This is the accepted version of the journal article: Bars Cortina, Francesc; Kamel, Mohamed; Schweizer, Andreas. «Bielliptic quotient curves of X0(N)». Mathematics of Computation, Vol. 92, Issue 340 (2023), p. 895-929. DOI 10.1090/mcom/3800 This version is available at https://ddd.uab.cat/record/270703 under the terms of the license Bielliptic quotient modular curves of X0(N ) Francesc Bars∗, Mohamed Kamel and Andreas Schweizer Abstract Let N ≥ 1 be a non-square free integer and let WN be a nontrivial 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 of genus gX ≥ 2 defined over a number field K. We also assume that X has at least one K-rational point. (All curves we investigate in this paper have this property.) The curve X is called bielliptic if it has an involution v such that X/v has genus 1. More precisely, X is called bielliptic over L for an extension L of K if v is defined over L. In this case X/v inherits an L-rational point from X and hence is an elliptic curve over L. Although for every finite extension L of K the set X(L) of L-rational points of X is finite by a famous theorem by Faltings, the set [ Γ2 (X, K) := X(L) [L:K]≤2 of quadratic points over K can be infinite. This happens for example if X is bielliptic over K and the elliptic curve X/v has positive rank over K, because then over each of the infinitely many K-rational points of X/v there is at least one quadratic point on X. By a similar argument Γ2 (X, K) is infinite if X is hyperelliptic, because then the hyperelliptic involution u is defined over K and X/u ∼ = P1 . Less obvious is that the converse also holds, i.e. Γ2 (X, K) is infinite if and only if X is hyperelliptic or X is bielliptic over K with elliptic quotient curve of positive K-rank. For details see [Bar18], also for the more complicated situation where X(K) = ∅. Note that the similar statement in [HS91] is slightly weaker in that (at least for gX ≤ 5) it is not stated over which field the bielliptic involution is defined. Therefore an investigation which curves of a certain type or in a certain family have infinitely many quadratic points often starts with determining the bielliptic ones, the determination of the hyperelliptic ones usually having been done decades earlier. This paper is the final installment in a series that investigates the two problems, biellipticity and infinitely many quadratic points over Q, for curves X0 (N )/WN where X0 (N ) is a modular ∗ First author is supported by PID2020-116542GB-I00. Key words: modular curve; Atkin-Lehner involution; bielliptic curve; quadratic points. 2020 Mathematics Subject Classification: 11G18; 14H45. 1 curve of Hecke type and WN is a subgroup of the group B(N ) of Atkin-Lehner involutions of X0 (N ). Note that the AL-involutions are defined over Q, and so X0 (N )/WN inherits a Q-rational cusp from X0 (N ). For the trivial subgroup WN = {1} it was already [Bar99] which determined all bielliptic curves X0 (N ) and those with infinite Γ2 (X0 (N ), Q). Later [Jeo18] solved the same problem for the curves X0+ (N ) = X0 (N )/wN where wN is the Fricke involution. Then [BG19] and [BG20] treated the curves X0∗ (N ) = X0 (N )/B(N ) for square-free resp. non-square-free N . After the case of proper subgroups WN was settled in [BGK20] for square-free N , we now deal with proper subgroups WN when N is not square-free. In Section 3, specializing a result by Harris and Silverman we note that X0 (N )/WN only has a chance of being bielliptic when X0∗ (N ) is bielliptic, hyperelliptic or of genus at most 1. So with [BG20], [Has97] and [GL98] we can restrict the candidates to a finite, explicit set of N , albeit with usually 7 curves for each such N . González and the first author in [BG20] provided a computational method by use of a theorem of Petri to decide whether a concrete curve is bielliptic or not, for modular curves after computing the Jacobian and the Galois conjugation basis. But the computations are tedious and not transparent to the reader. Therefore we first use several old and new criteria (see Section 4.1) to quickly discard as many candidates as possible. A bielliptic curve of genus smaller than 6 might have several bielliptic involutions, and, worse, it can happen that none of these is defined over Q. But if N is square-free, then all automorphisms of X0 (N )/WN are known to be defined over Q, which is quite helpful. For non-square-free N this does not necessarily hold. On the other hand, most non-square-free candidates N are divisible by 4 or 9, and in these cases there exist additional involutions of X0 (N ) coming from the normalizer of Γ0 (N ) in GL2 (R). This gives us more candidates for bielliptic involutions. Indeed, for practically all bielliptic X0 (N )/WN with N not square-free we can write down explicit bielliptic involutions, as matrices if one wants to. Our first main result is Theorem 1.1. Let N > 1 be a non square-free integer not a power of a prime. Assume that the genus of the modular curve X0 (N )/WN is at least 2 for a nontrivial subgroup WN of B(N ) different from hwN i. The curve X0 (N )/WN , denoted by the pair (N, WN ) is a bielliptic curve if and only if it appears below: (i) It is a pair (N, WN ) such that |WN | = 2 and N is in the set {40, 48, 52, 63, 68, 72, 75, 76, 80, 96, 98, 99, 100, 108, 124, 188}, or it is a pair (N, WN ) such that |WN | = 4 and N is in the set {84, 90, 120, 126, 132, 140, 150, 156, 220}. All these quotient modular curves are bielliptic over Q with an elliptic quotient given by X0∗ (N ) of genus 1, 2 (ii) or it is one of the following 29 pairs, ordered by genus Genus (N, WN ) 2 (44, hw4 i), (60, hw20 i), (60, hw4 , w3 i) 3 (56, hw8 i), (60, hw4 i) 4 (60, hw3 i), (60, hw5 i), (112, hw7 i), (168, hw3 , w56 i) 5 (84, hw4 i), (88, hw11 i), (90, hw9 i) (117, hw9 i), (120, hw15 i), (126, hw63 i), (168, hw8 , w7 i), (168, hw7 , w24 i), (180, hw4 , w9 i), (184, hw23 i), (252, hw4 , w63 i) 6 (104, hw8 i), (168, hw8 , w3 i) 7 (120, hw24 i), (136, hw8 i), (252, hw9 , w7 i) 9 (126, hw9 i), (171, hw9 i), (252, hw4 , w9 i) 10 (176, hw16 i) Remark 1.2. It turns out that almost all of the curves listed in Theorem 1.1 are bielliptic over Q. The only exceptions√ are the isomorphic curves X0 (126)/w63 and X0 (252)/hw4 , w63 i which are bielliptic over Q( −3). For more information see the tables in Appendix B where we list bielliptic involutions and the conductor of the corresponding elliptic quotient curve as well as the splitting of the Jacobian of X0 (N )/WN . The hyperelliptic X0 (N )/WN are already known (see [FH99]). Checking which of the non- hyperelliptic ones satisfy the condition from the theorem discussed at the beginning of this Introduction we obtain our second main result. Theorem 1.3. Let N > 1 be a non square-free integer not a power of a prime. Assume that the genus of the modular curve X0 (N )/WN is at least 2 for a nontrivial subgroup WN of B(N ) different from hwN i. Then the set Γ2 (X0 (N )/WN , Q) is infinite if and only if (N, WN ) appears in the following list: (i) It is a pair (N, WN ) that is an hyperelliptic curve, determined in [FH99] that we reproduce for the convenience of the readers: Genus (N, WN ) 2 (40, hw8 i), (40, hw5 i), (44, hw4 i), (48, hw16 i), (48, hw3 i), (52, hw4 i), (54, hw2 i), (60, hw20 i), (60, hw4 , w3 i), (60, hw5 , w12 i), (72, hw8 i), (84, hw4 , w3 i), (84, hw4 , w21 i) (84, hw3 , w28 i), (84, hw7 , w12 i), (90, hw9 , w5 i), (90, hw9 , w10 i), (90, hw2 , w45 i) (90, hw5 , w18 i), (100, w4 ), (120, hw8 , w15 i), (120, hw24 , w40 i), (126, hw2 , w63 i) (126, hw18 , w14 i), (132, hw4 , w11 i), (140, hw4 , w35 i), (150, hw6 , w50 i) (156, hw4 , w39 i) 3 (56, hw8 i), (60, hw4 i), (60, hw60 i), (63, hw9 i), (72, hw9 i), (120, hw5 , w24 i), (126, hw9 , w7 i), (126, hw9 , w14 i), 4 (60, hw12 i), (168, hw24 , w56 i), 5 (92, hw4 i). (ii) or for bielliptic curves that are not hyperelliptic corresponding to a pair (99, WN ) with |WN | = 2. 3 We observe that among the elliptic curves X0∗ (N ) the 38 ones with square-free N all have positive rank, whereas the 25 ones with non-square-free N with the exception of X0∗ (99) all have rank 0. We used also codes implemented in [Math] and [Magma] for obtaining and supporting above results. Such codes are available to math community at Quotient Modular Curves folders in https://github.com/FrancescBars 2 Notation Let N > 1 be an integer. We fix once and for all the following notation. (i) We denote by B(N ) the group of the Atkin-Lehner involutions of X0 (N ). So, the order of B(N ) is 2ω(N ) , where ω(N ) is the number of different primes dividing N . (ii) For N 0 |N , with (N 0 , N/N 0 ) = 1, B(N 0 ) denotes the subgroup of B(N ) formed by the Atkin-Lehner involutions wd such that d|N 0 and (d, N/N 0 ) = 1. In general, WN denotes a nontrivial 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 NewW N 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) Let h ∈ S2 (Γ0 (N )) be an eigenform of the form d|N/M cd f (q d ) for some f ∈ NewM P with M |N and cd ∈ Z. Since for every divisor d of N/M there is a morphism Bd from d 0 (N ) defined over Q sending every cusp form g ∈ S2 (M ) to g(q ) ∈ S2 (N ), the J0 (M ) to JP morphism d|N/M cd Bd provides an abelian variety Ah defined over Q attached to h and Q-isogenous to the abelian variety Af attached by Shimura to f . This abelian variety can be defined as the optimal quotient of J0 (N ) such that the pullback of Ω1Ah /Q is the vector space generated by the Galois conjugates of h(q) dq/q with rational q-expansion. This definition determines the Q-isomorphism class of Ah , although we are only interested in its Q-isogeny class. 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. That is, ψ(N ) = N p|N (1 + p−1 ), where Q the product is extended to all primes p dividing N . (x) We write #(w, X) for the number of fixed points of the automorphism w on the curve X.   1 1/d (kd) (xi) We write Sd = , and by wd ∈ GL2 (Z) a matrix which corresponds to a 0 1 lifting on level kd of the Atkin-Lehner involution wd of X0 (d) with d ≥ 2 and k ≥ 2 integers, (thinking wd ∈ GL2 (Z) with determinant d). 