This is the submitted version of the article: Bars Cortina, Francesc; González Rovira, Josep. «Bielliptic modular curves X0* (N) with square-free levels». Mathematics of Computation, Vol. 88, Issue 320 (April 2019), p. 2939-2957. DOI 10.1090/mcom/3424 This version is available at https://ddd.uab.cat/record/222678 under the terms of the license Bielliptic modular curves X0∗(N ) with square-free levels Francesc Bars∗and Josep González † Abstract Let N ≥ 1 be a square-free integer such that the modular curve X0∗ (N ) has genus ≥ 2. We prove that X0∗ (N ) is bielliptic exactly for 19 values of N , and we determine the automorphism group of these bielliptic curves. In particular, we obtain the first examples of nontrivial Aut(X0∗ (N )) when the genus of X0∗ (N ) is ≥ 3. Moreover, we prove that the set of all quadratic points over Q for the modular curve X0∗ (N ) with genus ≥ 2 and N square-free is not finite exactly for 50 values of N . 1 Introduction Let X be a smooth projective curve defined over a number field K of genus gX at least two. In [9], Faltings proved the finiteness of the set of points of X defined over K, denoted by X(K). After that, for a finite extension L/K, the natural object to consider was the set of points of X defined over all quadratic extensions of L, i.e. the set Γ2 (X, L) := ∪[F :L]≤2 X(F ). In [13], Harris and Silverman proved that the above set is not finite for some number field L if, and only if, X is hyperelliptic or bielliptic, i.e. the curve X ×K K admits a degree 2 map to the projective line or to an elliptic curve over a fixed algebraic closure K of K. Moreover, from the work of Abramovich, Harris and Silverman, in [5, Theorem 2.14] it is proved that the set Γ2 (X, L) is infinite if, and only if, X is hyperelliptic over L, i.e. there is a morphism φ : X → P1L of degree two defined over L, or X is bielliptic over L, i.e. there exist an elliptic curve E over L and a morphism φ : X → E of degree two defined over L, such that the L-rank of E is at least one. This made the study of bielliptic curves a matter of deep interest for Arithmetic Geometry. This was developed for the modular world, because L-points of modular curves have a moduli interpretation on elliptic curves. The first work concerned the modular curves X0 (N ). The levels N for which the set Γ2 (X0 (N ), Q) is finite are determined in [4]. Later, different results determining the bielliptic curves among some modular curves re- covering X0 (N ) can be found in [17] for X1 (N ), in [19] for X∆ (N ) and in [6] and [18] for X(N ). Two important tools to obtain such results are the following. First, if there is a morphism of curves X  X 0 such that X is bielliptic and the genus of X 0 is at least 2, then X 0 is hyperelliptic or bielliptic [13, Proposition 1]. One can use results about hyperelliptic modular curves, whose study has been widely treated in the last decades. Second, X is bielliptic if, and only if, there exists an involution of X fixing 2gX − 2-many points, where gX denotes de genus of X. ∗ First author is supported by MTM2016-75980-P and MDM-2014-0445 † The second author is partially supported by DGI grant MTM2012-34611. 1 In this paper we consider the modular curves X0∗ (N ). They are defined as the quotient of the modular curve X0 (N ) by the group of all Atkin-Lehner involutions, which is defined over Q. Its K-points, which are not cuspidal, correspond to K-curves with additional data from the level N . See [8] for further information. Here, we restrict our attention to the case where N is square-free. Under such assumption, the modular curve X0∗ (N ) corresponds to the quotient X0 (N )/ Aut(X0 (N )) and, thus, it does not have any natural automorphisms (in particular, involutions) coming from X0 (N ), except for N = 37. These curves have two properties that play an important role in the development of this article. On the one hand, all involutions of X0∗ (N ) are defined over Q. On the other hand, the endomorphism algebra End(Jac(X0∗ (N )) ⊗ Q is isomorphic to the product of totally real numbers fields (cf. [3, §2]). Any of these properties can fail when N is non square-free and the study of this case needs additional tools. We point out that if X0∗ (N ) is bielliptic, then a bielliptic quotient (i.e. the quotient of ∗ X0 (N ) by a bielliptic involution), is an elliptic curve E defined over Q of conductor M |N with odd analytic rank, because the attached newform is invariant under the Atkin-Lehner involution wM . Hence, it is expected that the algebraic rank of E is odd. In our case, the knowledge of the values of N for which X0 (N ) is bielliptic is not useful to obtain bielliptic curves X0∗ (N ), because, in this case, these curves have genus at most one or are hyperelliptic. A new approach is needed to deal with our case and, here, we use a method given in [11] to discard automorphisms of certain order in the automorphism group of a curve defined over a finite field. In particular, this method allows us to deduce that the automorphism group is trivial for such values of N , when such method works. This approach behaves well for odd square-free integers N , when N is the product of two or three primes. When it fails, we use the usual method of reducing modulo a prime p to discard some situations. For the remaining cases, using a Theorem of Petri [?], we implement a method to recognize whether a curve X0∗ (N ) is bielliptic and compute equations for the elliptic quotient. The main result of this article is the following. Theorem 1. Let N > 1 be a square-free integer. Assume that the genus of the modular curve X0∗ (N ) is at least 2. Then, 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 For these values of N , the automorphism group of X0∗ (N ) has order 2 when its genus is greater than two, otherwise it is the Klein group. Concerning Aut(X0∗ (N )), with N square-free and genus ≥ 2, it is known that it is an abelian 2-group (cf. [3]). Moreover, when N is prime this group is nontrivial if, and only if, the genus of the curve is 2 and, in this case, the group has order 2 (cf. [3, Theorem 1.1]). In fact, it is expected that this group is trivial for almost all square-free N . In this paper, we can observe that the Klein group appears naturally for genus two curves which are also bielliptic (cf. Remark 10). Moreover, we point out that a bielliptic curve could have several involutions and also more than one bielliptic involution when its genus is ≤ 5 (cf. [5, Prop.2.10]). Nevertheless, this does not happen in our case when the genus is > 2. As a by-product of this work, we also obtain the following result. 2 Proposition 2. The automorphism group of X0∗ (N ) is trivial for the following values of N : 185, 202, 259, 262, 267, 273, 282, 301, 305, 310, 326, 354, 393, 394, 395, 399, 426, 427, 429, 445, 458, 462, 489, 505, 518, 546, 553, 570, 581, 582, 602, 627, 638, 710, 754, 786, 795, 814, 826, 854, 903, 915, 1001, 1015 . Moreover, the group Aut(X0∗ (366)) has order 2, and the quotient curve has genus 2. For many of these values (see Propositions 11 and 21), this result is obtained by using the method to discard the existence of involutions, which was mentioned above. A Magma code to be applied in our case can be found in http://mat.uab.cat/~francesc/programmesXoestrellaMagma.html (this html page, also contains different codes in Magma for computing the genus of X0∗ (N ) and its Fpk -points). For the remaining values of the above proposition (see Propositions 16, 17, 23 and 24), Theorem of Petri is the main tool. As for quadratic points, by the work of Hasegawa and Hashimoto (cf. [15]), we know that X0∗ (N ) is hyperelliptic with N square-free if, and only if, the curve has genus 2. When X0∗ (N ) is bielliptic with genus > 2, the rank of the elliptic quotient turns out to be one. So we conclude Theorem 3. Let N > 1 be a square-free integer. Assume that the genus of the modular curve X0∗ (N ) is ≥ 2. Then, the set Γ2 (X0∗ (N ), Q) is infinite if, and only if, N lies 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} . Theorem 3 holds when we replace Γ2 (X0∗ (N ), Q) with Γ2 (X0∗ (N ), K), where K is any number field. This is due to the fact that if X0∗ (N ) is hyperelliptic or bielliptic over K, then it is hyperelliptic or biellitic over Q (cf. Lemma 4) 2 Preliminary results Let N > 1 be an integer. We fix once and for all the following notation. We denote by gN ∗ and gN the genus of X0 (N ) and X0∗ (N ), respectively, and n is the number of primes dividing N . For any 1 ≤ d|N with (d, N/d) = 1 we have an involution wd ∈ Aut(X0 (N )), called the Atkin-Lehner involution attached to d, and we denote by B(N ) the group of all Atkin-Lehner involutions. We denote by NewN the set of normalized newforms in S2 (Γ0 (N )), and New∗N is the subset of NewN consisting of the newforms invariant under the action of the group B(N ). For an integer m ≥ 1 and a newform f ∈ NewN , am (f ) is the m-th Fourier coefficient of f . For an eigenform f ∈ S2 (Γ0 (N ), Ag denotes the abelian variety defined over Q attached by Shimura to g. As usual, ψ is the Dedekind psi function. In the sequel, N is square-free. We recall the following result of Baker and Hasegawa. Lemma 4 (Corollary 2.6 in [3]). The group Aut X0∗ (N ) is elementary 2-abelian and every automorphism of X0∗ (N ) is defined over Q. From now on, we assume that X0∗ (N ) has a bielliptic involution u. Let us denote by E the elliptic quotient X0∗ (N )/hui and by π the nonconstant morphism X0 (N ) → X0∗ (N ) → E, which has degree 2n+1 and is defined over Q. Let M be the conductor of E. It is well-known that M |N and there exist 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 is the eigenform 3 wd∗ (f ) ∈ S2 (Γ0 (N ))B(N ) and wd stands for the Atkin-Lehner involution attached to P 1≤d|N/M d. More precisely, g(q) = 1≤d|N/M d fE (q d ). P Since the gonality of X0∗ (N ) is ≤ 4, by applying Proposition 4.4 in [2], we obtain gN ∗ ≤ 35. Nevertheless, this fact is not very useful in order to determine a finite set of possible values for N . The following lemma helps us to achieve this goal. Note that for a prime p - M , |E(Fp2 )| ≤ (p + 1)2 , but |E(Fp2 )| = (p + 1)2 − a2p (fE ) due to the congruence of Eichler-Shimura. Lemma 5. The following inequalities hold: (i) If p - N , then ψ(N ) 2|E(Fp2 )| − 1 ∗ |E(Fp2 )| |E(Fp2 )| (a) n ≤ 12 · , (b) gN ≤2 , (c) gN ≤ 2n+1 . 2 p−1 p−1 p−1 ∗ (ii) If p | N , then gN/p ≤ 1 or ψ(N/p) 2 |E(Fp2 )| − 1 |E(Fp2 )| |E(Fp2 )| n−1 ≤ 12 ∗ , gN/p ≤2 , gN/p ≤ 2n , if p - M , 2 p−1 p−1 p−1 ψ(N/p) 2p2 + 1 ∗ p2 + 1 2 np + 1 ≤ 12 , gN/p ≤ 2 , gN/p ≤ 2 , if p|M . 2n−1 p−1 p−1 p−1 Proof. Assume p - N . We generalize the argument used by Ogg in [20]. Indeed, X0 (N )(Fp2 ) contains 2n cusps and at least (p − 1) ψ(N 12 ) many supersingular points (cf. [2, 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 ) which has degree 2n+1 , |X0 (N )(Fp2 )| ≤ 2n+1 |E(Fp2 )|. Parts (b) and (c) in (i) are obtained applying [2, Lemma 3.25]. If p|N , then X0 (N )/Fp is the copy of two curves X0 (N/p)/Fp , and the normalization of X0 (N )/Fp is the curve X0∗ (N/p)/Fp (cf. [10]). If the reduction of the involution u is the identity, ∗ ∗ then gN/p is the genus of E/Fp . Otherwise, |X0∗ (N/p)(Fp2 )| is at most 2(p2 + 1) or |E(Fp2 )|, depending on whether p | M or not.  Remark 6. The above conditions imply n ≤ 4, when N is odd, and n ≤ 5 in the even case. ∗ The values N for which gN ≤ 1 can be found in [12, Proposition 3.1 and 3.2], and those for ∗ which gN = 2 can be found in [14, Theorem 2 ]. Keeping the above notation, we present the following lemma, which will used to discard some elliptic curves E for a value N . Lemma 7. 