This is the submitted version of the article: Bars Cortina, Francesc; González Rovira, Josep; Kamel, Mohamed. «Bielliptic quotient modular curves with N square-free». Journal of Number Theory, Vol. 216 (November 2020), p. 380-402. DOI 10.1016/j.jnt.2020.03.010 This version is available at https://ddd.uab.cat/record/240665 under the terms of the license Bielliptic quotient modular curves with N square-free Francesc Bars∗, Josep González †and Mohamed Kamel Abstract Let N ≥ 1 be an square free integer and let WN be a non-trivial subgroup of the group of the Atkin-Lehner involutions of X0 (N ) such that the modular curve X0 (N )/WN has genus at least two. We determine all pairs (N, WN ) such that X0 (N )/WN is a bielliptic curve and the pairs (N, WN ) such that X0 (N )/WN has an infinite number of quadratic points over Q. 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 not finite 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 )/WN defined as the quotient of the modular curve X0 (N ) by a proper non-trivial subgroup WN of the group B(N ) of all Atkin- Lehner involutions. For the trivial subgroup WN = {1}, the study concerning quadratic points for X0 (N ) is already done in [Bar99] and for WN = B(N ) the study can be found in [BG19a] and [BG19b]. In this paper we restrict to square-free levels N . Under this assumption, all automorphisms of the modular curve X0 (N )/WN are defined over Q (see §2) and, therefore, if X0 (N )/WN is bielliptic over a number field, then it is already bielliptic over the Q. The pairs (N, WN ) where X0 (N )/WN is an hyperelliptic curve are known (cf. [FH99] and [Has95]).The pairs (N, B(N )) where X0∗ (N ) = X0 (N )/B(N ) is a bielliptic curve can be found in [BG19a] and the pairs (N, hwN i), where wN is the Fricke involution, for which the modular curve X0 (N )/hwN i is bielliptic are given in [Jeo18] (cf. Table 4 in §7). We only consider non-trivial subgroups WN of B(N ) different from the subgroup hwN i. There are two main differences between the curves X0∗ (N ) and X0 (N )/WN . On the one hand, for N 6= 37 we have that Aut(X0 (N )) = B(N ) and a non-trivial involution of X0∗ (N ) does not come from an involution of X0 (N ). Nevertheless, the non-trivial group B(N )/WN provides a subgroup B 0 of Aut(X0 (N )/WN ) and, also, involutions of X0 (N )/WN , when B 0 6= {1}. Since the normalizer of the group Γ0 (N ) in SL2 (R) is the group B(N ) when N is square-free, an ∗ First author is supported by MTM2016-75980-P and MDM-2014-0445. † The third author is partially supported by DGI grant MTM2015-66180-R. 1 automorphism u ∈ Aut(X0 (N )/WN )\B 0 is called exceptional, in the sense that it does not come from an automorphism acting on the complex upper half-plane. On the other hand, the endomorphism algebra of the jacobian of X0∗ (N ) is isomorphic to the product of totaly real number fields and the jacobian of X0 (N )/WN can have a non-commutative endomorphism algebra. In particular, if E/Q is an elliptic curve quotient of X0∗ (N ), there is an only vector subspace of Ω1X ∗ (N )/Q of dimension 1 that is the pullback of Ω1E/Q under any 0 projection X0∗ (N ) → E. This fact can fail for elliptic quotients of the curve X0 (N )/WN . Next, we summarize the procedure used to determine all pairs (N, WN ) such that X0 (N )/WN is bielliptic. By a result of Silverman-Harris, for a finite map C → C 0 of smooth projective curves over K, one has that if C is bielliptic , then C 0 is bielliptic or hyperelliptic or its genus is ≤ 1. We consider the natural map X0 (N )/WN → X0∗ (N ) := X0 (N )/B(N ) . It is known the finite list M of levels N such that the curve X0∗ (N ) is bielliptic [BG19a], hyperelliptic [HH96] or has genus ≤ 1 [GL98]. The levels N ∈ M are the product of at most four primes (the case of one prime is cover by [Jeo18]). For such a value N ∈ M, we only will need to consider subgroups WN of B(N ) for which the decomposition of the jacobian of X0 (N )/WN has an elliptic curve E. In §3, by reducing modulo a suitable prime p with p - N , we can reduce the finite list of pairs (N, WN ) to be considered. Next, §4, §5 and §6 are devoted to study the cases that N is the product of two, three or four primes respectively. The hyperelliptic case can be solved easily by determining the automorphism group. To deal with the non-hyperelliptic case we use a theorem of Petri and the same proposition that the one applied in [BG19b] in order to conclude if the curve X0 (N )/W corresponding to pair (N, WN ) is bielliptic or not. The novelty here with respect to [BG19a] is that in the Jacobian decomposition of X0 (N )/WN via Q-isogeny, would appear repeated elliptic factors in the endomorphism algebra of the Jacobian, see §2 for further explanations (this is a phenomena that does not appears for X0∗ (N ) with N square-free). The main results are the two following theorems. Theorem 1.1. Let N > 1 be a square-free integer. Assume that the genus of the modular curve X0 (N )/WN is at least 2 for a non-trivial subgroup WN of B(N ) different from hwN i. The curve X0 (N )/WN is a bielliptic curve if, and only if, there exists v ∈ B(N ) \ WN such that the genus of X0 (N )/hWN , vi is one, except for the following quotient modular curves of genus 4: X0 (154)/hw2 , w77 i, X0 (285)/hw3 , w95 i and X0 (286)/hw2 , w143 i. The complete list of these bielliptic quotient modular curves is exhibited in Table 2 of §7. Remark 1.2. The genus 4 bielliptic modular curves X0 (154)/hw2 , w77 i, X0 (285)/hw3 , w95 i and X0 (286)/hw2 , w143 i are the only curves X0 (N )/WN , under the assumption of Theorem 1.1, such that their bielliptic involutions must be exceptional. Remark 1.3. In [Ken76], Kenku proves that if, for any subgroup WN of the Atkin-Lehner involutions of X0 (N ), the modular quocient X0 (N )/WN is elliptic, then N is smaller than e60 . As a by-product of the results, we obtain the complete list of all square-free values such that the curve X0 (N )/WN is elliptic. Since X0 (N )/WN has a Q-point corresponding to a cusp (always assume that the genus of X0 (N )/WN is ≥ 2), the curve X0 (N )/WN will have an infinite number of quadratic points over Q if, and only if, X0 (N )/WN is hyperelliptic (see Table 1 in §7) or is bielliptic with a bielliptic quotient with positive Q-rank. We obtain then, Theorem 1.4. Let N > 1 be a square-free integer. Assume that the genus of the modular curve X0 (N )/WN is at least 2 for a non-trivial subgroup WN of B(N ) different from hwN i. Then, the set Γ2 (X0 (N )/W, Q) is infinite if, and only if, the pair (N, WN ) is in Table 3 of §7. 