4 (xii) To be able to simultaneously describe the Atkin-Lehner involutions of different curves Q (for example in the tables in Appendix A) we use the notation $i = wpi ei where N = si=1 pi ei and p1 < p2 < . . . < ps are the different primes dividing N . Recall, if X0 (N )/WN is bielliptic, there is an involution u ∈ Aut(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 a number field K (over Q if u is unique). Since X0 (N )/WN (Q) is not empty, the genus 1 curve has a rational point and, therefore, it is an elliptic curve E over K, called a bielliptic quotient of X0 (N )/WN . In particular, such an elliptic curve is a K-isogeny factor for J0 (N )WN . 3 Selecting candidate bielliptic curves X0∗(N ) The starting point of our selection is based on the following result, which follows from [HS91, Proposition 1] by considering the natural projection map X0 (N )/WN → X0∗ (N ). Lemma 3.1. Let N be an integer, and take X0 (N )/WN of gW ≥ 2. If it is bielliptic, then X0∗ (N ) is bielliptic, hyperelliptic or has genus at most one. We assume once and for all N not square-free and N not a power of a prime. Theorem 3.2. Consider X0 (N )∗ with N a non-square free level not a power of a prime, then ∗ (i) (González-Lario, [GL98]) gN = 0 if and only if N ∈ {12, 18, 20, 24, 28, 36, 44, 45, 50, 54, 56, 60, 92} ∗ (ii) (González-Lario, [GL98]) gN = 1 if and only if N ∈ {40, 48, 52, 63, 68, 72, 75, 76, 80, 84, 90, 96, 98, 99, 100, 108, 120, 124, 126, 132, 140, 150, 156, 188, 220} ∗ (iii) (Hasegawa, [Has95]) gN = 2 if and only if N ∈ {88, 104, 112, 116, 117, 135, 147, 153, 168, 180, 184, 198, 204, 276, 284, 380} ∗ (iv) (Hasegawa, [Has97]) gN > 2 and X0∗ (N ) is hyperelliptic if and only if N appears next: ∗ gN N 3 136; 171; 207; 252; 315; 4 176; 5 279. (v) (Bars-González, [BG20]) X0∗ (N ) is bielliptic if and only if N appears in the following table ∗ gN N 2 88; 112; 116; 153; 180; 184; 198; 204; 276; 284; 380; 3 144; 152; 164; 189; 196; 207; 234; 236; 240; 245; 248; 252; 294; 312; 315; 348; 420; 476; 4 148; 160; 172; 200; 224; 225; 228; 242; 260; 264; 275; 280; 300; 306; 342; 5 364; 444; 495; 7 558. 5 4 Selecting possible bielliptic quotient curves 4.1 General criteria to discard bielliptic curves Here we collect some general criteria that allow to prove comparatively easily that certain curves are not bielliptic. Lemma 4.1. (special form of the Castelnuovo inequality [Acc94, Theorem 3.5]) Let φ : X → Y be a morphism of degree d of curves. If X has a bielliptic involution v, then g(X) ≤ dg(Y ) + d + 1 or the morphism φ factors over X/v. In particular: A hyperelliptic curve of genus g ≥ 4 cannot be bielliptic. A trigonal curve of genus strictly bigger than 4 cannot be bielliptic. A curve of genus g ≥ 6 has at most one bielliptic involution. Proposition 4.2. [JKS20, Proposition 3.2] Let X be a bielliptic curve of genus g ≥ 6 defined over a field K of characteristic 0. Then the bielliptic involution is unique, defined over K and lies in the center of Aut(X). Proof. The uniqueness was just mentioned in Lemma 4.1. Acting on this bielliptic involution with Aut(X) by conjugation resp. with Gal(K/K), the uniqueness implies the other properties. The following Lemmas 4.3 and 4.4 appeared in [Sch01] and [JKS20] respectively for genus ≥ 6. Here we present a slightly different proof for general genus. Lemma 4.3. Let w be an involution of X with more than 8 fixed points. Then either w is a bielliptic involution or X is not bielliptic Proof. Let g and h be the genera of X and X/w. Since w has more than 8 fixed points, by the Hurwitz formula we have g > 2h + 3. If v is a bielliptic involution, from Castelnuovo we get the contradiction that g cannot be bigger than 2h + 3, unless the two involutions factor over a common curve, i.e. are the same. Lemma 4.4. Let X be a curve of genus g with a bielliptic involution v and let G be a subgroup of Aut(X) such that the curve Y = X/G has genus h ≥ 2. (a) If the map φ : X → Y is ramified, i.e. if g − 1 > |G|(h − 1), and g ≥ 6, then Y must be hyperelliptic and v induces the hyperelliptic involution on Y . (b) (unramified covering criterion) If Y is not hyperelliptic, then it must be bielliptic and the map φ : X → Y must be unramified, i.e. g − 1 = |G|(h − 1). Proof. (a) Obviously v 6∈ G because h > 1. Since g ≥ 6, we know from Proposition 4.2 that v is central. So v induces an involution ve on Y , and G induces a group G e (isomorphic to G) of automorphisms on X/v. If φ is ramified, at least one nontrivial element of G has at least one fixed point. So the same holds for G.e Hence the Hurwitz formula for the covering from X/v to (X/v)/G e = Y /ev shows that the latter curve has genus 0. (b) Since Y is not hyperelliptic we have h ≥ 3. Assume that φ is ramified. Then the Hurwitz formula implies g ≥ 6, and part (a) leads to a contradiction. 