼ E310a , f4 ∈ New310 , dim Af4 = 2 , Q Q5 Q Q Q Q J0∗ (780) ∼ A2f1 i=2 A fi , Af1 ∼ E65a , Af2 ∼ E26b , Af3 ∼ E130c , Af4 ∼ E390a , Q Af5 ∼ E390e , After computing, we obtain N = 332 N = 336 N = 564 N = 620 N = 780 dim L2,3 = 3 dim L2,6 = 3 dim L2,3 = 2 dim L2,3 = 2 dim L2,4 = 1 Hence, the Q-isogeny class of an elliptic curveAfi withi > 1 can not be a bielliptic quotient of a1 b 1 X0∗ (N ). In all cases, there is no matrix A = ∈ GL2 (Q) satisfying a2 b 2 Q2 (a1 x1 + a2 x2 , b1 x1 + b2 x2 , x3 , · · · , x6 ) = Q2 (−a1 x1 + a2 x2 , −b1 x1 + b2 x2 , x3 , · · · , x6 ) for all Q2 ∈ L2 . So, the Q-isogeny class of the elliptic curve Af1 can not be a bielliptic quotient of X0∗ (N ). Hence, all these curves are not bielliptic.  22 ∗ 8.2 gN =7 Proof. Now dim L2 = 10. The splitting of the Jacobian over Q for the 12 cases to be considered is as follows: A2f1 6i=2 Afi , Af1 ∼ E20a , Q Q Q Q Q J0∗ (320) ∼ Q Af2 ∼ E32a , Af3 ∼ E80b , Af4 ∼ E160a , Q Q Af5 ∼ E160b Af6 ∼ E320b Q Q6 Q Q Q Q J0∗ (324) ∼ A2f1 i=2 A fi , Af1 ∼ E27a , Af2 ∼ E54a , Af3 ∼ E54b , Af4 ∼ E162a , Q Q Af5 ∼ E162b Af6 ∼ E162d Q Q5 Q Q Q Q J0∗ (360) ∼ A2f1 A2f2 i=3 A fi , Af1 ∼ E20a , Af2 ∼ E30a , Af3 ∼ E36a , Af4 ∼ E90b , Q Af5 ∼ E120b Q Q7 Q Q Q Q J0∗ (450) ∼ i=1 A fi , Af1 ∼ E15a , Af2 ∼ E30a , Af3 ∼ E75b , Af4 ∼ E75c , Q Q Q Af5 ∼ E39a , Af6 ∼ E225a , Af7 ∼ E450f , Q Q6 Q Q Q Q J0∗ (456) ∼ A2f1 i=2 A fi , Af1 ∼ E57a , Af2 ∼ E38b , Af3 ∼ E76a , Af4 ∼ E114c , Q Q Af5 ∼ E152a , Af6 ∼ E228a , Q Q5 Q Q Q Q J0∗ (492) ∼ A2f1 i=2 Afi , Af1 ∼ E123b , Af2 ∼ E82a , Af3 ∼ E246d , Af4 ∼ E246g , B(41) f5 ∈ New82 , dim Af5 = 2 , Q Q6 Q Q Q Q J0∗ (504) ∼ A2f1 i=2 A fi , Af1 ∼ E21a , Af2 ∼ E36a , Af3 ∼ E42a , Af4 ∼ E84b , Q Q Af5 ∼ E168d , Af6 ∼ E504a Q Q5 Q Q Q B(22) J0∗ (550) ∼ i=1 A fi , Af1 ∼ E55a , Af2 ∼ E275a , Af3 ∼ 550a , f4 ∈ New110 , dim Af4 = 2 , f5 ∈ New∗275 , dim Af5 = 2 , Q4 f3 ∈ New∗93 , Q Q Q J0∗ (558) ∼ i=1 A fi , Af1 ∼ 186c , Af2 ∼ E558a , dim Af3 = 2 , B(31) f4 ∈ New93 , dim Af4 = 3 , Q Q6 Q Q Q Q J0∗ (636) ∼ A2f1 i=2 A fi , Af1 ∼ E53a , Af2 ∼ E106a , Af3 ∼ E106b , Af4 ∼ E106c , Q Q Af5 ∼ E318a , Af6 ∼ E318c , Q3 f4 ∈ New∗165 , Q Q Q Q J0∗ (660) ∼ i=1 Afi A2f4 , Af1 ∼ E66c , Af2 ∼ E110b , Af3 ∼ E330d , dim Af4 = 2 , Q Q6 Q Q Q Q J0∗ (924) ∼ A2f1 i=2 A fi , Af1 ∼ E77a , Af2 ∼ E42a , Af3 ∼ E66b , Af4 ∼ E154a , Q Q Af5 ∼ E154b Af6 ∼ E462a We get N = 320 N = 450 N = 456 dim L2,3 = 3 dim L2,1 = 3 , dim L2,3 = 4 dim L2,4 = 4 , dim L2,6 = 6 N = 492 N = 504 dim L2,3 = dim L2,4 = 4 dim L2,3 = 6 , dim L2,4 = 4 N = 550 N = 558 N = 660 N = 924 . dim L2,1 = dim L2,2 = dim