2 2 Notation and preliminary results Let N > 1 be an square-free 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 ). So, the order of B(N ) is 2ω(N ) , where ω(N ) is the number of primes dividing N . (ii) For N 0 |N , B(N 0 ) denotes the subgroup of B(N ) formed by the Atkin-Lehner involutions wd such that d|N 0 . In general, WN denotes a non-trivial subgroup of B(N ). ∗ (iii) The integers gN , gWN and gN are the genus of X0 (N ), X0 (N )/WN and X0∗ (N ) = X0 (N )/B(N ) respectively. (iv) We denote by NewN the set of normalized newforms in S2 (Γ0 (N )). The sets NewWN N and WN S2 (N ) are the subsets of NewN and S2 (Γ0 (N )) formed by the cusp forms invariant under the action of the group WN . (v) J0 (N ) and J0 (N )WN are the jacobians Jac(X0 (N )) and Jac(X0 (N )/WN ) respectively. (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. (x) For an integer M > 1 and a letter x, EM x denotes the Q-isogeny class of an elliptic curve of conductor M and labeled with the letter x in Cremona notation. We recall that the Q-decomposition for J0 (N ) is of the form n Q Y Y J0 (N ) ∼ Af f , M |N f ∈NewM /GQ where nf is the number of positive divisors of N/M and GQ denotes the absolute Galois group Gal(Q/Q). Each newform f ∈ NewM provides a nf dimensional vector subspace of S2 (N ) generated by {f (q d ) : 1 ≤ d|N/M }. To determine the Q-decomposition for J0 (N )WN m Q Y Y J0 (N )WN ∼ Af f , M |N f ∈NewM /GQ we need to control which Af appears in this decomposition and the precise exponent 0 < mf ≤ nf . The next lemma allows us to determine a basis of S2 (N )WN and, in particular, the splitting of J0 (N )WN (see [BG19b, Lemma 2.1]). Lemma 2.1. Let M and N be positive integers such that M |N . Let M1 be a positive divisor of M such that and 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 (q) + ε d f (q d ) ∈ S2 (Γ0 (N )) is an eigenvector of the Atkin-Lehner involution wM1 d with eigenvalue ε(f ) · ε. 3 Again, consider the Q-decomposition for J0 (N )WN m Q Y Y J0 (N )WN ∼ Af f . M |N f ∈NewM /GQ Now, for f ∈ NewM if mf > 0 then f is necessarely fix by the Atkin-Lehner involutions wd ∈ WN , with d|M , and f provides mf -eigenforms gi ∈ S2 (N )WN lying in the vector space generated by {f (q d ) : 1 ≤ d|N/Mi }. The integer mf is determined by using Lemma 2.1. For example, consider the case such that N is the product of two primes p1 and p2 . w Without lost of generality, we can assume WN = hwp1 i. If fi ∈ Newp1p1 , then mi = 2 because fi provides in S2 (N )WN the two dimensional vector space generated by the two cusp forms fi (q) ± p2 fi (q p2 ). If fi ∈ Newp2 then mi = 1 and we have only one modular form associated to w S2 (N )WN corresponding to fi (q) + p1 fi (q p1 ). Finally, if fi ∈ NewNp1 , then mi = 1. In any case, each f ∈ NewM with M |p1 p2 provides at least a Q-factor Af in the splitting of J0 (p1 p2 )WN if f ∈ Neww M with d = gcd(M, p1 ). d We can provide also a similar ad-hoc argument when N is a product of 3 or 4 primes (with a fix subgroup WN ) in order to control the factors and the power of such factor in the Q-decomposition of J0 (N )WN . For N square-free, we recall the following result [Rib75, Proposition 3.1]: Theorem 2.2. If N is square-free, then all endormorphisms of J0 (N ) are defined over Q. Corollary 2.3. Assume N square-free. Then all endomorphisms of J0 (N )WN are defined over Q and, in particular, all automorphism of X0 (N )/WN are defined over Q. If the Q- decomposition of J0 (N )WN satisfies that mf ≤ 1 for all f ∈ NewM \GQ , then Aut(X0 (N )/WN ) is an elementary 2-abelian group. Proof. The first part is an immediate consequence of Theorem 2.2. If mf ≤ 1 for all f ∈ NewM , then the statement follows form the fact that End(J0 (N )WN ) ⊗ Q is isomorphic to the product of totaly real numbers field. Now, if X0 (N )/WN is bielliptic, there is an involution u ∈ AutQ (X0 (N )/WN ), called biel- liptic involution, which is unique if gWN ≥ 6 [HS91]) such that (X0 (N )/WN )/u is a genus 1 curve defined over Q. Sinced X0 (N )/WN (Q) is not empty, the genus 1 curve has a rational point and, therefore, it is an elliptic curve E over Q, called a bielliptic quotient of X0 (N )/WN . In particular, such an elliptic curve is a Q-isogeny factor for J0 (N )WN . If X0 (N )/WN is hyperelliptic, we know an equation (see [HH96], [Has95]) and Magma computes the automorphism group over Q. In this case, if X0 (N )/WN has a non hyperelliptic involution, we can compute the genus of the quotient curve by using [Ogg74, Proposition 1] and to determine if X0 (N )/WN is bielliptic or not. Consider X be a non-hyperelliptic curve of genus g ≥ 3 defined over a subfield K of the complex field C. 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. By using a theorem of Petri in [BG19a, Lemma 13], it is characterized the existence of a bielliptic involution of X0∗ (N ) with N square-free and not hyperelliptic. Later, [BG19b, Proposition 2.6] generalizes this result to any non hyperelliptic curve of genus > 2. K Proposition 2.4. 