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 D of the modular parametrization πN : X0 (N ) → E 0 divides 2n+1 . Proof. The statement follows from the optimality of πN and the fact that the degree of π is 2n+1 .  Remark 8. The degree D can be found in [7, Table 5]. 4 3 Odd case If N is odd, applying Lemma 5 for p = 2, we have ψ(N )/2n ≤ 204. This fact implies n ≤ 3, except for the values N ∈ {3 · 5 · 7 · 11, 3 · 5 · 7 · 13}. The case n = 1 can be discarded, since Aut(X0∗ (p)) is trivial for all p except for gp∗ = 2 and, in this particular hyperelliptic case, the ∗ automorphism group has order two (cf. [3, Theorem 1.1]). We assume also that gN ≥ 2 and n apply Lemma 5 for the values gcd(N, 3) for which ψ(N )/2 ≤ 186. There are exactly 146 values for odd N such that n > 1, ψ(N ) ≤ 204 · 2n (or ≤ 186 · 2n+1 if 6 - N ) and all of these satisfy 1 < gN ≤ 9 · 2n+1 . More precisely, we have 100 cases for n = 2, 44 for n = 3 and 2 for n = 4. We can reduce this list by applying Lemma 5 for each candidate N , we consider the pairs (N, E), where E is the Q-isogeny class of the elliptic curves of conductor M |N such that its attached newform fE lies in New∗M . From [7, Table 5,Table 3], we obtain the degree D of πN , when M = N , and a2 (fE ), a3 (fE ) and a5 (fE ). In particular, we know |E(F4 )|, |E(F9 )| when 3 - M , and |E(F25 )| when 5 - M . For n > 1, we can discard the pairs (N, E) that do not satisfy the conditions in Lemmas 5 ∗ and 7. When gN = 2, if an elliptic quotient of X0∗ (N ) is not bielliptic, then we can discard N (cf. Remark 10). In particular, we discard N = 285 because the elliptic quotient of X0∗ (285) with conductor 258 does not satisfy Lemma 7. In Table 1, we present the remaining possibilities, where the label of the elliptic curve E is the one in Cremona tables. ∗ N gN gN M Label E 273 = 3 · 7 · 13 33 4 91 a ∗ 385 = 5 · 7 · 11 45 4 77 a N gN gN M Label E 399 = 3 · 7 · 19 49 4 57 a 129 = 3 · 43 13 2 N a 429 = 3 · 11 · 13 53 3 143 a 43 a 435 = 3 · 5 · 29 57 5 145 a 183 = 3 · 61 19 3 61 a 455 = 5 · 7 · 13 53 3 65 a 185 = 5 · 37 17 3 37 a 91 a 215 = 5 · 43 21 2 N a 465 = 3 · 5 · 31 61 5 155 c 43 a 555 = 3 · 5 · 37 73 5 185 c 237 = 3 · 79 25 5 79 a 615 = 3 · 5 · 41 81 6 123 b 249 = 3 · 83 27 3 N b 645 = 3 · 5 · 43 85 5 215 a 83 a 129 a 259 = 7 · 37 23 4 37 a 705 = 3 · 5 · 47 88 8 141 d 267 = 3 · 89 29 4 89 a 715 = 5 · 11 · 13 81 8 143 a 301 = 7 · 43 27 6 43 a 65 a 303 = 3 · 101 33 3 101 a 795 = 3 · 5 · 53 105 10 265 a 305 = 5 · 61 29 4 61 a 53 a 393 = 3 · 131 43 5 131 a 861 = 3 · 7 · 41 109 7 123 b 395 = 5 · 79 39 4 79 a 903 = 3 · 7 · 43 113 11 129 a 415 = 5 · 83 41 8 83 a 987 = 3 · 7 · 47 125 9 141 d 427 = 7 · 61 39 8 61 a 1001 = 7 · 11 · 13 109 8 143 a 445 = 5 · 89 43 7 89 a 1015 = 5 · 7 · 29 117 11 145 a 581 = 7 · 83 55 8 83 a 1155 = 3 · 5 · 7 · 11 185 8 77 a 1365 = 3 · 5 · 7 · 13 237 9 455 a 65 a Table 1 5 First, we examine the hyperelliptic cases in Table 1, which correspond to those such that ∗ gN = 2 (cf. [10, Theorem 2]). Proposition 9. The curves of genus two X0∗ (129) and X0∗ (215) are bielliptic. Proof. For N = 129 and N = 215, the jacobian of X0∗ (N ) is isogenous over Q to the product of two elliptic curves E1 × E2 , where E1 has conductor N and E2 has conductor M = 43. Hence, there exist two normalized newforms f1 ∈ New∗N and f2 ∈ New∗M such that the elliptic curves Af1 and Af2 are isogenous over Q to E1 and E2 respectively. The set of the regular differentials ω1 = f1 (q)d q/q , ω2 = (f2 (q) + `f2 (q ` ))d q/q , with ` = N/M , ω2 dx is a basis of Ω1X ∗ (N )/Q . The functions x = and y = on X0∗ (N ) satisfy the equations 0 ω1 ω1 N equations 2 129 4 y = x6 − 11 x4 + 35 x2 − 9 215 4 y 2 = −x6 − 5 x4 − 3 x2 + 25 For N = 129 and 215, it is clear that the curves have two bielliptic involutions (x, y) → 7 (−x, ±y).  Remark 10. Assume that X0∗ (N ) has genus two and has an elliptic quotient. Then, Jac(X0∗ (N )) is isogenous over Q to the product of two non isogenous elliptic curves E1 and E2 . If X0∗ (N ) has a bielliptic involution u, then u∗ (Ei ) = Ei and their regular differentials ω1 and ω2 are eigenvec- tors of u. Hence, u∗ (ω1 ) must be ±ω1 and u∗ (ω2 ) = ∓ω2 . Therefore u∗ (x) = u∗ (ω1 /ω2 ) = −x and u∗ (y) is y or −y depending on whether u∗ (ω2 ) is −ω2 or not. In any case, y 2 = P (x2 ) for a degree three polynomial P ∈ Q[x], and the automorphism group of the curve is the Klein group generated by u and the hyperelliptic involution w. Moreover, X0∗ (N ) has two bielliptic involutions u and u · w and both elliptic curves are bielliptic quotients. Now, we will apply two sieves to discard some values of N . Both are based on the values of |X0∗ (N )(Fpn )| for a prime p - N . The first of them uses [11, Theorem 2.1], which allows us to detect some curves X/Q without involutions defined over Q (see [?, Prop.10.3.38]). More precisely, for a prime p of good reduction for a curve X/Q and an integer n ≥ 1, consider the sequence X Pp (n) := mod [( µ(n/d)|X(Fpn )|)/n, 2] d|n where mod [r, 2] denotes 0 or P1 depending on whether r is even or not, and µ is the Moebius function. Set Qp (2k + 1) = kn≥0 (2n + 1)Pp (2n + 1). If X0∗ (N ) has an involution defined over Q, then ∗ Qp (2k + 1) ≤ 2gN + 2 for all k ≥ 0 . Proposition 11. The curve X0∗ (N ) is not bielliptic and, moreover, Aut(X0∗ (N )) is the trivial group for the following values of N : 259, 267, 301, 305, 393, 395, 427, 445, 581, 795, 903, 1001, 1015 . Proof. For f ∈ NewM , denote by K the number field Q({an (f )}n≥1 ). Let p be a prime not dividing M . By the Eichler-Shimura congruence, the characteristic