L2,3 = 4 dim L2,2 = 10 dim L2,2 = 4 dim L2,7 = 4   a1 b 1 Moreover, there is no matrix A = ∈ GL2 (Q) satisfying a2 b 2 Q2 (a1 x1 + a2 x2 , b1 x1 + b2 x2 , x3 , x4 , x5 , x6 , x7 ) = Q2 (−a1 x1 + a2 x2 , −b1 x1 + b2 x2 , x3 , x4 , x5 , x6 , x7 ) 23 for all Q2 ∈ L2 when N = 320, 324, 360, 456, 492, 504, 636, 924 or Q2 (x1 , x2 , a1 x3 + a2 x4 , b1 x3 + b2 x4 , x5 , x6 , x7 ) = Q2 (x1 , x2 , −a1 x3 + a2 x4 , −b1 x3 + b2 x4 , x5 , x6 , x7 ) for all Q2 ∈ L2 when N = 360. Thus, the Q-isogeny class of the elliptic curve Af1 (resp. Af3 ) can not be a bielliptic quotient of X0∗ (N ) for N = 320, 324, 360, 456, 492, 504, 636, 924 (resp. N = 360). Hence, the curve X0∗ (558) is the only bielliptic curve of genus 7.  ∗ 8.3 Case gN =8 Proof. In this case dim L2 = 15. The splitting of the Jacobian over Q for the five cases to be considered is as follows: B(17) A2f1 5i=2 Afi , Af1 ∼ 102a , Q Q Q Q J0∗ (408) ∼ Q Af2 ∼ E34a , Af3 ∼ E204a , f4 ∈ New68 , dim Af4 = 2 f5 ∈ New∗408 dim Af5 = 2 ∗ Q 2 Q 3 2 Q Q Q B(13) J0 (468) ∼ Af1 i=2 Afi Af4 , Af1 ∼ E26b , Af2 ∼ E234b , Af3 ∼ E234c , f4 ∈ New39 , dim Af4 = 2 , Q 7 Q Q Q Q J0∗ (480) ∼ A2f1 i=2 Afi , Af1 ∼ E20a , Af2 ∼ E24a , Q Af3 ∼ E80b , Af4 ∼ E160a , Q Q Q Af5 ∼ E240b , Af6 ∼ E240c , Af7 ∼ E480a , Q Q5 Q Q Q Q J0∗ (540) ∼ A2f1 i=2 Afi A2f6 , Af1 ∼ E45a , Af2 ∼ E30a , Af3 ∼ E54b , f4 ∼ E90b , Q Q Af5 ∼ E270b , Af6 ∼ E135a , Q Q6 Q Q Q Q J0∗ (990) ∼ i=1 Afi , Af1 ∼ E30a , Af2 ∼ E66a , Af3 ∼ E99a , Af4 ∼ E990a , B(55) f5 ∈ New∗165 , dim Af5 = 2 f6 ∈ New165 , dim Af6 = 2 , N = 468 N = 480 N = 540 N = 990 dim L2,3 = dim L2,4 = 8 , dim L2,3 = dim L2,4 = dim L2,5 = 8 dim L2,4 = 10 dim L2,2 = 8 , dim L2,3 = 10   a1 b 1 There is no matrix A = ∈ GL2 (Q) satisfying a2 b 2 Q2 (a1 x1 +a2 x2 , b1 x1 +b2 x2 , x3 , x4 , x5 , x6 , x7 , x8 ) = Q2 (−a1 x1 +a2 x2 , −b1 x1 +b2 x2 , x3 , x4 , x5 , x6 , x7 , x8 ) for all Q2 ∈ L2 when N = 408, 468, 480, 540.  ∗ 8.4 Case gN =9 Proof. In this case dim L2 = 21. The splitting of the Jacobian over Q for the values to be considered are as follows: A2f1 7i=2 Afi , Af1 ∼ E70a , Af2 ∼ E20a , Q Q Q Q Q J0∗ (560) ∼ Q Af3 ∼ E56a , Af4 ∼ E112a , Q Q B(35) Af5 ∼ E140b , Af6 ∼ E280a , f7 ∈ New280 , dim Af7 = 2 , Q Q7 Q Q Q Q J0∗ (1140) ∼ A2f1 · A2f2 i=3 Afi , Af1 ∼ E285b , Af2 ∼ E57a , Af3 ∼ E38b , Af4 ∼ E114c , Q Q Q Af5 ∼ E190b , Af6 ∼ E570a , f7 ∼ E570g , For N = 560, dim L2,4 = dim L2,7 = 13, andfor N =1140, dim L2,7 = 14. Moreover for a1 b 1 N = 560 and 1140 and there is no matrix A = in GL2 (Q) satisfying a2 b 2 Q2 (a1 x1 + a2 x2 , b1 x1 + b2 x2 , x3 , · · · , x9 ) = Q2 (−a1 x1 + a2 x2 , −b1 x1 + b2 x2 , x3 , · · · , x9 ) for all Q2 ∈ L2 .  