6 If X0 (N )/WN is bielliptic over Q and p - N , then the morphism reduces modulo p and we have that |X0 (N )/WN (Fpn )| ≤ 2|E(Fpn )| (4.1) for all n ≥ 1, which can be computed. In particular we reproduce similar results as [BG19, Lemmas 5,7]: Lemma 4.5. Assume X0 (N )/WN is bielliptic over Q, and p - N . Then the following equality ψ(N ) 2|E(Fp2 )| − 1 ψ(N ) 2(p + 1)2 − 1 holds: ≤ 12 · . In particular we have, ≤ 12 · . |WN | p−1 |WN | p−1 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 |, we get |X0 (N )(Fp2 )| ≤ 2 · |WN ||E(Fp2 )|. Similarly, for optimal quotients with conductor M = N for elliptic quotient (related with the strong Weil parametrization) it is easy to derive the following lemma. Lemma 4.6. 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]. Theorem 4.7. Let X be a curve over a field of characteristic 0. Then (a) The stabilizer in Aut(X) of any point of X is cycylic. (b) Distinct involutions in Aut(X) have disjoint fixed points. Proof. (a) is proved in [FK80, Corollary III.7.7], and (b) follows immediately from (a). Proposition 4.8. Let X be a curve of genus g at least 6. Assume that Aut(X) has a subgroup H of order 2t such that 2t does not divide 2(g − 1). Then either the bielliptic involution of X is contained in H or X is not bielliptic. Proof. Let v (outside H) be the bielliptic involution. Since v commutes with the elements of H, H acts on the 2(g − 1) fixed points of v. If |H| does not divide 2(g − 1), there must be an orbit whose length is not |H|. So there must be an involution in H that fixes a point in this orbit. But v also fixes this point, contradicting Theorem 4.7. The curves X0 (N )/WN always have such a subgroup H (isomorphic to B(N )/WN ), and we can discard directly even genus quotient curves when B(N )/WN is of order at least 4. 4.2 Criteria for genus 5 bielliptic curves For curves of genus g ≤ 5 it can be difficult to decide whether they are bielliptic, because there might be several bielliptic involutions, but none of them is guaranteed to be defined over Q. The following results will be helpful for several of the curves we will encounter. Lemma 4.9. [Acc94, p.50] A genus 5 curve that is a degree two covering of a hyperelliptic genus 3 curve is either hyperelliptic or bielliptic. Lemma 4.10. [KMV11, Lemma 2.3] Let X a bielliptic curve of genus 5. Then it has 1, 2, 3 or 5 bielliptic involutions, and these involutions commute and generate a group of exponent 2 and order 2, 4, 8 and 16 respectively. Moreover, the product of any two bielliptic involutions is an involution whose quotient curve is of genus 3 and hyperelliptic. 7 Lemma 4.11. Let X be a bielliptic genus 5 curve with an involution u, such that X/u has genus 2. (a) If X has an odd number of bielliptic involutions, then there is a unique one among them, call it v, that commutes with u. If moreover u is defined over Q, then v is also defined over Q. (b) If X has exactly 2 bielliptic involutions, neither one of them will commute with u. Proof. The involution u acts on the bielliptic involutions by conjugation, so they come in pairs of conjugated ones plus the ones which commute with u. This already shows that there is at least one such v (resp. two or none if X has exactly 2 bielliptic involutions). Now we show that there can be at most one such v. Since v commutes with u and the map X → X/u is ramified, we can argue as in the proof of Theorem 4.4 (a) with the role of G being played by hui. Again we obtain that v must induce the hyperelliptic involution on X/u. So the only possibility for a bielliptic involution other than v that commutes with u would be uv. But they cannot both be bielliptic, because then u = (uv)v would have quotient genus 3 by Lemma 4.10. If u is defined over Q and commutes with v, then from the action of the absolute Galois group of Q it also commutes with all Galois conjugates of v, which by the uniqueness must be equal to v. Lemma 4.12. Let X be a curve of genus 5 with an involution w such that X/w is of genus 3 and non-hyperelliptic. If X is bielliptic, then X has exactly one or exactly 3 bielliptic involutions. Proof. First we treat the case where X has 5 bielliptic involutions. Then by [KMV11, Lemma 2.3 and Remark 3.2] the bielliptic involutions generate a group H of order 16 in which all other elements are involutions which have a quotient that is hyperelliptic of genus 3. In particular, w is not in H. Pick a bielliptic involution v1 that does not commute with w and define v2 as its conjugate under w. If w commutes with all bielliptic involutions, just pick any two v1 and v2 . In either case w commutes with the involution v1 v2 , which has quotient genus 3. By a classical result (see for example [Acc94, Lemma 5.10]) a genus 3 curve that is a degree 2 cover of a genus 2 curve must be hyperelliptic. So since X/w is not hyperelliptic, X/hv1 v2 , wi cannot have genus 0 or 2, and hence has genus 1. From g(X/w) + g(X/v1 v2 ) + g(X/v1 v2 w) = g(X) + 2g(X/hv1 v2 , wi) = 7 (see for example [Acc94, p.49]) we see that X/v1 v2 w also has genus 1. Therefore v1 v2 w is in H. With v1 v2 in H we get the contradiction w ∈ H. So we have proved that X has at most 3 bielliptic involutions. Now assume that X has exactly 2 bielliptic involutions. Call them v1 and v2 . By the same argument as before we get a third bielliptic involution. Corollary 4.13. Let X be a curve of genus 5 over Q that has involutions