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 (Q) the identity matrix. Take the basis {ωi } such tat ω1 , · · · , ωm 4 and ωm+1 , · · · , ωg are base of the pullbacks of Ω1E m /K and Ω1A/Q 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 Remark 2.5. The K-vector space L0i is the set of homogenous polynomials in K[x1 , · · · , xg ] of degree i such that Q(ω10 , · · · , ωm 0 , ωm+1 , · · · ωg ) = 0, where (ω10 , · · · , ωm 0 ) = A−1 (ω1 , · · · , ωm ). Remark 2.6. We recall that if g = 3, then dim L4 = 1 and the condition (2.1) can be restricted to i = 4. When g > 3, dim L2 = (g − 3)(g − 1)/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 [BG19a], for j ≤ g we introduce the K-vector space L2,j = {Q ∈ L2 : Q(x1 , · · · , xj−1 , −xj , xj+1 , · · · , xn ) ∈ L2 } . By using that the polynomials in L2 are irreducible, in [BG19a] it is proved that L2,j = {Q ∈ L2 : Q(x1 , · · · , xj−1 , xj , xj+1 , · · · , xn ) = Q(x1 , · · · , xj−1 , −xj , xj+1 , · · · , xn )} . A similar result is obtained when g = 3 and we replace L2,j with L4,j . Remark 2.7. We have J0 (N )WN ∼ 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 )WN such that J0 (N )WN ∼ rj=1 Agj , where r = si=1 ni and g1 , · · · , gr are all of Q Q P these eigenforms. The basis of the Galois conjugated of the newforms fi together the exponents ni allow us to compute |X0 /WN (Fpn )| 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 /WN by use of a theorem of Petri in the non-hyperelliptic case. Now, we assume that X0 (N )/WN has a bielliptic involution u defined over Q. Let us denote by E the elliptic quotient X0 (N )/hWN , ui and by π the non constant morphism X0 (N ) → X0 (N )/WN → E of degree 2 · |WN | 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 )/(WN ∩ Γ0 (M ))  E and a normalized newform W ∩Γ (M ) ∗ fE ∈ NewM 0 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 )WN is an eigenform in the vector space generated by {fE (q i ) : 1 ≤ i|N/M } which is invariant by WN . Note that for a prime p - M , |E(Fp2 )| ≤ (p + 1)2 , but due to the congruence of Eichler-Shimura |E(Fp2 )| = (p + 1)2 − a2p (fE ). Lemma 2.8. Assume X0 (N )/WN is bielliptic over Q, and p - N . Then the following equalities hold: ψ(N ) 2|E(Fp2 )| − 1 |E(Fp2 )| (a) ≤ 12 · , (b) gWN ≤ 2 , |WN | p−1 p−1 in particular we have, ψ(N ) 2(p + 1)2 − 1 (p + 1)2 (a) ≤ 12 · , (b) gWN ≤ 2 . |WN | p−1 p−1 5 Proof. Assume p - N . We generalize the argument used by Ogg in [Ogg74]. Indeed, X0 (N )(Fp2 ) contains 2ω(N ) cusps and at least (p − 1) ψ(N 12 ) many supersingular points (cf. [BGGP05, Lemma 3.20 and 3.21]). Since there is a nonconstant morphism defined over Q from X0 (N ) to an elliptic quotient E of X0 (N )/W which has degree 2 · |WN |, |X0 (N )(Fp2 )| ≤ 2 · |WN ||E(Fp2 )|. Part (b) is obtained applying [BGGP05, Lemma 3.25]. Similarly as in [BG19a, Lemma 7], for optimal quotients with conductor M = N for elliptic quotient (related with strong Weil parametrization) it is easy to derive the following lemma. Lemma 2.9. Let E 0 be the optimal elliptic curve in the Q-isogeny class of the bielliptic quotient E of X0 (N )/WN with conductor M = N . Then the degree D of the modular parametrization πN : X0 (N ) → E 0 divides 2 · |WN |. Note that the degree D can be found in [Cre17, Table 5]. 3 Selecting candidate bielliptic curves X0(N )/WN The starting point of our selection is based on the following result. Lemma 3.1. Let N be an integer and let WN0 and WN be subgroups of B(N ) such that WN0 ⊂ WN . If X0 (N )/WN0 is bielliptic, then X0 (N )/WN is hyperelliptic or bielliptic or has genus at most one. In particular, X0∗ (N ) is hyperelliptic or bielliptic or has genus at most one. Proof. We have the natural morphisms X0 (N )/WN0 → X0 (N )/WN → X0∗ (N ). The statement follows from [HS91, Proposition 1].  From the results of Hasegawa for hyperelliptic curves and by Bars-González [BG19a] for bielliptic curves for X0∗ (N ) we have the following result Theorem 3.2. Let N be an square-free integer. Then, ∗ (i) (González-Lario) gN = 0 if, and only if, N ∈ {2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 38, 39, 41, 42, 46, 47, 51, 55, 59, 62, 66, 69, 70, 71, 78, 87, 94, 95, 105, 110, 119} ∗ (ii) (González-Lario) gN = 1 if, and only if, N ∈ { 37, 43, 53, 57, 58, 61, 65, 74, 77, 79, 82, 83, 86, 89, 91, 101, 102, 111, 114, 118, 123, 130, 131, 138, 141, 142, 143, 145, 155, 159, 174, 182, 190, 195, 210, 222, 231, 238} (iii) (Hasegawa) X0∗ (N ) is hyperelliptic if, and only if, N ∈ { 67, 73, 85, 93, 103, 106, 107, 115, 122, 129, 133, 134, 146, 154, 158, 161, 165, 166, 167, 170, 177, 186, 191, 205, 206, 209, 213, 215, 221, 230, 255, 266, 285, 286, 287, 299, 330, 357, 390}. In all these cases ∗ gN = 2. (iv) (Bars-González) X0∗ (N ) is bielliptic with gN ∗ ≥ 2 if and only if N ∈ { 106, 122, 129, 158, 166, 178, 183, 215, 246, 249, 258, 290, 303, 318, 370, 390, 430, 455, 510}. Let M be the set of the square-free integers N that appear in Theorem 3.2 which are not primes. Recall that we are assuming WN 6= hwN i. To discard curves X0 (N )/WN with N ∈ M that are not bielliptic, we apply the strategy used in [BG19a], that is summarized in Proposition 3.3. 6 Indeed, for a prime p - N , (X0 (N )/WN )(Fpk ) denote the set of the Fpk -points of the reduction modulo p of X0 (N )/WN . Moreover, if E is a bielliptic quotient over Q of X0 (N )/WN , then n(N, E, pE kE ) := |X0∗ (N )(Fpk )| − 2|E(Fpk )| ≤ 0 for all k > 0. From previous arguments and Lemmas 2.8 and 2.9, it follows the next result: Proposition 3.3. Suppose that the curve X0 (N )/WN with gWN ≥ 2 satisfies one of the three following conditions: ψ(N ) 2(p+1)2 −1 (i) There exist a prime p - N such that |WN | > 12 · p−1 . (ii) Every elliptic curve E defined over Q of conductor NE which appear in the Q-decomposition of J0 (N )WN 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 2 · |WN |. Then, the curve X0 (N )/WN is not bielliptic over Q. All objects appeared in Proposition 3.3 can be computed by Magma (and Cremona tables). We refer to the web page http://mat.uab.cat/∼francesc/JacobiandecompositionquotientsXoN.pdf for the programmes in Magma computing (which we introduced small modification of the ones used by X0∗ (N ) in [BG19a], following the previous section) the different objects involved in Proposition 3.3, and the list of the Q-decomposition of the Jacobian of X0 (N )/WN for N square-free with N ∈ M with the computation of different n(N, E, pE kE ). After applying Proposition 3.3, we obtain a list of pairs (N, WN ) to study if the modular quotient curve X0 (N )/WN is bielliptic. These candidate are presented in the next sections according to the number of primes dividing N . 4 N ∈ M with N = p1p2 a product of two primes Assume N = p1 p2 ∈ M and set WN = hwk i with k equal to p1 or p2 . We collect the candidates ∗ ∗ ∗ ∗ in three blocks depending on wether if the genus gN satisfies gN ≥ 2, gN = 1 or gN = 0. ∗ • For gN ≥ 2, if X0 (N )/WN is bielliptic, then the pairs (N, hwk i) that should be considered are in the following list. gWN (N, WN ) 6 (122, hw61 i),(129, hw3 i),(133, hw7 i),(166, hw83 i) 7 (129, hw43 i) 10 (158, hw2 )i ∗ Table II-1, candidates N with two primes and gN ≥2 ∗ • For gN = 1, if gWN > 1, then the Fricke involution wN provides a bielliptic involution of X0 (N )/WN with bielliptic quotient X0∗ (N ). We get the following result. 