24 ∗ 8.5 Case gN = 10 Proof. Now, dim L2 = 28. We have to consider twoQcases X0∗ (840) and X0∗ (1050). For N = 840, the Jacobian splits over Q as A2f1 9i=2 Afi , where Q Q Q Q Q Q Af1 ∼ E210d , Af2 ∼ E20a , Af3 ∼ E42a , Af4 ∼ E70a , Af5 ∼ E84b , Af6 ∼ E140b , Q Q Q Af7 ∼ E280a , Af8 ∼ E420a and Af9 ∼ E840a.   a1 b 1 We get dim L2,3 = dim L2,7 = 19 and there is no matrix A = in GL2 (Q) a2 b 2 satisfying Q2 (a1 x1 + a2 x2 , b1 x1 + b2 x2 , x3 , · · · , x10 ) = Q2 (−a1 x1 + a2 x2 , −b1 x1 + b2 x2 , x3 , · · · , x10 ) for all Q2 ∈ L2 . For N = 1050, the Jacobian splits over Q as 6i=1 Afi × Af7 × Af8 , where Q Q Q Q Q Q Q Af1 ∼ E15a , Af2 ∼ E175b , Af3 ∼ E210d , Af4 ∼ E35a , Af5 ∼ E525a , Af6 ∼ E1050a , B(7) f7 ∈ New35 , dim Af7 = 2 , f8 ∈ New∗525 , dim Af8 = 2. We get dim L2,2 = 19.  ∗ 8.6 Case gN = 11 ∗ Proof. Now, dim L2 = 36. We have to consider only the Q9case X0 (672). 2 2 For N = 672, the Jacobian splits over Q as Af1 Af2 i=3 Afi , where Q Q Q Q Q Q Af1 ∼ E84b , Af2 ∼ E42a , Af3 ∼ E24a , Af4 ∼ E56a , Af5 ∼ E112a , Af6 ∼ E112c , Q Q Q Af7 ∼ E224a , Af8 ∼ E336a and Af9 ∼ E672a. We get dim L2,7 = dim L2,8 = 26 .  ∗ 8.7 Case gN = 13 Proof. Now, dim L2 = 55. We have to consider only theQcase X0∗ (1260). For N = 1260, the Jacobian splits over Q as A2f1 A2f2 ( 7i=3 Afi ) · A2f8 , where Q Q Q Q Q Q Af1 ∼ E21a , Af2 ∼ E70a , Af3 ∼ E30a , Af4 ∼ E42a , Af5 ∼ E90b , Af6 ∼ E210d , Q B(315) Af7 ∼ E630g , Af8 ∈ New315 and dim Af8 = 2. We are interested for 21a, 70a, 90b and 210d. We get dim L2,7 = dim L2,8 = 43, thus the pair (1260, 90b),(1260, 210d) are not bielliptic. For theremaining  (1260, 21a) and (1260, 70a) they a1 b 1 are not bielliptic because there is no matrix A = in GL2 (Q) satisfying a2 b 2 Q2 (a1 x1 + a2 x2 , b1 x1 + b2 x2 , x3 , · · · , x13 ) = Q2 (−a1 x1 + a2 x2 , −b1 x1 + b2 x2 , x3 , · · · , x13 ) for all Q2 ∈ L2 , or respectively Q2 (x1 , x2 , a1 x3 +a2 x4 , b1 x3 +b2 x4 , x5 , · · · , x13 ) = Q2 (x1 , x2 , −a1 x3 +a2 x4 , −b1 x3 +b2 x4 , x5 , · · · , x13 )  Hence, part (i) of Theorem 1.2 is proved . 25 8.8 A remark for levels 4M with M odd After the paper was completed, Andreas Schweizer mentioned us the following argument, which helps in some levels to obtain that the modular curve X0∗ (N ) is not bielliptic. Lemma 8.2 (Schweizer). Let M be odd and let the genus of X0∗ (4M ) be at least 6. If X0∗ (4M ) is bielliptic, then the genus of X0∗ (2M ) must be 1 or 2. ∗ Proof. Assume that X0∗ (4M ) is bielliptic and g4M ≥ 6. By Proposition 