u and w as in the two lemmas. with u defined over Q. If X is bielliptic, then it has one or three bielliptic involutions and at least one bielliptic involution is defined over Q. More precisely, if it has 3 bielliptic involutions, then there is a unique one that commutes with u, and that one is defined over Q. (The other two might be defined over Q or not.) 8 4.3 Formulas for the genus of a quotient curve by certain involutions The most natural way to prove that a curve is bielliptic is to exhibit a bielliptic involution. We will do this for practically all the bielliptic curves X0 (N )/WN . The most obvious candidates are the Atkin-Lehner involutions outside WN . If N is divisible by 4 or 9, then X0 (N ) has additional involutions, which under certain conditions induce involutions on X0 (N )/WN . See below for details. When v is an involution of X0 (N ) that induces an involution ve on X0 (N )/WN , it is usually quicker not to bother about the fixed points of ve on X0 (N )/WN but to determine the genus of (X0 (N )/WN )/v from the fixed points of the elements of G = hWN , vi on X0 (N ) by applying the Hurwitz formula to the covering X0 (N ) → X0 (N )/G. In the situations we encounter the group G is of the form G ∼ = Z/2 × · · · × Z/2. Then by Theorem 4.7 the stabilizer of a point is trivial or Z/2. Moreover, all involutions in G have disjoint fixed points, and the Hurwitz formula takes the following simple shape. Lemma 4.14. Let G be a subgroup of Aut(X0 (N )) in which all the nontrivial elements are involutions. Then the fixed points of these involutions are disjoint and the genus of X0 (N )/G is obtained by the formula X |G|(2g(X0 (N )/G) − 2) + #(w, X0 (N )) = 2g(X0 (N )) − 2. w∈G Formulas for the number of fixed points of the Atkin-Lehner involutions on X0 (N ) are in [Ogg74]. For the other involutions we will review and expand the known results in this section.   1 1/k Recall Sk = . The following result is well-known (see [FH99] or [Bar99]). 0 1 Proposition 4.15. Let N = 2α M with α ≥ 2 and M odd. (a) Then S2 is an involution of X0 (N ), defined over Q, and commutes with all Atkin-Lehner involutions wr for which r is odd. Hence, V2 = S2 w2α S2 also is an involution of X0 (N ), defined over Q, and commutes with all wr for which r||M . (b) If α ≥ 3, then V2 also commutes with w2α . So V2 w2α is an involution, and consequently S2 w2α has order 4. In fact, hS2 , w2α i ∼ = D4 . (c) If α = 2, then hS2 , w4 i is non-abelian of order 6 with V2 = S2 w4 S2 = w4 S2 w4 being the third involution and S2 w4 and w4 S2 having order 3. Lemma 4.16. [Bar99, Proposition 3.5] If N = 2α M with α ≥ 2 and M odd, then X0 (N )/w2α S2 w2α = X0 (N/2). Proof. An easy calculation shows that w2α S2 w2α lies in Γ0 (N/2) but not in Γ0 (N ). Lemma 4.17. Let u and v be two commuting involutions on a curve X. Then uv is also an involution and #(uv, X) = 2#(u, X/v) − #(u, X). Proof. By Theorem 4.7 , the fixed points of the three involutions u, v and uv are disjoint, and each fixed point has ramification index 2 in the degree 4 covering X → X/hu, vi. This implies that every fixed point of u or uv must be ramified in X/v → X/hu, vi. Con- versely, if a point of X/v is ramified in X/v → X/hu, vi, then the two points of X lying above it must have ramification group hui or huvi. So, all in all #(u, X) + #(uv, X) = 2#(u, X/v). 9 Lemma 4.18. Let N = 2α M with α ≥ 2 and M odd. Also let r||M . (a) [Bar99, Proposition 3.9] #(V2 , X0 (N )) = #(w2α , X0 (N )) and #(V2 wr , X0 (N )) = #(w2α wr , X0 (N )). (b) #(S2 , X0 (N )) = #(w2α S2 w2α , X0 (N )) = (2g(X0 (N )) − 2) − 2(2g(X0 (N/2)) − 2). (c) [Bar99, Proposition 3.6] #(S2 wr , X0 (N )) = #(w2α S2 w2α wr , X0 (N )) = 2#(wr , X0 (N/2)) − #(wr , X0 (N )). (d) [Bar99, Proposition 3.10] If α ≥ 3, then #(V2 w2α , X0 (N )) = 2#(S2 , X0 (N/2)) − #(S2 , X0 (N )) and #(V2 w2α wr , X0 (N )) = 2#(S2 wr , X0 (N/2)) − #(S2 wr , X0 (N )). Proof. Note that conjugate involutions have the same number of fixed points. This proves (a) and together with Lemma 4.16 also (b). The other formulas are special cases of Lemma 4.17, also using Lemma 4.16.   1 1/3 Lemma 4.19. Let 9||N and S3 = . 0 1 (a) S3 √ normalizes Γ0 (N ) and induces an automorphism of X0 (N ) of order 3 defined over Q( −3). Its Galois conjugate is S32 . Moreover, S3 commutes with the Atkin-Lehner involutions wr with r ≡ 1 mod 3, whereas for r ≡ 2 mod 3 we have wr S3 = S32 wr and w9 S3 has order 3. (b) V3 = S3 w9 S32is an involution of X0 (N ). With respect to Atkin-Lehner involutions we V3 wr if r ≡ 1 mod 3 or r = 9 and have wr V3 = V3 w9 wr if r ≡ 2 mod 3 Moreover, if r ≡ 2 mod 3 then hV3 , wr i ∼ = D4 and V3 wr has order 4 with (V3 wr )2 = w9 . √ √ (c) V3 as an involution of X0 (N ) is defined over Q( −3). Its Gal(Q( −3)/Q)-conjugate is V3 w9 . In particular, V3 and V3 w9 have the same number of fixed points on X0 (N ). (d) More generally we have #(V3 w9 , X0 (N )) = #(V3 , X0 (N )) = #(w9 , X0 (N )) and for r ≡ 1 mod 3 also #(V3 w9 wr , X0 (N )) = #(V3 wr , X0 (N )) = #(w9 wr , X0 (N )). (e) V3 as an involution of X0 (N )/W is defined over