polynomial of Frobp acting on the Tate module of Af is Y (x2 − ap (f σ ) + p) , σ : K,→Q 6 where σ runs over the set of all Q-embeddings of K into Q a fixed algebraic closure of Q. The jacobian of X0∗ (N ) is isogenous over Q to the product 1≤M |N f ∈New∗ /GQ Af , where GQ Q M denotes the absolute Galois group Gal(Q/Q). To compute |X0∗ (N )(Fpn )|, we proceed as follows. By using Magma, we determine New∗M for all M |N and, then, the characteristic polynomial Rp (x) of Frobp acting on the Tate module of Jac(X0∗ (N )) is obtained as follows ∗ 2 gN Y Y Y Rp (x) = (x2 − ap (f ) x + p) = (x − αi ) . ∗ i=1 M |N f ∈NewM Finally, ∗ 2 gN X |X0∗ (N )(Fpn )| =p +1−n αin . i=1 The statement follows from these computations: ∗ N Qp (2k + 1) 2gN +2 259 = 7 · 37 Q2 (9) = 17 10 267 = 3 · 89 Q2 (7) = 15 10 N Qp (2k + 1) ∗ 2gN +2 301 = 7 · 43 Q2 (11) = 22 14 795 = 3 · 5 · 53 Q2 (13) = 27 22 305 = 5 · 61 Q2 (7) = 15 10 903 = 3 · 7 · 43 Q2 (13) = 31 24  393 = 3 · 131 Q2 (11) = 23 12 1001 = 7 · 11 · 13 Q2 (11) = 22 18 395 = 5 · 79 Q2 (7) = 15 10 1015 = 5 · 7 · 29 Q2 (13) = 30 24 427 = 7 · 61 Q2 (11) = 27 18 445 = 5 · 89 Q3 (9) = 22 16 581 = 7 · 83 Q2 (13) = 20 18 The second sieve is based on the following fact. For a degree two morphism of curves X → Y defined over Q and a prime p of good reduction for X, one has |X(Fpk )| − 2|Y (Fpk )| ≤ 0 , for all k > 0 . Proposition 12. The pairs (N, E) in the set {(273, 91a), (385, 77a), (415, 83a), (429, 143a), (435, 145a), (455, 91a), (465, 155c), (555, 185c), (615, 123b), (705, 141d), (715, 65a), (715, 143a), (861, 123b), (987, 141d), (1155, 77a), (1365, 65a), (1365, 455a)} are not bielliptic. In particular, the curve X0∗ (N ) is not bielliptic for the following values of N : 273, 385, 415, 429, 435, 465, 555, 615, 705, 715, 861, 987, 1155, 1365 . . 7 Proof. N pk E |X0∗ (Fpk )| − 2|E(Fpk )| 273 8 91a 3 385 9 77a 3 415 9 83a 4 429 16 143a 5 455 4 91a 1 435 8 145a 2 465 7 155c 2 555 2 185c 1  615 11 123b 4 705 4 141d 1 715 9 65a 7 715 9 143a 1 861 4 123b 1 987 25 141d 10 1155 2 77a 1 1365 4 65a 1 1365 2 455a 3 After applying the two sieves, the following possibilities for the pairs (N, E), ordered by the genus, remain. ∗ N gN E ∗ 183 3 61a N gN E 185 3 37a 399 4 57a 249 3 83a 237 5 79a 249b 645 5 129a 303 3 101a 215a 455 3 65a Table 2 Finally, in order to decide which values N in Table 2 correspond to bielliptic curves, we shall use equations. We recall that, for a nonhyperelliptic curve X defined over C with genus g > 2, the image of the canonical map X → Pg−1 is the common zero locus of a set of homogeneous polynomials of degree 2 and 3, when g > 3, or of a homogenous polynomial of degree 4, if g = 3. More precisely, assume that X is defined over Q and choose a basis ω1 , · · · ωg of Ω1X/Q . For any integer i ≥ 2, let us denote by Li the Q-vector space of homogeneous polynomials Q ∈ Q[x1 , · · · , xg ] of degree i that satisfy Q(ω1 , · · · , ωg ) = 0. Of course, dim Li ≤ dim Li+1 because one has xj · Q ∈ Li+1 for all Q ∈ Li and for 1 ≤ j ≤ g. If g = 3, then dim L2 = dim L3 = 0 and dim L4 = 1. Any generator ofLL4 provides an equation for X. For g > 3, dim L2 = (g − 2)(g − 3)/2 > 0 and a basis of L2 L3 provides a system of equations for X. When X is neither trigonal nor a smooth plane quintic (g = 6), it suffices to take a basis of L2 . For the curve X0∗ (N ) there exists a set of normalized eigenforms g1 , · · · , gk ∈ S2 (Γ0 (N ))B(N ) Q Q such that Jac(X0∗ (N )) ∼ Ag1 × · · · × Agk , where the symbol ∼ means isogenous over Q. These abelian varieties are simple and pairwise nonisogenous over Q. Hence, any involution u of the 8 curve leaves stable Agi and acts on Ω1Agi as the product by −1 or the identity, because the endomorphism algebra EndQ Agi ⊗ Q is isomorphic to a (totally real) number field. We choose a basis {ω1 , · · · , ωgN∗ } of Ω1X ∗ (N )/Q obtained as the union of bases of all Ω1Ag /Q . 0 i An involution u of X0∗ (N ) induces a linear map u∗ : Ω1X ∗ (N )/Q → Ω1X ∗ (N )/Q sending (ω1 , · · · , ωgN∗ ) 0 0 ∗ to (ε1 ω1 , · · · , εn ωgN∗ ) with εi = ±1 for all i ≤ gN and satisfying Q(ε1 x1 , · · · , εgN∗ xgN∗ ) ∈ Li for all Q ∈ Li and for all i . (3.1) Conversely, for a linear map u∗ as above satisfying condition (3.1), only one of the two maps ±u∗ comes from an involution of the curve, because we are assuming that X is nonhyperelliptic. We particularize this fact to our case. Lemma 13. Assume X0∗ (N ) is nonhyperelliptic. Let ω1 , · · · , ωgN∗ be a basis of Ω1X ∗ (N )/Q as 0 above, such that ω1 is the differential attached to an elliptic curve E. Then, the pair (N, E) is bielliptic if, and only, if Q(−x1 , x2 , · · · , xgN∗ −1 , xgN∗ ) ∈ Li for all Q ∈ Li and for all i . (3.2) Proof. If u is an involution of X0∗ (N ) such that E is Q-isogenous to X0∗ (N )/hui, then u∗ (ω1 ) = ω1 and u∗ (ωi ) = −ωi for i > 1. Hence, condition (3.2) is satisfied. Conversely, since the curve is nonhyperelliptic the condition (3.2) implies that only one of the two linear maps (ω1 , ω2 , · · · , ωgN∗ −1 , ωgN∗ ) 7→ ±(−ω1 , ω2 , · · · , ωgN∗ −1 , ωgN∗ ) comes from an involution u of the curve. The genus gu of the curve X0∗ (N )/hui agrees with ∗ the number of differentials ωi invariant under the action of u. When gN > 3, it follows that gu ∗ ∗ must be 1 because it cannot be gN − 1 due to Riemann-Hurwitz formula. For gN = 3, the genus gu must be different from 2, since otherwise the curve would be hyperelliptic (cf. [1, Lemma 5.10]).  ∗ Remark 14. When gN > 4, dim L2 > 1. If ωj is the differential attached to an elliptic curve, we need to check that the vector space L2,j := {Q ∈ L2 : Q(x1 , · · · , −xj , · · · , xgN∗ ) ∈ L2 } is L2 . Note that L2,j = {Q ∈ L2 : Q(x1 , · · · , xj , · · · , xgN∗ ) = Q(x1 , · · · , −xj , · · · , xgN∗ )} Indeed, if Q ∈ L2,j , then H := Q(x1 , · · · , xj , · · · , xgN∗ ) − Q(x1 , · · · , −xj , · · · xgN∗ ) ∈ L2 . There- fore, H = xj P for an homogenous polynomial P ∈ Q[x1 , · · · , xgN∗ ] of degree at most 1. Hence, P must be 0, otherwise P (ω1 , · · · , ωgN∗ ) = 0. Remark 15. Recall that, for each one of the normalized eigenforms gi ∈ S2 (Γ0 (N ))B(N ) , there is fi ∈ New∗M such that gi = d|N/M d fi (q d ) and Agi ∼ Afi . To get a basis of Ω1Ag /Q we can P Q i proceed as follows. If dim Afi = 1, we take as basis gi (q) dq/q. When dim Afi = r > 1, the endomorphism algebra EndQ (Afi ) ⊗ Q is generated by Hecke operators and is isomorphic to the totally real number field Ki := Q({an (fi )}n>0 ) of degree r. Let I be the set of Q-embeddings of Ki into a fixed algebraic closure of Q. For every a ∈ Ki there is a Hecke operator T such that 9 T (fiσ ) = aσ fiσ for all σ ∈ I. The two cusp-form h = σ∈I fiσ is nonzero because the coefficient P of q is g. Hence, taking T such that a is a primitive element of Ki , the set X fj0 = T j (h) = (aj )σ fiσ ∈ Q[[q]], 0 ≤ j ≤ r − 1 , σ∈I is a basis of the vector space generated by fiσ . Therefore, X ωj0 = ( d fi0 (q d )) dq/q 0 ≤ j ≤ r − 1 d|N/M is a basis of Ω1Ag /Q . One can take a as the value provided by Magma in the q-expansion of fi i and, in this case, fj0 ∈ Z[[q]]. The curve X0∗ (N ) is determined by the first Fourier coefficients of the chosen basis for Ω1X ∗ (N )/Q (cf [2, Proposition 2.8]). In order to get shorter equations, it 0 is suitable to replace the basis fj0 with a basis of the Z-module Q(f10 , · · · , fg0 ) ∩ (Z[[q]] dq). Proposition 16. Among the curves of genus three X0∗ (183), X0∗ (185), X0∗ (249), X0∗ (303), and X0∗ (455), only X0∗ (183), X0∗ (249), X0∗ (303) and X0∗ (455) are bielliptic. The corresponding ellip- tic quotients are labeled as E61a1, E249b1, E101a1 and E65a1, respectively. In all these cases, the automorphism group has order 2. The automorphism groups of the remaining curves are trivial. Proof. For these values of N , the splitting of the jacobian of X0∗ (N ), J0∗ (N ), is as follows: Q2 f2 ∈ New∗183 , Q Q J0∗ (183) ∼ A fi , Af1 ∼ E61a , dim Af2 = 2 , Q Qi=1 3 Q Q Q J0∗ (185) ∼ A fi , Af1 ∼ E37a , Af2 ∼ E185b , Af3 ∼ E185c , Q Q3i=1 Q Q Q J0∗ (249) ∼ Afi , Af1 ∼ E83a , Af2 ∼ E249a , Af3 ∼ E249b , Qi=1 2 f2 ∈ New∗303 , Q Q J0∗ (303) ∼ A fi , Af1 ∼ E101a , dim Af2 = 2 , Q Qi=1 3 Q Q Q J0∗ (455) ∼ i=1 Afi , Af1 ∼ E65a , Af2 ∼ E91a , Af3 ∼ E455a . We take a basis of Ω1X ∗ (N )/Q following the order exhibited in the splitting of its jacobian and 0 we obtain the following generators Q ∈ L4 : N Q 183 x − 10x y + 9y − 24x2 yz + 24y 3 z + 32y 2 z 2 + 32yz 3 − 16z 4 4 2 2 4 185 243x4 − 1728x2 y 2 − 768xy 3 − 2816y 4 − 324x3 z − 1728x2 yz − 1152xy 2 z +5888y 3 z − 918x2 z 2 + 1152xyz 2 − 192y 2 z 2 − 852xz 3 + 2624yz 3 + 571z 4 249 16x4 − 64x3 y − 12x2 y 2 + 44xy 3 + 97y 4 − 180x2 z 2 + 36xyz 2 − 18y 2 z 2 + 81z 4 303 x4 + 2x2 y 2 − 3y 4 − 16y 2 z 2 + 16yz 3 − 16z 4 455 81x − 162x y − 79y − 324x2 yz + 244y 3 z + 192y 2 z 2 + 64yz 3 − 16z 4 4 2 2 4 By Lemma 13, only the curves corresponding to N = 183, 249, 303, 455 are bielliptic with an only bielliptic involution u. The affine equations for the bielliptic quotients are N X0∗ (N )/hui 183 x − 10x + 9 − 24xz + 24z + 32z 2 + 32z 3 − 16z 4 2 =0 249 97 + 44x − 12x2 − 64x3 + 16x4 − 18z + 36xz − 180x2 z + 81z 2 =0 303 x2 + 2x − 3 − 16z 2 + 16z 3 − 16z 4 =0 455 81x2 − 162x − 79 − 324xz + 244z + 192z 2 + 64z 3 − 16z 4 =0 10 have genus one and their j-invariants are − 912673 61 , 357911 249 , 262144 101 and 117649 65 . They correspond to the elliptic curves E61a1, E249b1, E101a1 and E65a1. Taking into account the splitting of their jacobians and their equations, all their automor- phism groups have order 2. The remaining curves have trivial automorphism groups. For instance, for N = 249, the linear map (ω1 , ω2 , ω3 ) 7→ (ω1 , ω2 , −ω3 ) is the only option to be con- sidered and Q(−x, y, z) ∈/ L4 . For N = 185, none of the polynomials Q(−x, y, z),Q(x, −y, z), Q(x, y, −z) lies in L4 .  Proposition 17. The curves of genus four X0∗ (399) is not bielliptic and its automorphism group is trivial. Proof. The splitting of J0∗ (399) is: f3 ∈ New∗133 , dim Af3 = 2 . Q Q Q J0∗ (399) ∼ Af1 × Af2 × Af3 , Af1 ∼ E57a , Af2 ∼ E399a , In this case dim L2 = 1. As in the previous proposition, we take a basis of Ω1X ∗ (399)Q following 0 the order exhibited in the splitting of its jacobian. Next, we show a generator Q(x, y, z, t) ∈ L2 : N Q 399 −99t2 + 90tx + 125x2 + 189ty + 80xy − 151y 2 + 306tz − 105xz + 42yz + 9z 2 Since Q(−x, y, z, t) ∈ / L2 , the curve is not bielliptic. The conditions Q(x, −y, z, t), Q(−x, −y, z, t) ∈ / L2 imply that the curve does not have any nontrivial involutions.  Proposition 18. The curves of genus five X0∗ (237) and X0∗ (645) are not bielliptic. Proof. The splitting of J0∗ (N ) is: Q2 f2 ∈ New∗237 , Q Q J0∗ (237) ∼ Afi , Af1 ∼ E79a , dim Af2 = 4 , Q4i=1 f4 ∈ New∗645 , Q Q Q Q J0∗ (645) ∼ i=1 Afi , Af1 ∼ E43a , Af2 ∼ E129a , Af3 ∼ E215a , dim Af4 = 2 . Now, dim L2 = 3. For N = 237, dim L2,1 = 0 and for N = 645 we also have dim L2,2 = dim L2,3 = 0.  As a consequence, we obtain the statement of Theorem 1 for N odd. Corollary 19. For N odd, X0∗ (N ) is bielliptic if, and only if, N ∈ {129, 183, 215, 249, 303, 455}. ∗ For these values of N , the automorphism group has order 2 when gN > 2, otherwise it is the Klein group. 4 Even case By applying Lemma 5, we determine a finite set of possible values of N . Then, we proceed as in the odd case and we obtain the pairs (N, E) exhibited in Table 3 together the genera of X0 (N ) and X0∗ (N ). As in the odd case, for gN∗ = 2 we can discard N = 154, 286 because both ∗ curves X0 (N ) have an elliptic quotient of conductor N that does not satisfy Lemma 7. Table 3 is divided into 4 cases: 3 - N , 6 | N and 30 - N , 30 | N and 210 - N and, finally, 210 | N . 