3.6, X0∗ (4M ) is isomor- phic over Q to the curve X := X0 (2M )/hB(M )i. It is well-known that the Fricke involution W2M always has fixed points. By the unramified covering criterion, [JKS17, Lemma 3.10], applied to the Galois covering X → X/W2M = X0∗ (2M ), we get that X0∗ (2M ) must be hyperel- liptic or of genus ≤ 1. But X0∗ (2M ) cannot have genus 0, because then X0∗ (4M ) would also be hyperelliptic, which is not possible for genus bigger or equal than 6 (cf. [Has97, Theorem B]). ∗ Since 2M is congruent to 2 mod 4, again applying [Has97, Theorem B] we get g2M < 3. Corollary 8.3 (Schweizer). Let M be odd and let the genus of X0∗ (4M ) be at least 8. Then, X0∗ (4M ) is bielliptic if and only if X0∗ (2M ) has genus 1. ∗ Proof. The condition g2M > 1 contradicts the Castelnuovo Inequality. These results could be applied, without the use of Petri’s theorem and Proposition 2.6, for the followings levels N = 324, 468, 492, 540, 564, 620, 636, 660, 924, 1140, 1260. 9 Quadratic Points In this section, we determine the values N such that the set Γ2 (X0∗ (N ), Q) is infinite. The values N square-free are in part (ii) of Theorem 1.1. Assume that N is not square-free. All hyperelliptic curves X0∗ (N ) are over Q and, thus, have infinite number of points over Q. These ∗ values can be found in [Has97]. More precisely, for gN = 2 the values N are in Table 4 of the ∗ Appendix and for gN ≥ 3 one has ∗ gN 3 4 5 N 136, 171, 207, 252, 315 176 279 When X0∗ (N ) is bielliptic over Q and not hyperelliptic, we select the values N such that X0∗ (N ) have at least a bielliptic quotient with positive rank. We get the following cases. ∗ gN (N, E) 3 (128, E128a), (152, E152a), (164, E82a), (234, E234c), (236, E118a), (240, E240c), (248, E248a), (312, E312b), (348, E58a), (420, E210d), (476, E238b) 4 (148, E37a), (172, E43a), (224, E224a), (228, E57a), (225, E225a), (260, E65a), (264, E88a), (280, E280a), (342, E342e). 5 (364, E91a), (444, E37a), (495, E99a). 7 (558, E558a). This concludes the proof of part (ii) of Theorem 1.2. 26 10 Appendix Here, we present the list of curves X0∗ (N ) with gN ∗ ≤ 5. We developed a programme in Magma ∗ to obtain the genus of X0 (N ) for any N . It can be found in http://mat.uab.cat/∼francesc/Computegenusquotientsx0N.pdf In particular, we can list all the modular curves X0∗ (N ) of genus ≤ 5. For genus 0 and 1, in [GL98] are listed the cases with N square-free. For genus 2, the values N can be found in Hasegawa [Has95]. For genus 3 or 4, the curve X0∗ (N ) is trigonal or hyperelliptic (see Remark ∗ 3.13), then the