7 ∗ Proposition 4.1. Assume gN = 1. The curve X0 (N )/WN is bielliptic over Q if, and only if, the genus gWN > 1 and, in this case, the elliptic curve X0∗ (N ) is a biellitic quotient. This occurs for the following pairs (N, WN ) gWN (N, WN ) 2 (57, hw3 )i,(58, hw29 i),(142, hw71 i) 3 (57, hw19 i),(58, hw2 i),(65, hw5 i),(65, hw13 i),(77, hw7 )i,(82, hw41 i),(91, hw13 i), (118, hw59 i),(123, hw41 i), (141, hw47 i) 4 (74, hw2 i),(74, hw37 i),(77, hw11 i),(82, hw2 i),(86, hw43 i),(91, hw7 i),(145, hw29 i) 5 (86, hw2 i),(111, hw3 i),(155, hw31 i) 6 (111, hw37 i),(143, hw13 i),(145, hw5 )i,(159, hw53 i) 7 (118, hw2 i),(123, hw3 i),(143, hw11 i) 8 (141, hw3 i),(155, hw5 i) 9 (142, hw2 i),(159, hw3 i). ∗ • If gN = 0, then the only quotients curves X0 (N )/WN that may be bielliptic correspond to the following pairs (N, WN ): gWN (N, WN ) 2 (33, hw3 i),(38, hw2 i) 3 (46, hw2 i),(51, hw3 i),(55, hw5 i) ∗ Table II-2, candidates N with two primes and gN =0 Now we proceed case by case for the above possible pairs (N, hwk i) in Tables II-1 and II-2 to determine if X0 (N )/hwk i is bielliptic or not. 4.1 Hyperelliptic case We recall that among the candidates to be studied, only are hyperelliptic curves the five quotient curves corresponding to the pairs in Table II-2. In the paper of Hasegawa [Has95], there are equations of the form y 2 = f (z) for such hyperelliptic curves. All automorphisms are defined over Q and by using Magma P:=PolynomialRing(Rationals()); H1:=HyperellipticCurve(f(z)); H1; AutomorphismGroup(H1); we obtain that the automorphism group over Q for these five curves have order 2. Therefore, all of them are not bielliptic. 4.2 Non hyperelliptic case It remains to study the quotient curves corresponding to the six pairs (N, WN ) in Table II-1. 4.2.1 Genus 6 For gWN = 6, we have the quotient curves corresponding to the four pairs (122, hw61 i), (129, hw3 i), (133, hw7 i) and (166, hw83 i) with possible bielliptic quotient E61a, E129a, E19a and E83a re- spectively. See all the results in http://mat.uab.cat/∼francesc/JacobiandecompositionquotientsXoN.pdf. Now we apply the theorem of Petri explained in §2 for each of this quotient curves and possible bielliptic quotients. 8 The splitting of the jacobians of these curves is Q J0 (122)hw61 i ∼ E61a2 × E122a × Af3 , dim Af3 = 3, f3 ∈ New(122)hw61 i , Q hw i J0 (129)hw3 i ∼ E43a × Af2 × E129a × Af4 , dim Af2 = dim Af4 = 2, f2 ∈ New43 , f4 ∈ New1293 , hw i J0 (133)hw7 i ∼ E19a × Af2 × Af3 , dim Af2 = 2, dim Af3 = 3, f2 ∈ New∗133 , f3 ∈ New1337 , Q Q hw i J0 (166)hw83 i ∼ E83a2 × E166a × Af3 , dim Af3 = 3, f3 ∈ New16683 . We know that dim(L2 ) = 6. We obtain by use of techniques in §2 concerning a theorem of Petri. (129, hw3 i) (133, hw7 i) dim L2,3 = 3 dim L2,1 = 1 Therefore X0 (129)/hw3 i and X0 (133)/hw7 i are not bielliptic. For the situations where the possible elliptic quotient E appears in the decomposition of the Jacobian with an exponent n > 1, we need to study if there is matrix in GLn (Q) satisfying Proposition  2.4. For the  levels N = 122 or 166, if X0 (N )/WN is bielliptic it should exist a a1 b 1 matrix A = ∈ GL2 (Q) satisfying the condition 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 . After computing, we can check that such a matrix A does no exist. Lemma 4.2. The genus 6 curves X0 (122)/hw61 i, X0 (129)/hw3 i, X0 (133)/hw7 i, X0 (166)/hw83 i are not bielliptic. 4.2.2 Genus ≥ 7 Only remains to study two modular quotient curves. The genus 7 curve X0 (129)/hw43 i, where J0 (129)hw43 i ∼ E43a2 × E129a × E129b × Af4 , dim Af4 = 3 Q with E43a as possible bielliptic quotient. By computing similarly as in the previous subsection, we get that there is no a matrix A ∈ GL2 (Q) satisfying Q2 (a1 x1 + a2 x2 , b1 x1 + b2 x2 , x3 , . . . , x6 , x7 ) = Q2 (−a1 x1 + a2 x2 , −b1 x1 + b2 x2 , x3 , . . . , x6 , x7 ) for all Q2 ∈ L2 . Hence, X0 (129)/hw43 i is not bielliptic. Consider the genus 10 curve X0 (158)/hw2 i, J0 (158)hw2 i ∼ E79a × E158b × E158d × Af4 × Af5 , dim Af4 = 5, dim Af5 = 2 Q with E158b as possible bielliptic quotient. Now dim L2 = 28 and we obtain dim L2,2 = 20, obtaining that X0 (158)/hw2 i is not bielliptic and concluding the proof of Theorem 1.1 when the level is a product of two primes. 9 4.2.3 Quadratic points for X0 (p1 p2 )/WN with gWN ≥ 2 The Birch and Swinnerton-Dyer conjecture has been checked for all elliptic curves E defined over hW i Q contained in Cremona tables (cond(E) < 500000). It is well-known that for f ∈ NewM M , ∗ its L-function has an odd vanishing order at s = 1. Therefore, for N square-free, when gN =1 ∗ one has that the Q-rank of the elliptic curve X0 (N ) is odd and, thus, at least one. Theorem 1.4 for levels N that are product of two primes follows from previous computations. Effectively, every bielliptic curve X0 (N )/WN satisfies that one bielliptic quotient is X0∗ (N ) and has positive rank and §7 contains the list of hyperelliptic quotient curves X0 (N )/W . These curves are the only quotient modular curves with gWN ≥ 2 which have an infinite number of quadratic points in any field extension of Q, in particular over Q. ∗ Remark 4.3. We computed all bielliptic quotients for the curves X0 (N )/WN when gN = 1. When gWN = 2 there are two bielliptic quotients, otherwise there is an only bielliptic quo- tient. Next, we present the table (N, hwp i, E, rank E), where E denotes a bielliptic quotient of X0 (N )/hwp i: gWN (N, hwp i, E, rank E) 2 (57, hw3 i, E19a, 0),(57, hw3 i, E57a, 1),(58, hw29 i, E58a, 1),(58, hw29 i, E58b, 0), (142, hw71 i, E142b, 1),(142, hw71 i, E142d, 0) 3 (58, hw3 i, E58a, 1),(65, hw5 i, E65a, 1),(65, hw13 i, E65a, 1),(77, hw7 i, E77a, 1), (82, hw41 i, 82a, 1),(91, hw13 i, E91a, 1),(123, hw41 i, E123b, 1),(141, hw47 i, E141d, 1) 4 (74, hw37 i, 37a, 1),(77, hw11 i, E77a, 1),(82, hw2 i, E82a, 1),(86, hw43 i, E43a, 1), (91, hw7 i, E91a, 1),(145, hw29 i, E145a, 1), 5 (86, hw2 i, E43a, 1),(111, hw3 i, 37a, 1),(155, hw31 i, 155c, 1) 6 (111, hw37 i, 37a, 1),(143, hw13 i, 143a, 1),(145, hw5 i, E145a, 1),(159, hw53 i, E53a, 1) 7 (118, hw2 i, E118a, 1),(123, hw3 i, E123b, 1),(143, hw11 i, E143a, 1) 8 (141, hw3 i, E141d, 1),(155, hw5 i, E155c, 1), 9 (159, hw3 i, E53a, 1). 