11 ∗ 6|N, 30 - N gN gN M Label E N, 3 - N gN ∗ gN M Label E 246 = 2 · 3 · 41 39 3 82 a 106 = 2 · 53 12 2 106 b 123 b 53 a 258 = 2 · 3 · 43 41 3 43 a 122 = 2 · 61 14 2 122 a 129 a 61 a 282 = 2 · 3 · 47 45 3 141 d 158 = 2 · 79 19 2 158 b 318 = 2 · 3 · 53 51 3 53 a 79 a 106 b 166 = 2 · 83 20 2 166 a 354 = 2 · 3 · 59 57 4 118 a 83 a 366 = 2 · 3 · 61 59 4 61 a 178 = 2 · 89 21 3 89 a 122 a 202 = 2 · 101 24 4 101 a 402 = 2 · 3 · 67 65 5 201 a 262 = 2 · 131 32 4 131 a 201 c 394 = 2 · 197 48 10 197 a 426 = 2 · 3 · 71 69 4 142 b 458 = 2 · 229 56 10 229 a 438 = 2 · 3 · 73 71 5 219 a 219 c 290 = 2 · 5 · 29 41 3 58 a 474 = 2 · 3 · 79 77 7 79 a 145 a 158 b 310 = 2 · 5 · 31 45 3 155 c 498 = 2 · 3 · 83 81 7 83 a 370 = 2 · 5 · 37 53 4 185 a 166 a 185 c 249 a 370 a 249 b 410 = 2 · 5 · 41 59 5 82 a 534 = 2 · 3 · 89 87 8 89 a 430 = 2 · 5 · 43 63 3 43 a 582 = 2 · 3 · 97 95 7 291 c 215 a 606 = 2 · 3 · 101 99 8 101 a 530 = 2 · 5 · 53 77 7 106 b 642 = 2 · 3 · 107 105 9 214 b 265 a 786 = 2 · 3 · 131 129 11 131 a 574 = 2 · 7 · 41 81 5 82 a 590 = 2 · 5 · 59 87 6 118 a 462 = 2 · 3 · 7 · 11 89 3 77 a 154 a 602 = 2 · 7 · 43 85 9 43 a 546 = 2 · 3 · 7 · 13 105 4 91 a 710 = 2 · 5 · 71 105 7 142 b 714 = 2 · 3 · 7 · 17 137 5 102 a 742 = 2 · 7 · 53 105 10 371 a 238 b 770 = 2 · 5 · 7 · 11 137 5 77 a 798 = 2 · 3 · 7 · 19 153 5 57 a 154 a 399 a 910 = 2 · 5 · 7 · 13 161 5 65 a 858 = 2 · 3 · 11 · 13 161 6 143 a 91 a 286 c 455 a 966 = 2 · 3 · 7 · 23 185 8 138 a 1190 = 2 · 5 · 7 · 17 209 8 238 b 1122 = 2 · 3 · 11 · 17 209 9 374 a 1254 = 2 · 3 · 11 · 19 233 12 57 a ∗ 30|N, 210 - N gN gN M Label E ∗ 30|N, 210 - N gN gN M Label E 1230 = 2 · 3 · 5 · 41 245 10 123 b 390 = 2 · 3 · 5 · 13 77 2 65 a 615 a 390 a 1290 = 2 · 3 · 5 · 43 257 10 129 a 510 = 2 · 3 · 5 · 17 101 3 102 a 215 a 570 = 2 · 3 · 5 · 19 113 4 57 a 1410 = 2 · 3 · 5 · 47 281 14 141 d 190 b 705 a 285 b 1590 = 2 · 3 · 5 · 53 317 14 53 a 690 = 2 · 3 · 5 · 23 137 6 138 a 265 a 870 = 2 · 3 · 5 · 29 173 7 58 a 795 a ∗ 145 a 210|N gN gN M Label E 290 a 2310 = 2 · 3 · 5 · 7 · 11 561 12 77 a 930 = 2 · 3 · 5 · 31 185 8 155 c 1155 a 1110 = 2 · 3 · 5 · 37 221 9 185 c 2730 = 2 · 3 · 5 · 7 · 13 657 14 65 a 455 a Table 3 As in the odd case, first we examine the hyperelliptic curves. Proposition 20. All the curves of genus two appearing in Table 3, i.e. X0∗ (106), X0∗ (122), X0∗ (158), X0∗ (166) and X0∗ (390), are bielliptic. Proof. For N ∈ {106, 122, 154, 158, 166, 286, 390}, the jacobian of X0∗ (N ) is isogenous over Q to the product of two elliptic curves E and F of conductors N and M < N , respectively. We have that M = N/6 for N = 390 and M = N/2 otherwise. Let fE ∈ New∗N and fF ∈ New∗M be the corresponding newforms attached to these elliptic curves. The functions fE (q) 2qdx x := P d , y := P d , d|N/M d fF (q ) d|N/M d fF (q ) provide the following equations X0∗ (106) : y2 = 4x6 + 17x4 − 6x2 + 1 , X0∗ (122) : y2 = 4x6 + x4 + 10x2 + 1 , X0∗ (158) : y2 = −2x6 + 11x4 + 8x2 − 1 , X0∗ (166) : y2 = −4x6 + 17x4 + 2x2 + 1 , X0∗ (390) : y2 = −(3x2 + 1)(4x4 − 7x2 − 1) . 12 In all cases, one has the involutions (x, y) 7→ ±(−x, ±y).  Next, we use the sieve based on [11, Theorem 2.1]. Proposition 21. The curve X0∗ (N ) is not bielliptic and its automorphism group is trivial for the following values of N : 394, 458, 582, 602, 710, 786, . Proof. For a prime p - N , let Qp (2k + 1) be as in Proposition 11. After the following computations, ∗ N Qp (2k + 1) 2gN +2 394 = 2 · 197 Q3 (9) = 25 22 458 = 2 · 229 Q5 (15) = 36 22 582 = 2 · 3 · 97 Q13 (9) = 18 16 602 = 2 · 7 · 43 Q5 (11) = 31 20 710 = 2 · 5 · 71 Q3 (9) = 25 16 786 = 2 · 3 · 131 Q5 (11) = 27 24 the statement follows.  Now, we apply the sieve based on the values of |X0∗ (N )(Fpn )| − 2|E(Fpn )| and a modification for primes p dividing the conductor N . Proposition 22. The pairs (N, E) in the set {(290, 58a), (370, 37a), (370, 185a), (402, 201a), (410, 82a), (438, 219c), (474, 79a), (474, 158b), (498, 83a), (498, 166a), (498, 249a), (498, 249b), (530, 106b), (530, 265a), (534, 89a), (574, 82a), (590, 118a), (606, 101a), (642, 214b) (714, 102a), (742, 371a), (770, 77a), (770, 154a), (798, 57a), (870, 58a), (910, 65a), (930, 155c), (966, 138a), (1122, 374a), (1190, 238b), (1230, 123b), (1230, 615a), (1254, 57a), (1290, 129a), (1290, 215a), (1410, 141d), (1410, 705d), (1590, 53a), (1590, 265a), (1590, 795a), (2310, 77a), (2310, 1155a), (2730, 65a), (2730, 455a)} are not bielliptic. In particular, the curve X0∗ (N ) is not bielliptic for the following values of N : 410, 474, 498, 530, 534, 574, 590, 606, 642, 742, 770, 930, 966, 1122, 1190, 1230, 1254, 1290, 1410, 1590, 2310, 2730 . Proof. We put n(N, E, pk ) = |X0∗ (N )(Fpk )| − 2|E(Fpk )|. N pk E n(N, E, pk ) 290 9 58a 5 N pk E n(N, E, pk ) N pk E n(N, E, pk ) 370 9 37a 6 606 5 101a 2 1230 49 123b 19 3 185a 2 642 7 214b 9 11 615a 2 402 5 201a 2 714 25 102a 6 1254 5 57a 1 410 13 82a 1 742 9 371a 4 1290 49 129a 3 438 25 219c 12 49 215a 3 474 25 79a 10 770 9 77a 9 3 154a 1 1410 49 141d 20 11 158b 7 7 705d 10 498 11 83a 5 798 25 57a 3 1590 49 53a 54 7 166a 3 870 13 58a 1 7 265a 6 7 249a 1 910 9 65a 3 7 795a 10 13 249b 5 930 7 155c 2 2310 17 77a 1 530 25 106b 35 966 25 138a 7 17 1155a 9 7 265a 8 1122 5 374a 5 2730 11 65a 7 534 5 89a 2 1190 3 238b 1 11 455a 3 574 9 82a 4 590 9 118a 2 13 1 .  