Q if and only if w9 ∈ W . Proof. Most of this (and some more) is already in [Bar99], [FH99] or [Bar08]. So we only prove (c). √ V3 = S3 w9 S32 is √ defined over Q( −3) because S3 is. Now let σ be the nontrivial Galois automorphism of Q( −3). Then σ(V3 ) = σ(S3 w9 S32 ) = S32 w9 S3 . So V3 σ(V3 ) = w9 (w9 S3 )3 = w9 by part (a). Remark 4.20. From Lemma 4.19(e), we obtain the result in [BG20] that V3 as an involution of X0∗ (N ) is always defined over Q. 10 4.4 Useful isomorphisms on non-square free quotient curves For later use we refine [FH99, Proposition 4]. Proposition 4.21. Suppose 4||N and write N = 4M . Let W 0 be a subgroup of B(N ) generated by w4 , wm1 , . . . , wms with mi ||M . Then we have X0 (N )/W 0 ∼ = X0 (N )/hS2 w4 S2 , wm1 , . . . , wms i = X0 (N )/hw4 S2 w4 , wm1 , . . . , wms i = X0 (2M )/hwm1 , . . . , wms i. Hence if A ∈ GL2 (R) is a bielliptic involution of X0 (2M )/hwm1 , . . . , wms i, then S2 AS2 normalizes hΓ0 (N ), W 0 i and induces a bielliptic involution on X0 (N )/W 0 . Proof. The isomorphism comes from conjugating with S2 , the first equality from part (c) of Proposition 4.15 and the final equality from Lemma 4.16. Proposition 4.22. [FH99, Proposition 5] Assume 9||N . Let W 0 be a subgroup of B(N ) e(n ) generated by wn1 , . . . , wnt (ni ||N ) and let W 00 = h{wni w9 i }i∈{1,...,t} i where e(m) = 0 if m ≡ 1 mod 3 or if 9||m and m/9 ≡ 1 mod 3, and e(m) = 1 otherwise. Then V3 induces an isomorphism X0 (N )/W 0 ∼= X0 (N )/W 00 . 5 Jacobian decomposition, field of bielliptic involutions, Petri theorem. 5.1 On the field where bielliptic involutions may be defined Let X0 (N )/WN be a quotient curve. We want to control if the automorphism, or more con- cretely if a candidate to bielliptic involution is defined over Q or a number field. In order to control the number field K (when gWN ≤ 5), we have the following results in [BG20]: Proposition 5.1. Let A be a modular abelian variety defined over Q such that A ∼ m ni Q Q i=1 Afi for some fi ∈ NewNi , where Afi are pairwise non-isogenous over Q. All endomorphisms of A are defined over Q if, and only if, for every nontrivial quadratic Dirichlet character χ, the newform fi ⊗ χ is different from any Galois conjugates of fj for all i and j. Remark√ 5.2. If χ is the quadratic Dirichlet character attached to the quadratic number field K = Q( D), then there is an isogeny between the abelian varieties Af and Af ⊗χ defined over K. Also, the following result specific for modular forms non-corresponding to elliptic curves [Pyl04] clarifies the possible elliptic quotient that could appear: Proposition 5.3. When dim Af > 1 and f does not have complex multiplication (CM), i.e. f 6= f ⊗ χ for all quadratic Dirichlet characters, a necessary condition for Af to have an elliptic quotient over Q is ap (f )2 ∈ Z (the p-th Fourier coefficient of the modular form f ) for all primes p. 11 5.2 On the Jacobian decomposition of quotient modular curves We recall that the Q-decomposition for J0 (N ) has 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 an 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 , and next results allow us, once fixed (N, WN ), to determine a basis of S2 (N )WN and, in particular, the splitting of J0 (N )WN (see [BG20, Lemma 2.1, Prop.2.2]). Lemma 5.4. Let M and N be positive integers such that M |N . Let M1 be a positive di- visor of M such that gcd(M, M/M1 ) = 1 and let d be a positive divisor of N/M such that gcd(M1 d, N/(M1 d)) = 1. If f ∈ S2 (Γ0 (M )) is an eigenvector of the Atkin-Lehner involution wM1 with eigenvalue ε(f ) and ε ∈ {−1, 1}, then f (q) + ε d f (q d ) ∈ S2 (Γ0 (N )) is an eigenvector of the Atkin-Lehner involution wM1 d with eigenvalue ε(f ) · ε. Proposition 5.5. Assume that N = pk · M , where k ≥ 1, p is a prime and M is an integer coprime to p. For 0 ≤ i < k, let f ∈ S2 (Γ0 (pi ·M ))W be such that wpi (f ) = ε·f with W ≤ B(M ) (clearly ε = 1 when i = 0). Let S be the vector subspace of S2 (Γ0 (pk · M ))W generated by the k − i + 1 linearly independent, Q-isogenous to f , eigenforms {f, Bp (f ), · · · , Bpk−i (f )}. Then, (i) The following normalized eigenforms g0 = (1 + pBp )k−i f, · · · , gj = (1 + pBp )k−i−j (1 − pBp )j f, · · · , gk−i = (1 − pBp )k−i f , are a basis of S (recall Bp is the morphism sending a modular form g(q) to g(q p )). (ii) Every gj is an eigenvector of wpk with eigenvalue (−1)j ε. 