5 N ∈ M with N = p1p2p3 Assume that N ∈ M is the product of 3 primes, i.e N = p1 p2 p3 . Now, B(N ) has order 8. As ∗ in the previous section, we split the cases in three blocks depending on the value of gN . By use of the Jacobian decomposition with Prop.3.3 and the Prop.3.1, the only candidates to bielliptic quotient modular curves X0 (N )/WN are presented in next tables and, moreover, the quotient curves that are already bielliptic by use Prop.3.1 joint with genus computations. See web page links for the Jacobian decomposition and genus computations: http://mat.uab.cat/∼francesc/JacobiandecompositionquotientsXoN.pdf http://mat.uab.cat/∼francesc/GenusquotientXoN.pdf ∗ • Assume gN ≥ 2. All candidate pairs (N, WN ) obtained satisfy |WM | = 4. 10 gWN (N, WN ) 3 (165, hw11 , w15 i) 4 (154, hw2 , w77 i),(170, hw2 , w85 i), (285, hw3 , w95 i),(286, hw2 , w143 i) 5 (154, hw2 , w7 i),(154, hw14 , w22 i), (165, hw3 , w55 )i,(165, hw5 , w33 )i, (170, hw17 , w10 i), (170, hw2 , w17 i),(186, hw6 , w62 i),(230, hw5 , w46 i) (230, hw2 , w115 i),(285, hw15 , w57 i),(357, hw7 , w17 i), 6 (154, hw2 , w11 i),(170, hw2 , w5 i),(186, hw31 , w6 i),(255, hw3 , w85 i) (266, hw7 , w38 i),(285, hw5 , w19 i),(357, hw3 , w119 )i 7 (186, hw2 , w3 i),(255, hw3 , w17 i),(246, hw3 , w41 i),(246, hw41 , w6 i), (258, hw6 , w86 i), (290, hw29 , w10 i),(318, hw3 , w53 i),(318, hw2 , w159 i) 8 (258, hw2 , w129 i),(290, hw5 , w58 i),(318, hw6 , w106 )i 9 (255, hw5 , w17 i), (266, hw2 , w7 i),(285, hw3 , w5 i),(286, hw11 , w26 i) 11 (357, hw7 , w51 i) ∗ Table III-1, candidates N with three primes and gN ≥ 2. ∗ • Assume gN = 1 (i.e. N ∈ {102, 114, 130, 138, 174, 182, 190, 195, 222, 231, 238}). For WN with |WN | = 4 we obtain that X0 (N )/W → X0∗ (N ) is a degree two morphism to the elliptic curve X0∗ (N ), therefore Proposition 5.1. Fix N ∈ {102, 114, 130, 138, 174, 182, 190, 195, 222, 231, 238} and let WN be any of the seven subgroup of order 4 of B(N ). Then, X0 (N )/WN is a bielliptic curve. In particular we obtain 77 bielliptic modular curves X0 (N )/W with WN of order 4 and N = p1 p2 p3 . For WN with |WN | = 2 (recall we are assuming WN 6= hwN i), we can reduce to study the modular quotient curves corresponding to the following pairs: gWN (N, WN ) 5 (102, hw51 i),(138, hw23 i) 6 (102, hw17 i), (114, hw38 i),(130, hw26 i) 7 (102, hw2 i) 8 (102, hw6 i), (102, hw34 i),(114, hw3 i),(114, hw57 i),(130, hw10 i) 9 (114, hw6 i),(138, hw69 i ∗ Table III-2, candidates N with three primes and gN = 1. ∗ • Assume gN = 0 (i.e. N ∈ {30, 42, 66, 70, 78, 105, 110}). If for some subgroup W such that |W | = 4, the curve X0 (N )/W has genus one, then for w ∈ B(N ) \ W the modular quotient curve X0 (N )/ < w > is a bielliptic curve. The next proposition provides all these cases. Proposition 5.2. The following 35 quotient curves X0 (N )/WN with |WN | = 2 of genus ≥ 2 are bielliptic curves. Each of them maps to a bielliptic quotient which corresponds to X0 (N )/W with |W | = 4 for certain W ≤ B(N ) with WN ≤ W . gWN (N, WN ) 2 (30, hw2 i),(30, hw3 i),(30, hw10 i) (h42, w3 i),(h42, w6 i), (42, hw21 i) (66, hw11 i),(70, hw35 i),(78, hw39 i) 3 (42, hw2 i),(42, hw7 i),(70, hw14 i),(78, hw26 i),(105, hw35 i) 4 (66, hw2 i),(66, hw33 i),(70, hw5 i),(70, hw10 i),(110, hw55 i) 5 (66, hw3 i),(66, hw22 i),(70, hw2 i), (70, hw7 i),(78, hw3 i),(105, hw5 i), (105, hw21 i),(110, w11 i) 6 (78, hw2 i),(78, hw13 ) 7 (105, hw3 i),(105, hw7 )i,(105, hw15 i),(110, hw10 i). 8 (110, hw2 i),(110, hw5 i) 11 For the values N ∈ {30, 42, 66, 70, 78, 105, 110}, it remains to to study the pairs gWN (N, WN ) 2 (66, hw2 , w3 i),(66hw6 , w22 i),(66, hw3 , w22 i), (70, hw2 , w5 i),(70, hw2 , w7 )i, (70, hw7 , w10 i), (78, hw2 , w13 i),(110, hw10 , w22 i). 3 (78, hw2 , w3 i),(78, hw13 , w6 i),(105, hw3 , w5 i), (105, hw3 , w7 i), (105, hw7 , w15 i),(110, hw2 , w11 i), (110, hw5 , w22 i). 4 (110, hw2 , w5 i),(66, hw6 i) 6 (78, hw6 i) 8 (110, hw22 i) ∗ Table III-3, candidates N with three primes and gN = 0. 5.1 Hyperelliptic case The hyperelliptic curves among the quotient curves corresponding to the pairs (N, WN ) in tables III-1, III-2 and III-3 are those such that gWN ≤ 3 and those that in Table III-3 satisfy gWN = 4 and gWN = 6 (genus 6). By [Has95], we know an equation for each of these hyperelliptic curves and, by using Magma, we get that Aut(X0 (N )/WN ) has order 2 except to the pairs (165, hw11 , w15 i) (genus 3), (66, hw6 i) (genus 4) and (78, hw6 i) (genus 6). Again, by using Magma, we get in these cases Aut(X0 (N )/WN ) is the Klein group. Lemma 5.3. The quotient curve corresponding to the pair (165, hw11 , w15 i) is the only curve among the curves corresponding to the pairs in the tables III-1, III-2 and III-3 that is hyperel- liptic and bielliptic. Moreover, the Q-isogeny class of the elliptic quotient is E55a. Proof. We have only to consider the three cases mentioned above. In [Ogg74, Proposition 1], Ogg proved that if X is a hyperelliptic curve over C with genus g ≥ 2 and u a nonhyperelliptic involution of X, then the genus gu of the curve X/hui satisfies gu = g/2 when g is even or gu ∈ {(g − 1)/2, (g + 1)/2} when g is odd. In any case, if v is the involution obtained as the product of u by the hypereliptic involution, then gu + gv = g. Hence, only X0 (165)/hw11 , w15 i is hw ,w i Q bielliptic. Since J0 11 15 ∼ E55a × Af with dim Af = 2, it follows that E55a is the Q-isogeny class of the bielliptic quotient of X0 (165)/hw11 , w15 i. 5.2 Non hyperelliptic case Proposition 5.4. Among the genus 4 curves in Table III-1, only the quotient curves corre- sponding to the pairs (154, hw2 , w77 i), (285, hw2 , w95 i) and (286, hw2 , w143 i) are bielliptic. Proof. We have the following decomposition over Q of J0 (N )WN , where we mark in bold the possible bielliptic quotients after applying Proposition 3.3 Q J0 (154)hw2 ,w77 i ∼ E11a × E14a × E77a × E154a , Q hw i J0 (170)hw2 ,w85 i ∼ E17a × Af2 × E170a , dim Af = 2, f ∈ New85 85 , Q J0 (285)hw3 ,w95 i ∼ E15a × E19a × E57a × E285b , Q J0 (286)hw2 ,w143 i ∼ E11a × E26a × E143a × E286c . We recall that a