After applying the two sieves, the following possibilities for the pairs (N, E), ordered by the genus, remain: ∗ N gN E N ∗ gN E ∗ 178 3 89a N gN E 402 5 201c 246 3 82a, 123b 202 4 101a 438 5 219a 258 3 43a, 129a 262 4 131a 714 5 238b 282 3 141d 354 4 118a 798 5 399a 290 3 145a 366 4 61a, 122a 910 5 91a, 455a 310 3 155c 370 4 185c, 370a 690 6 138a 318 3 53a, 106b 426 4 142b 858 6 143a, 286c 430 3 43a, 215a 546 4 91a 870 7 145a, 290a 462 3 77a, 154a 570 4 57a, 190b, 285b 510 3 102a 1110 9 185c Table 4 Proposition 23. Among the curves of genus three X0∗ (178), X0∗ (246), X0∗ (258), X0∗ (282), X0∗ (290), X0∗ (310), X0∗ (318), X0∗ (430), X0∗ (462) and X0∗ (510), only X0∗ (178), X0∗ (246), X0∗ (258), X0∗ (290), X0∗ (318),X0∗ (430) and X0∗ (510) are bielliptic. The corresponding elliptic quotients are labeled as E89a1, E82a1, E43a1, E145a1, E53a1, E43a1 and E102a1, respectively. In all these cases, the automorphism group of X0∗ (N ) has order 2. The automorphism groups of the remaining curves are trivial. Proof. For these values of N , the splitting of the jacobian of X0∗ (N ), J0∗ (N ), is as follows: Q2 f2 ∈ New∗178 , Q Q J0∗ (178) ∼ A fi , Af1 ∼ E89a , dim Af2 = 2 , Q Q3i=1 Q Q Q J0∗ (246) ∼ A fi , Af1 ∼ E82a , Af2 ∼ E123b , Af3 ∼ E246d , Q Qi=1 3 Q Q Q J0∗ (258) ∼ A fi , Af1 ∼ E43a , Af2 ∼ E129a , Af3 ∼ E258a , Qi=1 2 f2 ∈ New∗282 , Q Q J0∗ (282) ∼ Afi , Af1 ∼ E141d , dim Af2 = 2 , Q Qi=1 3 Q Q Q J0∗ (290) ∼ Afi , Af1 ∼ E145a , Af2 ∼ E58a , Af3 ∼ E290a , Q2i=1 f2 ∈ New∗310 , Q Q J0∗ (310) ∼ Afi , Af1 ∼ E155c , dim Af2 = 2 . Q Qi=1 3 Q Q Q J0∗ (318) ∼ A fi , Af1 ∼ E53a , Af2 ∼ E106b , Af3 ∼ E318c , Q Q3i=1 Q Q Q J0∗ (430) ∼ A fi , Af1 ∼ E43a , Af2 ∼ E215a , , Af3 ∼ E430a , Q Q3i=1 Q Q Q J0∗ (462) ∼ A fi , Af1 ∼ E77a , Af2 ∼ E154a , Af3 ∼ E462a , Q2i=1 f2 ∈ New∗85 , Q Q J0∗ (510) ∼ i=1 Afi , Af1 ∼ E102a , dim Af2 = 2 , 1 F:Bars, is a problem because we reduce modulo 2 a degree two morphism TO ARISE: Now, suppose that the pair (N, E) is bielliptic. If p|N , we know that X0 (N ) modulo p is the copy of two curves X0 (N/p)/Fp , and the normalization of X0∗ (N )/Fp is the curve X0∗ (N/p)/Fp (cf. [10]). If p does not divide the conductor of E, ∗ then E/Fp is an elliptic curve. If gN/p > 1, then this is the quotient curve of X0∗ (N/p)/Fp by an involution k defined over Fp . Therefore, n(N/p, E, p ) ≤ 0. Taking N = 1110, p = 2 and E with the label 185c, we get n(555, E, 2) = 1 and, thus, the pair (1110, E) can be discarded 14 We take a basis of Ω1X ∗ (N )Q following the order exhibited in the splitting of the jacobian, and 0 we obtain the following generators Q ∈ L4 : N Q 178 x − 2x y + y − 12x2 yz − 4y 3 z − 4x2 z 2 + 20y 2 z 2 + 32yz 3 4 2 2 4 246 81x4 − 16y 4 + 324x2 yz − 32y 3 z − 162x2 z 2 + 48y 2 z 2 − 260yz 3 + 17z 4 258 81x − 90x2 y 2 + 41y 4 + 288x2 yz − 176y 3 z − 36x2 z 2 + 84y 2 z 2 − 128yz 3 − 64z 4 4 282 x4 − 4x3 y + 6x2 y 2 − 4xy 3 + y 4 − 6x3 z − 18x2 yz − 27xy 2 z − 30y 3 z −9x2 z 2 + 45xyz 2 − 117y 2 z 2 + 54xz 3 + 108yz 3 290 27x4 + 36x2 y 2 + 36x2 yz + 32y 3 z − 126x2 z 2 + 12y 2 z 2 − 84yz 3 + 67z 4 310 x4 + 2x3 y − 3x2 y 2 − 4xy 3 + 4y 4 − 81xy 2 z − 81xyz 2 − 81y 2 z 2 − 54xz 3 + 54yz 3 + 81z 4 318 9x4 − 10x2 y 2 + y 4 − 28x2 yz + 12y 3 z + 20x2 z 2 − 4y 2 z 2 − 32yz 3 + 32z 4 430 81x4 + 54x2 y 2 − 7y 4 − 432x2 yz − 128y 3 z − 108x2 z 2 + 444y 2 z 2 + 64yz 3 + 32z 4 462 128x4 − 320x3 y + 264x2 y 2 − 44xy 3 − y 4 − 448x3 z + 48x2 yz + 492xy 2 z − 92y 3 z +840x2 z 2 + 12xyz 2 + 498y 2 z 2 − 972xz 3 − 972yz 3 + 567z 4 510 3x4 − 4x2 y 2 + y 4 + 18x2 yz − 10y 3 z + 14x2 z 2 + 18y 2 z 2 + 40yz 3 + 16z 4 By Lemma 13, only the curves corresponding to N = 178, 246, 258, 290, 318, 430, 510 are biel- liptic and only have a bielliptic involution u. The affine equations for the bielliptic quotients are N X0∗ (N )/hui 178 −4x + x2 + 32y − 12xy + 20y 2 − 2xy 2 − 4y 3 + y 4 = 0 246 17 − 162x + 81x2 − 260y + 324xy + 48y 2 − 32y 3 − 16y 4 = 0 258 −64 − 36x + 81x2 − 128y + 288xy + 84y 2 − 90xy 2 − 176y 3 + 41y 4 = 0 290 67 − 126x + 27x2 − 84y + 36xy + 12y 2 + 36xy 2 + 32y 3 = 0 318 32 + 20x + 9x2 − 32y − 28xy − 4y 2 − 10xy 2 + 12y 3 + y 4 = 0 430 32 − 108x + 81x2 + 64y − 432xy + 444y 2 + 54xy 2 − 128y 3 − 7y 4 = 0 510 16 + 14x + 3x2 + 40y + 18xy + 18y 2 − 4xy 2 − 10y 3 + y 4 = 0 have genus one and their j-invariants are − 117649 89 , 389017 16 , − 4096 43 , 2146689 145 , 3375 53 , − 4096 43 and 1771561 612 . They correspond to the elliptic curves in the statement. By the splitting of the jacobians and the equations of these curves, we obtain that all their automorphism groups have order 2. It is easy to check that the automorphism groups of the remaining curves are trivial.  Proposition 24. Among the curves of genus four X0∗ (202), X0∗ (262), X0∗ (354), X0∗ (366), X0∗ (370), X0∗ (426), X0∗ (546) and X0∗ (570), only X0∗ (370) is bielliptic. The corresponding quo- tient curve is the elliptic curve labeled as E370a1 and the automorphism group of X0∗ (370) has order 2. The automorphism groups of the curves X0∗ (202), X0∗ (262), X0∗ (354), X0∗ (426), X0∗ (546) and X0∗ (570) are trivial. The automorphism group of X0∗ (366) has order 2 and the quotient curve has genus 2. Proof. The splitting of J0∗ (N ) is: Q Q2 f2 ∈ New∗202 , Q J0∗ (202) ∼ i=1 Afi , Af1 ∼ E101a , dim Af2 = 3 , Q Q3 New∗262 ∗ Q Q J0 (262) ∼ i=1 Afi , Af1 ∼ E131a , Af2 ∼ E262b , f3 ∈ , dim Af3 = 2 , Q Q3 New∗177 Q Q J0∗ (354) ∼ i=1 Afi , Af1 ∼ E118a , Af2 ∼ E354b , f3 ∈ , dim Af3 = 2 , Q Q3 New∗183 Q Q J0∗ (366) ∼ i=1 Afi , Af1 ∼ E61a , Af2 ∼ E122a , f3 ∈ , dim Af3 = 2 , Q Q4 Q Q Q Q J0∗ (370) ∼ i=1 Afi , Af1 ∼ E370a , Af2 ∼ E185a , Af3 ∼ E185c , Af4 ∼ E37a , Q Q3 f3 ∈ New∗243 , Q Q J0∗ (426) ∼ i=1 Afi , Af1 ∼ E142a , Af2 ∼ E426b , dim Af3 = 2 , Q Q2 New∗273 Q J0∗ (546) ∼ i=1 Afi , Af1 ∼ E91a , f2 ∈ , dim Af2 = 3 , ∗ Q Q4 Q Q Q Q J0 (570) ∼ i=1 Afi , Af1 ∼ E57a , Af2 ∼ E190b , Af3 ∼ E285b , Af4 ∼ E570a . 