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 necessarily fixed 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/M }. The integer mf is determined by using Lemma 5.4 and Proposition 5.5. Read readme.md file in https://github.com/FrancescBars/Magma-functions-on-Quotient-Modular-Curves Jacobian decomposition allows us to compute |X0 (N )/WN (Fpn )| for all p - N thanks to the Eichler-Shimura congruence, see a MAGMA function code FpnpointsQuotientCurve in funcions.m and examples in the Readme.md in the github folder https://github.com/FrancescBars/Magma-functions-on-Quotient-Modular-Curves 12 5.3 The application of a result of Petri If X0 (N )/WN is hyperelliptic, we know an equation (see [FH99], [Has95]) and MAGMA com- putes the automorphism group over Q. In this case, if X0 (N )/WN has a non-hyperelliptic involution over Q, we can compute the genus of the quotient curve by using [Ogg74, Propo- sition 1] and determine if X0 (N )/WN is bielliptic or not over Q. For finite extensions of K we need to deal with the decomposition of the Jacobian and study the endomorphism algebra following Proposition 5.1 and to compute it in such number field, which is a quadratic field. MAGMA computes such automorphism group over quadratic fields. Thus for hyperelliptic quotient modular curves we can decide if they are bielliptic or not. Consider a non-hyperelliptic curve X 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, [BG19, Lemma 13] characterizes the existence of a bielliptic involution of X0∗ (N ) with N square-free for non-hyperelliptic curves. Later, [BG20, Proposition 2.6] generalizes this result to any non-hyperelliptic curve of genus > 2. K Proposition 5.6. With the above notation, assume that Jac(X) ∼ E m × A, where E is an elliptic curve and A an abelian variety such 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 that ω1 , · · · , ωm and ωm+1 , · · · , ωg are bases of the pullback 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 , (5.1)   A 0 where B is the matrix ∈ GLg (K) and L0i = {Q((x1 , x2 , · · · , xg ) · B)) : Q ∈ Li }. 0 Ig−m Remark 5.7. 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 5.8. We recall that if g = 3, then dim L4 = 1 and the condition (5.1) can be restricted to i = 4. When g > 3, dim L2 = (g − 3)(g − 1)/2. In this case, it suffices to check (5.1) only for i = 2, 3 and, in the particular case that X is neither a smooth quintic plane curve (g = 6) nor a trigonal curve, we can restrict the condition to i = 2. As in [BG19], for 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 [BG19] it is proved that L2,j = {Q ∈ L2 : Q(x1 , · · · , xj−1 , xj , xj+1 , · · · , xn ) = Q(x1 , · · · , xj−1 , −xj , xj+1 , · · · , xn )} . and a modular form corresponding to a dimension one abelian variety of genus > 3 with associated differential ωi , is bielliptic if and only if dim L2 = dim L2,i (here we are assuming not a smooth plane quintic curve, nor a trigonal curve). A similar result is obtained when g = 3 and we replace L2,j with L4,j . 13 Remark 5.9. 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 conjugates 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. Thus for any Atkin-Lehner quotient of a modular curve we can carry out all these compu- tations and decide if is bielliptic or not. For examples, see the proofs of Lemmas 7.5, 7.6, 8.7 and 8.16 and the github folder https://github.com/FrancescBars/Mathematica-files-on-Quotient-Modular-Curves for computations (and their source files) done by MATHEMATICA applying the above results from Petri’s theorem. ∗ 6 Bielliptic quotients with gN =0 From the tables in the Appendix we see that for these levels most quotients have genus 1 or even 0. So we only have to examine the following 13 curves. • X0 (44)/w4 ∼ = X0 (22) by Proposition 4.21. Since X0 (22) has the two bielliptic involutions (22) (22) w2 and w22 by [Bar99], from Proposition 4.21 we also see that S2 w2 S2 and S2 w22 S2 are bielliptic involutions of X0 (44)/w4 . • X0 (54)/w2 of genus 2, has Jacobian decomposition E27a × E54a, so all automorphisms are defined over Q. By MAGMA the automorphism group is Z/2Z, so the only involution is the hyperelliptic one and the curve is not bielliptic. • X0 (56)/w8 is bielliptic. Namely, by Lemma 4.18 the involution V2 w7 has 8 fixed points on X0 (56), so it has at least 4 fixed points on X0 (56)/w8 . Thus it is a bielliptic involution or the hyperelliptic involution. But the hyperelliptic involution obviously is w7 . • X0 (92)/w4 of genus 5 and X0 (92)/w92 of genus 4 are both hyperelliptic, so by the Castel- nuovo inequality they cannot be bielliptic. • X0 (60)/w12 , by the same argument, is hyperelliptic of genus 4 and thus not bielliptic. • X0 (60)/w4 ∼ = X0 (30) and X0 (60)/hw4 , w3 i ∼ = X0 (30)/w3 again by Proposition 4.21. As be- fore, the bielliptic involutions (compare [Bar99] and [BGK20]) conjugate back to the biel- (30) (30) (30) (30) liptic involutions w5 , S2 w6 S2 and S2 w30 S2 of X0 (60)/w4 resp. S2 w2 S2 and S2 w10 S2 of X0 (60)/hw4 , w3 i. • X0 (60)/w4 and X0 (60)/w60 each have a bielliptic Atkin-Lehner involution because they map of degree 2 to the elliptic curve X0 (60)/hw4 , w5 i resp. X0 (60)/hw3 , w20 i.(See also the previous item.) Noting that X0 (60)/hw3 , w5 i also has genus 1 we see that each of X0 (60)/w3 , X0 (60)/w5 and X0 (60)/w20 maps to two of these three elliptic curves and hence has two bielliptic AL-involutions. • X0 (60)/hw5 , w12 i has genus 2. Its Jacobian decomposition over Q is E20a × E30a, there- fore all automorphisms are defined over Q. By MAGMA its automorphism group over Q is Z/2Z, so there is no bielliptic involution. 