genus four curve is trigonal and satisfy dim L2 = 1 and dim L3 = 5. With the chosen order in the regular differentials (from the above Jacobian decomposition), following §2 we compute a non-zero polynomial Q2 ∈ L2 12 (N, WN ) Q2 (x, y, z, t) (154, hw2 , w77 i) t + 9x2 − 9y 2 + 4tz − 5z 2 2 (170, hw2 , w85 i) 3t + 2tx + x2 − 6y 2 − 16yz − 4z 2 2 (285, hw2 , w95 i) 7t2 + 9x2 − 9y 2 − 8tz + z 2 (286, hw2 , w143 i) 3t2 + 3x2 − 3y 2 − 4tz + z 2 and a polynomial Q3 ∈ L3 that is not multiple of Q2 : (N, WN ) Q3 (x, y, z, t) (154, hw2 , w77 i) 9t − 14tx − ty 2 − 4x2 z + 4y 2 z + 6tz 2 3 2 (170, hw2 , w85 i) −2txy − x2 y + 3y 3 − 8txz − x2 z + 13y 2 z + 20yz 2 + 4z 3 (285, hw2 , w95 i) t3 + 7tx2 − ty 2 + 8x2 z − 8y 2 z − 7tz 2 (286, hw2 , w143 i) 25t3 − 9tx2 − 9ty 2 − 24x2 z − 12y 2 z + 29tz 2 By [BG19a, Proof Prop.24], we know that the one dimensional elliptic curve E corre- sponding to ωi is a bielliptic quotient if, and only if, Q2 (. . . , xi , . . .) = Q2 (. . . , −xi , . . .) and Q3 (. . . , −xi , . . .) ∈ L3 . So, we obtain that • X0 (154)/hw2 , w77 i is bielliptic; one bielliptic quotient is E11a and another is E14a. • X0 (285)/hw3 , w95 i is bielliptic and has exactly two bielliptic quotient: E15a and E19a. • X0 (286)/hw2 , w143 i is bielliptic and has exactly two bielliptic quotient: E11a and E26a (corresponding to the variables x and y). • X0 (170)/hw2 , w85 i is not bielliptic (Q2 is not an even polynomial in the variable t). 5.2.1 genus 5 Proposition 5.5. There is not a bielliptic curve among all genus 5 curves that appears in Tables III-1 and III-2. Proof. We consider first the Jacobian decomposition for each of genus 5 curves involved in Tables III-1 and II-2, which are not hyperelliptic: Q Q4 Q Q Q Q J0 (102)hw51 i ∼ A2f1 i=2 Afi , Af1 ∼ E17a , Af2 ∼ E34a , Af3 ∼ E102a , Af4 ∼ E102b , Q Q4 Q Q Q Q J0 (138)hw23 i ∼ A2f1 i=2 Afi , Af1 ∼ E69a , Af2 ∼ E138a , Af3 ∼ E138b , Af4 ∼ E138c , Q Q5 Q Q Q Q Q J0 (154)hw2 ,w7 i ∼ i=1 Afi , Af1 ∼ E11a , Af2 ∼ E77a , Af3 ∼ E77c , Af4 ∼ E154a , Af5 ∼ E154c , f4 ∈ New Q Q4 Q Q Q J0 (154)hw14 ,w22 i ∼ i=1 Afi , Af1 ∼ E11a , Af2 ∼ E77a , Af3 ∼ E154a , 154 , dim Af4 = 2 , Q Q4 Q Q Q J0 (165)hw3 ,w55 i ∼ i=1 A fi , Af1 ∼ E11a , Af2 ∼ E15a , Af3 ∼ E33a , f4 ∈ New165 , dim Af4 = 2 , hw i f3 ∈ New Q Q3 Q J0 (165)hw5 ,w33 i ∼ i=1 A fi , Af1 ∼ E11a , f2 ∈ New55 5 , dim Af2 = 2 , 165 , dim Af3 = 2 , Q Q3 Q hw i J0 (170)hw17 ,w10 i ∼ i=1 A fi , Af1 ∼ E34a , f2 ∈ New∗85 , dim Af2 = 2 , f3 ∈ New85 17 , dim Af3 = 2 , Q Q3 hw i Q J0 (170)hw17 ,w2 i ∼ i=1 A fi , f1 ∈ New∗85 , dim Af1 = 2 , f2 ∈ New85 17 , dim Af2 = 2 , Af3 ∼ E170d , Q Q3 hw ,w i Q J0 (186)hw6 ,w62 i ∼ i=1 A fi , f1 ∈ New31 , dim Af1 = 2 , f2 ∈ New93 3 31 , dim Af2 = 2 , Af3 ∼ E186a , Q Q3 Q hw ,w i J0 (230)hw5 ,w46 i ∼ i=1 A fi , f1 ∈ New23 , dim Af1 = 2 , Af2 ∼ E115a , f3 ∈ New1155 23 , dim Af3 = 2 , Q Q4 Q Q Q J0 (285)hw15 ,w57 i ∼ i=1 A fi , Af1 ∼ E19a , Af2 ∼ E57a , Af3 ∼ E285b , f4 ∈ New285 , dim Af4 = 2 , f4 ∈ New Q Q4 Q Q Q J0 (357)hw7 ,w17 i ∼ i=1 A fi , Af1 ∼ E21a , Af2 ∼ E51a , Af3 ∼ E357c , 357 , dim Af4 = 2 , 13 For a quotient modular curve above, we fix the order of the regular differentials following the splitting presented of its jacobian. Denote by xni the variable corresponding to the differential regular associated to an elliptic curve Afi . First, we assume that this is not repeated in the Jacobian decomposition. In such non-repeated case, if Afi is a bielliptic quotient, then dim L2 = dim L2,ni = 3 (from Prop.2.4 with Remark 2.6). We compute such dimensions and we obtain (N, WN ) dim L2,ni (102, hw51 i) dim L2,2 = dim L2,4 = 1, dim L2,3 = 2 (138, hw23 i) dim L2,2 = dim L2,4 = 2, dim L2,3 = 1 (154, hw2 , w7 i) dim L2,i = 0, i = 1, 3, 5; dim L2,j = 1, j = 2, 4 (154, hw14 , w22 i) dim L2,2 = dim L2,3 = 1, dim L2,1 = 0 (165, hw3 , w55 i) dim L2,2 = dim L2,3 = 1, dim L2,1 = 0 (165, hw5 , w33 i) dim L2,1 = 0 (170, hw17 , w10 i) dim L2,1 = 0 (170, hw17 , w2 i) dim L2,3 = 0 (186, hw6 , w62 i) dim L2,3 = 0 (230, hw5 , w46 i) dim L2,2 = 1 (285, hw15 , w57 i) dim L2,2 = dim L2,3 = 1, dim L2,1 = 0 (357, hw7 , w17 i) dim L2,i = 0, i = 1, 2, 3 Therefore any of the non-repeated elliptic factors of the Jacobian decomposition is not a biel- liptic quotient. Now consider repeated elliptic factors, i.e. the pairs (102, hw51 i),  (138, hw23 i).  By using a1 a2 Prop.2.4 and Remark 2.6, we check that there is not a matrix A = ∈ GL2 (Q) b1 b2 satisfying the condition Q2 (a1 x1 + a2 x2 , b1 x1 + b2 x2 , x3 , x4 , x5 ) = Q2 (−a1 x1 + a2 x2 , −b1 x1 + b2 x2 , x3 , x4 , x5 ) (5.1) for all Q2 ∈ L2 . Therefore such quotient modular curves are not bielliptic. 5.2.2 Genus ≥ 6 Among the list of genus ≥ 6 in Tables III-1, III-2 and III-3, all pairs (N, WN ) 6= (78, hw6 i) are not hyperelliptic. By applying the arguments used in the previous subsection on genus 5 curves, we obtain the following result. Proposition 5.6. All curves in Tables III-1,III-2 and III-3 of genus ≥ 6 are not bielliptic. For completeness, one can consult of the computations in the rar file http://mat.uab.cat/∼francesc/productthreeprimes.rar . 5.2.3 Quadratic Points Because X0 (N )/WN has a Q-point and all automorphism are defined over Q, it has infinite number of quadratic points over Q if, and only if, X0 (N )/WN is hyperelliptic, or is bielliptic with a bielliptic quotient of positive Q-rank. First, we observe that each of the 77 bielliptic modular curves in Proposition 5.1 has an infinite number of quadratic points because the bielliptic quotient is the elliptic curve X0∗ (N ) which has positive Q-rank. 14 In Proposition 5.4 we determine all bielliptic quotients for the bielliptic (non-hyperelliptic) modular quotients corresponding to the pairs (154, hw2 , w77 i), (285, hw3 , w95 i) and (286, hw2 , w143 i). In each case there are two bielliptic quotients and, by [Cre17, Table 2], all have rank 0. There- fore, these three bielliptic modular curves of genus 4 have a finite set of quadratic points. Only remains 35 quotient bielliptic modular curves of Proposition 5.2, we consider only the not hyperelliptic ones. We obtain that all remaining quotient modular curves do not have elliptic curves in their Jacobians with positive Q-rank and, therefore, all of them have a finite set of