15 In all cases, dim L2 = 1. Next, we show a generator Q2 (x, y, z, t) ∈ L2 : N Q2 202 −9t + x + 9ty − 2xy + y 2 + 9tz − xz − 8yz − 2z 2 2 2 262 7t2 − tx + x2 − 4ty − xy − 4tz − xz + yz 354 −42t2 − 5tx + 2ty − 10xy + 22y 2 + 48tz − 15xz + 6yz − 3z 2 366 −4t2 + x2 + 2xy − y 2 − 2z 2 370 −27t + 144x + 72ty + 32y 2 − 54tz − 40yz − 127z 2 2 2 426 −108t2 + 165tx + 100x2 + 78ty + 55xy − 137y 2 + 27tz − 30xz + 84yz − 72z 2 546 388t2 + 36tx + x2 − 68ty − 10xy + 9y 2 − 420tz − 34xz + 58yz + 125z 2 570 −327t2 − 232tx + 176x2 + 36ty + 64xy + 192y 2 + 70tz − 40xz + 188yz − 127z 2 Only X0∗ (370) could be bielliptic. In this case, the curve is trigonal (see [16, Proposition 1]) and dim L3 = 5. By computing a polynomial Q3 ∈ L3 that is not multiple of Q2 , we get Q3 (x, y, z, t) = −81t3 − 3024tx2 + 3456x2 y + 1056ty 2 − 2016x2 z − 4056tyz − 1408y 2 z +1479tz 2 + 248yz 2 + 4346z 3 . Since Q3 (−x, y, z, t) ∈ L3 , the curve X0∗ (370) is bielliptic by Lemma 13. Let us check this result. Set P (X, Y ) := Resultant (Q2 (X, Y, Z, 1), Q3 (X, Y, Z, 1), Z). More precisely, P (X, Y ) = −592 + 12952X 2 − 109468X 4 + 45796X 6 + (10816X 2 + 60978X 4 )Y +(3304 − 6112X 2 − 1763X 4 )Y 2 + (3712 − 26412X 2 )Y 3 +(908 + 5066X 2 )Y 4 + 1210Y 5 − 395Y 6 . The curve determined by the equation P (X, Y ) = 0 has genus 4. Hence, it is a plane model for X0∗ (370). The model admits the involution u : (X, Y ) 7→ (−X, Y ). Replacing X 2 with X, we obtain a genus one curve, whose j-invariant is 15438249/2960. Checking [7, Table1], the elliptic quotient has conductor 370 and label a1. The polynomials Q2 and Q3 show that u is the only nontrivial involution of the curve. It is clear that the remaining curves, except X0∗ (366), have trivial automorphism group. Looking at the polynomial Q2 for N = 366, we may ask whether one of the two linear maps (ω1 , ω2 , ω3 , ω4 ) 7→ ±(−ω1 , −ω2 , ω3 , ω4 ) comes from an involution u of X0∗ (366). After determining L3 , the answer is affirmative. Hence, Jac(X0∗ (366)) is isogenous over Q to E61a × E122a or Af3 . After checking which of the vector subspaces hω1 , ω2 i or hω3 , ω4 i provides a hyperelliptic curve, the right answer is Af3 , and an equation for the quotient curve X0∗ (366)/hui is Y 2 = X 6 − 6X 5 + 23X 4 − 42X 3 + 53X 2 − 24X + 4 .  Remark 25. It is expected that the automorphism group of X0∗ (N ) is trivial for a large enough N and, thus, the genera of the quotients curves by nontrivial involutions are bounded. The curve X0∗ (366) shows that if this bound exists, then it is at least 2. Proposition 26. The curves X0∗ (402), X0∗ (438), X0∗ (714), X0∗ (798), X0∗ (910), X0∗ (690), X0∗ (858) and X0∗ (870) are not bielliptic. 16 Proof. The splitting of J0∗ (N ) for the curves of genus 5 in the statement is: Q4 f4 ∈ New∗67 , Q Q Q Q J0∗ (402) ∼ i=1 Afi , Af1 ∼ E201c , Af2 ∼ E201a , Af3 ∼ E402a , dim Af4 = 2 , Q4 f4 ∈ New∗73 , Q Q Q Q J0∗ (438) ∼ i=1 Afi , Af1 ∼ E219a , Af2 ∼ E219c , Af3 ∼ E438a , dim Af4 = 2 , Q4 f4 ∈ New∗357 Q Q Q Q J0∗ (714) ∼ i=1 A fi , Af1 ∼ E238b , Af2 ∼ E102a , Af3 ∼ E714a , dim Af4 = 2 , Q4 f4 ∈ New∗133 , Q Q Q Q J0∗ (798) ∼ i=1 Afi , Af1 ∼ E399a , Af2 ∼ E57a , Af3 ∼ E798a , dim Af4 = 2 , Q4 f4 ∈ New∗910 , Q Q Q Q J0∗ (910) ∼ i=1 A fi , Af1 ∼ E91a , Af2 ∼ E455a , Af3 ∼ E65a , dim Af4 = 2 . In all cases, dim L2,1 = 0. For N = 910, also dim L2,2 = 0. For the curves of genus 6, the splitting of J0∗ (N ) is: Q4 f3 ∈ New∗345 , f4 ∈ New∗115 , Q Q Q J0∗ (610) ∼ i=1 Afi , Af1 ∼ E138a , Af2 ∼ E690a , dim Af 3 = 2 dim Af4 = 2 , Q4 f3 ∈ New∗429 , f4 ∈ New∗898 Q Q Q J0∗ (858) ∼ i=1 Afi , Af1 ∼ E143a , Af2 ∼ E290a , dim Af3 = 2 dim Af4 = 2 , In all cases, dim L2,1 = 1. Hence, dim L2,1 = 1 < dim L2 = 6. For N = 898, also dim L2,1 = 1. Finally, the splitting for the curve X0∗ (870) of genus seven is: Q4 f4 ∈ New∗435 , Q Q Q Q J0∗ (870) ∼ i=1 Afi , Af1 ∼ E145a , Af2 ∼ E290a , Af3 ∼ E58a , dim Af4 = 4 . In this case, dim L2,1 = dim L2,2 = 4 < dim L2 = 10.  As a consequence of the previous results, we obtain the statement of Theorem 1 for N even. Corollary 27. For N even, the curve X0∗ (N ) is bielliptic exactly for the thirteen values of N in the set {106, 122, 158, 166, 178, 246, 258, 290, 318, 370, 390, 430, 510} . ∗ For these values of N automorphism group has order 2 when gN > 2, otherwise it is the Klein group. 5 Quadratic points Let us now prove Theorem 2. We know by [15] that if N is square-free and X0∗ (N ) is hyper- ∗ elliptic, then gN = 2. On the other hand, a genus two curve defined over a number field K is hyperelliptic over K and, thus, all genus two curves X0∗ (N ) are hyperelliptic over Q. The set ∗ of values of N in Theorem 2 are those for which gN = 2 and those such that X0∗ (N ) is bielliptic ∗ ∗ and gN ≥ 3. This is due to the fact that, when gN ≥ 3, the quotient curve is always an elliptic curve with rank equal to 1 (see [7, Table1]). Hence, all these values of N are exactly the values for which Γ2 (X0∗ (N ), Q) is infinite (cf. [5, Theorem 2.14]). 17 6 Appendix ∗ Here we list the values N such that gN ≤ 2. The table for genus 2 reproduces the one in [14]. The tables for genus 0 or 1 are taken from [12]. We note that the value N = 141, which does not appear in Proposition 1.1 of [12], is included in the appendix of this paper and here. ∗ gN =0 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 . ∗ gN = 1 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 . ∗ gN = 2 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 . References [1] R. D. M. Accola. Topics in the theory of Riemann surfaces, volume 1595 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1994. [2] 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. [3] M. H. Baker and Y. Hasegawa. Automorphisms of X0∗ (p). J. 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Bielliptic intermediate modular curves . Preprint, 2017. [20] A. P. Ogg. Hyperelliptic modular curves. Bull. Soc. Math. France, 102:449–462, 1974. 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 19