table follows from [HS00] and [Has97]. When gN = 5, we know that the gonality ∗ is ≤ 4. We prove that if X0 (N ) has gonality ≤ 4, then N ≤ 121337. This is made following an argument of Ogg in [Ogg74], that in [HS00, Lemma 1,Proposition 2] is used to find an upper bound for trigonal curves X0∗ (N ). Applying the program we can complete the list. In fact, with these tools we could obtain the complete list of curves X0∗ (N ) with a fixed genus. We obtain the following list, where for completeness of the paper, we mark by bold the square-free integers and with italic the integers that are a non trivial power of a prime. ∗ gN = 0 2, 3, 4, 5, 6,7,8,9, 10,11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 41, 42, 44, 45, 46, 47, 49, 50, 51, 54, 55, 56, 59, 60, 62, 66, 69, 70, 71, 78, 87, 92, 94, 95, 105, 110, 119 ∗ gN = 1 37, 40, 43, 48, 52, 53, 57, 58, 61, 63, 64, 65, 68, 72, 74, 75, 76, 77, 79, 80,81, 82, 83, 84, 86, 89, 90,91, 96, 98, 99, 100, 101, 102, 108, 111, 114, 118, 120, 123, 124, 126, 130, 131, 132, 138, 140, 141, 142, 143, 145, 150, 155, 156, 159, 174, 182, 188, 190, 195, 210, 220, 222, 231, 238 ∗ gN = 2 67, 73, 85, 88, 93, 103, 104, 106, 107, 112, 115, 116, 117, 121, 122, 125, 129, 133, 134, 135, 146, 147, 153, 154, 158, 161, 165, 166, 167, 168, 170, 177, 180, 184, 186, 191, 198, 204, 205, 206, 209, 213, 215, 221, 230, 255, 266, 276, 284, 285, 286, 287, 299, 330, 357, 380, 390. ∗ gN = 3 97, 109, 113, 127, 128, 136, 139, 144, 149, 151, 152, 162, 164, 169, 171, 175, 178, 179, 183, 185, 187, 189, 194, 196, 203, 207, 217, 234, 236, 239, 240, 245, 246, 248, 249, 252, 258, 270, 282, 290, 294, 295, 303, 310, 312, 315, 318, 329, 348, 420, 429, 430, 455, 462, 476, 510. ∗ gN = 4 137, 148, 160, 172, 173, 176, 199, 200, 201, 202, 214, 219, 224, 225, 228, 242, 247, 251, 254, 259, 260, 261, 262, 264, 267, 273, 275, 280, 300, 305, 306, 308, 311, 319, 321, 322, 334, 335, 341, 342, 345, 350, 354, 355, 366, 370, 374, 385, 395, 399, 426, 434, 483, 546, 570. ∗ gN = 5 157, 181, 192, 208, 212, 216, 218, 226, 227, 235, 237, 250, 253, 263, 278, 279, 302, 316, 323, 339, 364, 371, 376, 377, 378, 382, 391, 393, 396, 402, 406, 407, 410, 413, 414, 418, 435, 438, 440, 442, 444, 465, 494, 495, 551, 555, 572, 574, 595, 630, 645, 663, 714, 770, 798, 910. Table 4 27 Acknowledgements We thank to Andreas Schweizer for his comments and suggestions, and specially those concern- ing morphisms between modular curves with different level and the remark quoted in Subsection 8.8 for levels 4N . References [Acc94] Robert D. M. Accola. Topics in the