quadratic points over Q. 6 N ∈ M with N = p1p2p3p4 Assume that N ∈ M is the product of 4 primes p1 , p2 , p3 , p4 , with p1 < p2 < p3 < p4 . In such a case, N ∈ {210, 330, 390, 510}. We observe that B(N ) of order 16, and it has 15 subgroups of order exactly 2, 35 subgroups of order exactly 4, and 15 subgroups of order exactly 8. We begin to study the subgroups WN of order 8 of B(N ). We denote by H1, . . . , H15 the following sets H1 := {1, p1 , p2 , p3 , p1 p2 , p1 p3 , p2 p3 , p1 p2 p3 } , H2 := {1, p1 , p2 , p4 , p1 p2 , p1 p4 , p2 p4 , p1 p2 p4 } , H3 := {1, p4 , p2 , p3 , p4 p2 , p4 p3 , p2 p3 , p4 p2 p3 } , H4 := {1, p1 , p4 , p3 , p1 p4 , p1 p3 , p4 p3 , p1 p4 p3 } , H5 := {1, p1 p2 , p3 p4 , p2 p3 , p1 p4 , p2 p4 , p1 p3 , p1 p2 p3 p4 } , H6 := {1, p1 p2 p3 , p1 , p2 p3 , p4 , p1 p2 p3 p4 , p1 p4 , p2 p3 p4 } , H7 := {1, p1 p2 p3 , p2 , p1 p3 , p4 , p1 p2 p3 p4 , p2 p4 , p1 p3 p4 } , H8 := {1, p1 p2 p3 , p3 , p1 p2 , p4 , p1 p2 p3 p4 , p3 p4 , p1 p2 p4 } , H9 := {1, p1 p3 p4 , p1 , p3 p4 , p2 , p1 p2 p3 p4 , p1 p2 , p2 p3 p4 } , H10 := {1, p1 p3 p4 , p3 , p1 p4 , p2 , p1 p2 p3 p4 , p2 p3 , p1 p2 p4 } , H11 := {1, p2 p3 p4 , p3 , p2 p4 , p1 , p1 p2 p3 p4 , p1 p3 , p1 p2 p4 } , H12 := {1, p1 p2 p3 , p2 p3 p4 , p1 p3 p4 , p2 p4 , p1 p4 , p1 p2 , p3 } , H13 := {1, p1 p2 p3 , p2 p3 p4 , p2 p1 p4 , p1 p4 , p3 p4 , p1 p3 , p2 } , H14 := {1, p1 p2 p3 , p1 p3 p4 , p1 p2 p4 , p2 p3 , p2 p4 , p3 p4 , p1 } , H15 := {1, p1 p2 p4 , p2 p3 p4 , p1 p3 p4 , p1 p2 , p1 p3 , p2 p3 , p4 } . ∗ The groups Hi = {wd ∈ B(N ) : d ∈ Hi} are all subgroups of B(N ) of order 8. Since g210 = 1, we get the following. Proposition 6.1. The fifteen quotient modular curves X0 (210)/Hi are bielliptic and the curve X0∗ (210) is a bielliptic quotient. By applying proposition 3.3 for N ∈ {330, 390, 510} we can only claim that are not bielliptic the quotient curves (510, H1 ), (510, H2 ), (510, H4 ), (510, H5 ), (510, H7 ), (510, H15 ) In, http://mat.uab.cat/∼francesc/Productoffourprimes.rar, it can be found the computations of the equations for the reaming quotient curves (N, Hi ), and we can conclude Lemma 6.2. All quotient modular curves X0 (N )/Hi with N ∈ {330, 390, 510} are not bielliptic. By Lemma 3.1 with the results on hyperelliptic quotient curves, see Table 1 in appendix, we obtain 15 Corollary 6.3. The curves X0 (N )/WN with N ∈ {330, 390, 510} and |WN | < 16 are not bielliptic. Now, for N = 210 we have to study the 35 curves X0 (210)/WN , where |WN | = 4. By apply- ing proposition 3.3 to the pairs (210, WN ) with |WN | = 4, we can claim that (210, hw2 , w21 i), (210, hw21 , w15 i) are not bielliptic. The computations for the remainder 33 pairs can be found in the rar file with all the computations in http://mat.uab.cat/∼francesc/Productoffourprimes.rar. Finally, we conclude with the following result. Lemma 6.4. The curves X0 (210)/WN with |WN | ≤ 4 are not bielliptic. Remark 6.5. Instead of discarding the above quotient modular curves by use Petri theorem, which required computation, we could discard some of them by unramified criteria [JKS20], with the use that we know few automorphism for such quotient modular curves, such criteria and different ones will be presented in dealing with the non-square free levels in [BGKS19]. 6.1 Quadratic points when N is a product of four primes Observe that in this case, all bielliptic quotient modular curves are the curves described in Proposition 6.1. In all theses cases, the curves have infinite quadratic points because X0∗ (210) is an elliptic curve with positive rank. Theorem 1.4, follows from collecting the results of all subsections concerning to quadratic points. 7 Appendix This section is devoted to present tables of results. Again, N is square-free and WN is a non trivial subgroup of B(N ) different from hwN i. First, we present the list of the pairs (N, WN ) such that X0 (N )/WN is hyperelliptic (cf. [FH99] and [Has95]). gWN (N, WN ) 2 (30, hw2 i), (30, hw3 i), (30, hw10 i), (33, hw3 i), (35, hw7 i),(38, hw2 i), (39, hw13 i),(42, hw3 i),(42, hw6 i), (42, hw21 i), (57, hw3 i),(58, hw29 i),(66, hw11 i),(66, hw2 , w3 i),(66, hw3 , w22 i),(66, hw6 , w22 i), (70, hw35 i),(70, hw2 , w5 i), (70, hw2 , w7 i),(70, hw7 , w10 i),(78, hw39 i),(78, hw2 , w13 i),(87, hw29 i), (102, hw3 , w17 i), (102, hw2 , w51 i),(110, hw10 , w22 i),(114, hw3 , w38 i),(130, hw10 , w26 i), (138, hw3 , w23 i),(138, hw23 , w6 i),(142, hw71 i), (190, hw5 , w19 i),(210, hw6 , w10 , w14 i). 3 (46, hw2 i),(51, hw3 i),(55, hw5 i), (70, hw14 i),(78, hw26 i),(78, hw2 , w3 i),(78, hw13 , w6 i),(95, hw19 i), (105, hw3 , w5 i),(105, hw3 , w7 i),(105, hw7 , w15 i),(110, hw2 , w11 i),(110, hw5 , w22 i),(114, hw2 , w19 i), (130, hw2 , w13 i),(165, hw11 , w15 i),(195, hw5 , w39 i). 4 (62, hw2 i),(66, hw6 i),(69, hw3 i),(70, hw10 i),(110, hw2 , w5 i),(119, hw17 i). 5 (87, hw3 i),(95, hw5 i). 6 (78, hw6 i),(94, hw2 i),(119, hw7 i). Table 1, Hyperelliptic curves X0 (N )/WN 16 Bielliptic modular quotients X0 (N )/WN gWN (N = p1 p2 , WN ) 2 (57, hw3 i),(58, hw29 i),(142, hw71 i). 3 (57, hw19 i),(58, hw2 i),(65, hw5 i),(65, hw13 i),(77, hw7 i),(82, hw41 i),(91, hw13 i), (118, hw59 i), (123, hw41 i),(141, hw47 i). 4 (74, hw2 i),(74, hw37 i),(77, hw11 i),(82, hw2 i),(86, hw43 i),(91, hw7 i),(145, hw29 i). 5 (86, hw2 i),(111, hw3 i),(155, hw31 i). 6 (111, hw37 i),(143, hw13 i),(145, hw5 i),(159, hw53 i). 7 (118, hw2 i),(123, hw3 i),(143, hw11 i). 8 (141, hw3 i),(155, hw5 i). 9 (142, hw2 i),(159, hw3 i). (N = p1 p2 p3 , WN ) 2 (30, hw2 i),(30, hw3 i),(30, hw10 i),(42, hw3 i),(42, hw6 i),(42, hw21 i),(66, hw11 i),(70, hw35 i), (78, hw39 i),(102, hw3 , w17 i),(102, hw2 , w57 i),(114, hw3 , w38 i),(130, hw10 , w26 i), (138, hw3 , w23 i),(138, hw23 , w6 i),(190, hw5 , w19 i), 3 (42, hw2 i),(42, hw7 i),(70, hw14 i),(78, hw26 i), (102, hw2 , w17 i),(102, hw6 , w34 )i,(102, hw17 , w6 i), (105, hw35 i),(114, hw2 , w19 i),(114, hw6 , w38 i),(114, hw2 , w57 i),(130, hw2 , w13 i), (130, hw2 , w65 )i,(130, hw5 , w26 i),(138, hw2 , w23 i),(165, hw11 , w15 i),(174, hw3 , w29 i), (174, hw2 , w87 i),(182, hw14 , w26 i),(190, hw10 , w38 i),(190, hw2 , w95 i),(195, hw15 , w39 i), (195, hw3 , w65 i),(195, hw5 , w39 i),(222, hw6 , w74 i),(231, hw3 , w77 i),(238, hw7 , w17 i), (238, hw14 , w34 i),(238, hw2 , w119 i). 4 (66, hw2 i),(66, hw33 i),(70, hw5 i),(70, hw10 i),(110, hw55 i),(102, hw2 , w3 i),(102, hw3 , w34 i), (114, hw2 , w3 i),(114, hw3 , w19 i),(114, hw19 , w6 i),(130, hw13 , w10 i),(130, hw2 , w5 i), (130, hw5 , w13 i),(138, hw2 , w69 i),(154, hw2 , w77 i),(174, hw6 , w58 i),(174, hw29 , w6 i), (182, hw2 , w91 i),(182, hw7 , w26 i),(182, hw13 , w14 i),(222, hw2 , w111 i),(231, hw21 , w33 i), (231, hw7 , w33 i),(231, hw11 , w21 i),(285, hw2 , w95 i),(286, hw2 , w143 i). 5 (66, hw3 i),(66, hw22 i),(70, hw2 i),(70, hw7 i),(78, w3 i),(105, hw5 i),(105, hw21 i),(110, hw11 i), (138, hw6 , w46 i),(138, hw3 , w46 i),(182, hw2 , w13 i),(182, hw7 , w13 i),(190, hw19 , w10 i), (195, hw3 , w13 i),(195, hw5 , w13 i),(195, hw13 , w15 i),(222, hw3 , w37 i),(222, hw3 , w74 i). 6 (78, hw2 i),(78, hw13 i),(138, hw2 , w3 i),(174, hw2 , w29 i),(174, hw3 , w58 i),(182, hw2 , w7 i), (190, hw2 , w19 i), (231, hw7 , w11 i). 7 (105, hw3 i),(105, hw7 i),(105, hw15 i),(110, hw10 i),(174, hw2 , w3 i),(190, hw2 , w5 i), (190, hw5 , w38 i),(195, hw3 , w5 i),(222, hw2 , w37 i),(231, hw3 , w7 i),(231, hw3 , w11 i), (238, hw2 , w17 i),(238, hw7 , w34 i),(238, hw17 , w14 i). 8 (110, hw2 i),(110, hw5 i),(222, hw37 , w6 i). 9 (222, hw2 , w3 i),(238, hw2 , w7 i). (N = p1 p2 p3 p4 , WN ) 2 (210, hw6 , w10 , w14 i) 3 (210, hw3 , w5 , w7 i),(210, hw2 , w5 , w7 i),(210, hw5 , w7 , w6 i),(210, hw2 , w3 , w35 i) (210, hw5 , w6 , w14 i),(210, hw3 , w10 , w14 i) 4 (210, hw2 , w3 , w7 i),(210, hw2 , w7 , w15 i),(210, hw3 , w5 , w14 i), (210, hw2 , w5 , w21 i),(210, hw2 , w15 , w21 i) 5 (210, hw2 , w3 , w5 i),(210, hw3 , w7 , w10 i),(210, hw7 , w6 , w10 i). Table 2, Bielliptic modular curves X0 (N )/WN 17 Infinite quadratic points of X0 (N )/WN gWN (N = p1 p2 , WN ) 2 (33, hw3 i),(35, hw7 i),(38, hw2 i),(39, hw13 i), (57, hw3 i),(58, hw29 i),(87, hw29 i),(142, hw71 i). 3 (46, hw2 i),(51, hw3 i),(55, hw5 i),(57, hw19 i),(58, hw2 i),(65, hw5 i),(65, hw13 i),(77, hw7 i), (82, hw41 i), (91, hw13 i),(95, hw19 i),(118, hw59 i), (123, hw41 i),(141, hw47 i). 4 (62, hw2 i),(69, hw3 i),(74, hw2 i),(74, hw37 i),(77, hw11 i),(82, hw2 i),(86, hw43 i),(91, hw7 i), (119, hw17 i),(145, hw29 i). 5 (86, hw2 i), (87, hw3 i),(95, hw5 i),(111, hw3 i),(155, hw31 i). 6 (94, hw2 i),(111, hw37 i),(119, hw7 i),(143, hw13 i),(145, hw5 i),(159, hw53 i). 7 (118, hw2 i),(123, hw3 i),(143, hw11 i). 8 (141, hw3 i),(155, hw5 i). 9 (142, hw2 i),(159, hw3 i). (N = p1 p2 p3 , WN ) 2 (30, hw2 i),(30, hw3 i),(30, hw10 i),(42, hw3 i),(42, hw6 i),(42, hw21 i),(66, hw11 i),(66, hw2 , w3 i), (66, hw3 , w22 i),(66, hw6 , w22 i), (70, hw35 i),(70, hw2 , w5 i),(70, hw2 , w7 i),(70, hw7 , w10 i), (78, hw39 i), (78, hw2 , w13 i),(102, hw3 , w17 i),(102, hw2 , w57 i),(110, hw10 , w22 i),(114, hw3 , w38 i),(130, hw10 , w26 i), (138, hw3 , w23 i),(138, hw23 , w6 i),(190, hw5 , w19 i). 3 (70, hw14 i),(78, hw26 i),(78, hw2 , w3 i),(78, hw6 , w13 )i, (102, hw2 , w17 i),(102, hw6 , w34 )i, (102, hw17 , w6 i),(105, hw3 , w5 i),(105, hw7 , w15 i),(105, hw3 , w7 i),(110, hw5 , w22 i),(110, hw2 , w11 i), (114, hw2 , w19 i),(114, hw6 , w38 i),(114, hw2 , w57 i),(130, hw2 , w13 i),(130, hw2 , w65 )i,(130, hw5 , w26 i), (138, hw2 , w23 i),(165, hw11 , w15 i),(174, hw3 , w29 i),(174, hw2 , w87 i),(182, hw14 , w26 i),(190, hw10 , w38 ), (190, hw2 , w95 i),(195, hw15 , w39 i),(195, hw3 , w65 i),(195, hw5 , w39 i),(222, hw6 , w74 i),(231, hw3 , w77 i), (238, hw7 , w17 i), (238, hw14 , w34 i),(238, hw2 , w119 i). 4 (66, hw6 i),(70, hw10 i),(102, hw2 , w3 i),(102, hw3 , w34 i),(110, hw2 , w5 i), (114, hw2 , w3 i), (114, hw3 , w19 i),(114, hw6 , w19 i),(130, hw13 , w10 i),(130, hw2 , w5 i),(130, hw5 , w13 i),(138, hw2 , w69 i), (174, hw6 , w58 i),(174, hw29 , w6 i),(182, hw2 , w91 i),(182, hw7 , w26 i),(182, hw13 , w14 i),(222, hw2 , w111 i), (231, hw21 , w33 i),(231, hw7 , w33 ),(231, hw11 , w21 i). 5 (138, hw6 , w46 i),(138, hw3 , w46 i),(182, hw2 , w13 i),(182, hw7 , w13 i),(190, hw19 , w10 i),(195, hw3 , w13 i), (195, hw5 , w13 i), (195, hw13 , w15 i),(222, hw3 , w37 i),(222, hw3 , w74 i). 6 (78, hw6 i),(138, hw2 , w3 i),(174, hw2 , w29 i),(174, hw3 , w58 i),(182, hw2 , w7 i),(190, hw2 , w19 i), (231, hw7 , w11 i). 7 (174, hw2 , w3 i),(190, hw2 , w5 i),(190, hw5 , w38 i),(195, hw3 , w5 i),(222, hw2 , w37 i), (231, hw3 , w7 i), (231, hw3 , w11 i),(238, hw2 , w17 i),(238, hw7 , w34 i),(238, hw17 , w14 i). 8 (222, hw37 , w6 i). 9 (222, hw2 , w3 i),(238, hw2 , w7 i). (N = p1 p2 p3 p4 , WN ) 2 (210, hw6 , w10 , w14 i). 3 (210, hw3 , w5 , w7 i),(210, hw2 , w5 , w7 i),(210, hw5 , w7 , w6 i),(210, hw2 , w3 , w35 i)(210, hw5 , w6 , w14 i), (210, hw3 , w10 , w14 i) 4 (210, hw2 , w3 , w7 i),(210, hw2 , w7 , w15 i),(210, hw3 , w5 , w14 i),(210, hw2 , w5 , w21 i),(210, hw2 , w15 , w21 i). 5 (210, hw2 , w3 , w5 i),(210, hw3 , w7 , w10 i),(210, hw7 , w6 , w10 i). Table 3, Infinite quadratic poinst of X0 (N )/WN 18 From Theorem 1.1 in [Jeo18], we have that for N square-free, the curve X0+ (N ) = X0 (N )/hwN i is bielliptic if, and only if, N is in the following list 42,57,58,66,70,74,77,78,82,85,86,91,105,110,111,118,123,141,142,143,145,155,159 . Table 4, Bielliptic curves X0 (N )/hwN i Table 2 provides all bielliptic curves X0 (N )/W for all non trivial subgroups W of B(N ) with W 6= hwN i, and Table 3 provides all quotient curves X0 (N )/W with an infinite number of quadratic points for all non-trivial subgroups W of B(N ) with W 6= hwN i (the case W = hwN i, see [Jeo18, Theorem1.2]). References [BGGP05] M. H. Baker, E. González-Jiménez, J. González, and B. Poonen. Finiteness results for modular curves of genus at least 2. Amer. J. Math., 127(6):1325–1387, 2005. [Bar99] F. Bars. Bielliptic modular curves. J. Number Theory, 76(1):154–165, 1999. [Bar18] F. 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. [BG19a] F. Bars and J. González. Bielliptic modular curves X0∗ (N ) with square-free levels. Mathematics of Computation, 88:2939–2957, 2019. [BG19b] F. Bars and J. 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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 Mohamed Gamal Kamel Asraan Youssef Department of Mathematics, Faculty of Science, Cairo University, 12613 Giza-Egypt mohgamal@sci.cu.edu.eg 20