theory of Riemann surfaces, volume 1595 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1994. [Bar18] Francesc Bars. On quadratic points of classical modular curves. In Number theory related to modular curves—Momose memorial volume, volume 701 of Contemp. Math., pages 17–34. Amer. Math. Soc., Providence, RI, 2018. [BG19] Francesc Bars and Josep González. Bielliptic modular curves X0∗ (N ) with square- free levels. Math. Comp., 88(320):2939–2957, 2019. [BGGP05] Matthew H. Baker, Enrique González-Jiménez, Josep González, and Bjorn Poo- nen. Finiteness results for modular curves of genus at least 2. Amer. J. Math., 127(6):1325–1387, 2005. [GJU12] Josep González and Jorge Jiménez-Urroz. The Sato-Tate distribution and the values of Fourier coefficients of modular newforms. Exp. Math., 21(1):84–102, 2012. [GL98] Josep González and Joan-C. Lario. Rational and elliptic parametrizations of Q- curves. J. Number Theory, 72(1):13–31, 1998. [Gon16] Josep González. Automorphism group of split Cartan modular curves. Bull. Lond. Math. Soc., 48(4):628–636, 2016. [Gon17] Josep González. Constraints on the automorphism group of a curve. J. Théor. Nombres Bordeaux, 29(2):535–548, 2017. [Has95] Yuji Hasegawa. Table of quotient curves of modular curves X0 (N ) with genus 2. Proc. Japan Acad. Ser. A Math. Sci., 71(10):235–239 (1996), 1995. [Has97] Yuji Hasegawa. Hyperelliptic modular curves X0∗ (N ). Acta Arith., 81(4):369–385, 1997. [HS91] Joe Harris and Joe Silverman. Bielliptic curves and symmetric products. Proc. Amer. Math. Soc., 112(2):347–356, 1991. [HS00] Yuji Hasegawa and Mahoro Shimura. Trigonal modular curves X0∗ (N ). Proc. Japan Acad. Ser. A Math. Sci., 76(6):83–86, 2000. [Jeo18] Daeyeol Jeon. Bielliptic modular curves X0+ (N ). J. Number Theory, 185:319–338, 2018. [JKS17] Daeyeol Jeon, Chang Heon Kim, and Andreas Schweizer. Bielliptic intermediate modular curves . Preprint, 2017. 28 [Ogg74] A. P. Ogg. Hyperelliptic modular curves. Bull. Soc. Math. France, 102:449–462, 1974. [Pyl04] Elisabeth E. Pyle. Abelian varieties over Q with large endomorphism algebras and their simple components over Q. In Modular curves and abelian varieties, volume 224 of Progr. Math., pages 189–239. Birkhäuser, Basel, 2004. Francesc Bars Cortina Departament Matemàtiques, Edif. C, Universitat Autònoma de Barcelona 08193 Bellaterra, Catalonia francesc@mat.uab.cat Josep González Rovira Departament de Matemàtiques, Universitat Politècnica de Catalunya EPSEVG, Avinguda Vı́ctor Balaguer 1, 08800 Vilanova i la Geltrú, Catalonia josep.gonzalez@upc.edu 29