14 7 Bielliptic quotient curves when X0∗(N ) has genus 1 Part (i) of Theorem 1.1 is almost self-evident, as these are exactly the values of N and WN for which X0∗ (N ) has genus 1 and there is a degree 2 map from X0 (N )/WN to it. From the tables in the Appendix we see that with the exception of the elliptic curves X0+ (40), X0+ (48), X0+ (63) and X0+ (75) these curves X0 (N )/WN do have genus at least 2. ∗ In the remainder of this section we finish the case gN = 1 by deciding the curves X0 (N )/wd ∗ where X0 (N ) is elliptic and N has 3 different prime divisors. Lemma 7.1. The following 48 quotient curves X0 (N )/wd are not bielliptic N (N, wd ) 84 (84, w7 ); (84, w28 ); (84, w21 ) 90 (90, w2 ); (90, w10 ); (90, w18 ) 120 (120, w5 ); (120, w8 ); (120, w3 ); (120, w40 ) 126 (126, w2 ); (126, w7 ); (126, w18 ) 132 (132, wd ), d||132 140 (140, wd ), d||140 150 (150, wd ), d||150 156 (156, wd ), d||156 220 (220, wd ), d||220 Proof. From the tables in the Appendix we see that for each such curve X0 (N )/wd there exists a suitable Atkin-Lehner involution wm such that g(X0 (N )/wd ) ≥ 2g(X0 (N )/hwd , wm i) and X0 (N )/hwd , wm i is not subhyperelliptic. So X0 (N )/wd cannot be bielliptic by Lemma 4.4. Remark 7.2. There are several other methods by which one could prove a large subset of the 48 curves in the previous lemma to be not bielliptic. Note that all these curves have a map of degree 4 to the genus 1 curve X0∗ (N ). So if g(X0 (N )/wd ) > 10 a hypothetical bielliptic map would by the Castelnuovo inequality have to factor over a common quotient curve with this degree 4 map, i.e. the bielliptic involution would have to be an Atkin-Lehner involution, which by the tables does not exist. Alternatively, again because Aut(X0 (N )/wd ) has a subgroup of order 4, Proposition 4.8 shows that those with even genus g ≥ 6 are not bielliptic. Finally, one could also use Lemma 4.5 with p = 3 to exclude all curves with N = 220. Lemma 7.3. The curve X0 (126)/w9 is bielliptic with bielliptic involution V3 w7 . The genus 5 curve √ X0 (126)/w63 has at least two bielliptic involutions, namely V3 and V3 w9 , both defined over Q( −3). The two genus 7 curves X0 (126)/w14 and X0+ (126) are isomorphic and not bielliptic. Proof. From Lemma 4.19 we see that V3 has the same number of fixed points as w9 , namely none. And V3 w9 , being a Galois conjugate of V3 , also has the same number of fixed points. See Lemma 4.19. By exactly the same arguments each of the involutions w63 , V3 w7 and V3 w63 has 16 fixed points. Now with Lemma 4.14 one easily checks that the modular curves X0 (126)/hw9 , V3 w7 i, X0 (126)/hw63 , V3 i and X0 (126)/hw63 , V3 w7 i have genus 1. By Proposition 4.22 the curves X0 (126)/w14 and X0+ (126) are isomorphic, and by [Jeo18] the latter one is not bielliptic. Lemma 7.4. The involution V2 w40 induces on each of the curves X0 (120)/w15 , X0 (120)/w24 and X0+ (120) a bielliptic involution. Moreover, X0 (120)/w15 has exactly two more bielliptic involutions, namely S2 and w8 S2 w8 . 15 Proof. From Lemma 4.18 we obtain the following table: v #(v, X0 (120)) v #(v, X0 (120)) id - V2 0 w8 0 V2 w8 0 w3 0 V2 w3 8 w5 0 V2 w5 0 w24 8 V2 w24 0 w40 0 V2 w40 16 w15 16 V2 w15 8 w120 8 V2 w120 0 Now the genus of X0 (120)/hwd , V2 w40 i can be easily calculated using Lemma 4.14. Since X0 (120)/w15 maps with degree 2 to the elliptic curve X0 (60)/w15 isomorphic to X0 (120)/hw8 S2 w8 , w15 i (compare Lemma 4.16), w8 S2 w8 is also a bielliptic involution of X0 (120)/w15 . Furthermore, S2 is conjugate to w8 S2 w8 in the automorphism group of X0 (120)/w15 , and hence also a bielliptic involution. Finally, since S2 (w8 S2 w8 )V2 w40 = w5 and X0 (120)/hw15 , w5 i is non- hyperelliptic of genus 3, by [KMV11, Remark 3.2] there are no further bielliptic involutions of X0 (120)/w15 . Lemma 7.5. V3 w10 is a bielliptic involution of X0 (90)/w9 . And X0+ (90) actually has (at least) two, namely V3 and V3 w9 . The curves X0 (90)/w5 and X0 (90)/w45 on the other hand are not bielliptic. Proof. As before we get from Lemma 4.19 that each of w9 , V3 and V3 w9 has 4 fixed points, and each of w90 , V3 w10 , V3 w90 has 8. Then we use Lemma 4.14 to check the genus of the quotient curves in question. We have J0 (90)hw5 i ∼Q (E30a)2 × (E45a)2 × E90b J0 (90)hw45 i ∼Q (E15a)2 × E30a × E90b × E90c For J0 (90)hw45 i there is a quadratic twist E30a ∼Q(√−3) E90c. Over the rationals we have for p = 11 and E = E30a or E90c does not satisfy (4.1) because |#X0 (90)/w45 (F11 )−2∗#E(F11 )| = 2, and dim L2,E90b < dim L2 . So remains if E15a is or nota bielliptic quotient over Q, but is a1 b 1 not possible because there does not exist any matrix A = ∈ GL2 (Q) satisfying the a2 b 2 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 ) (7.1) for all Q2 ∈ L2 .√ Now for Q( −3) (by the quadratic twist) we have the Jacobian decomposition J0 (90)hw45 i ∼Q(√−3) (E15a)2 × (E30a)2 × E90b √   a1 b 1 but there does not exist any matrix A = ∈ GL2 (Q( −3)) satisfying the condition a2 b 2 (7.1) and Q2 (x1 , x2 , a1 x3 + a2 x4 , b1 x3 + b2 x4 , x5 ) = Q2 (x1 , x2 , −a1 x3 + a2 x4 , −b1 x3 + b2 x4 , x5 ) (7.2) for all Q2 ∈ L2 . 16 For J0 (90)hw5 i we obtain that there are no quadratic twists, thus any automorphism of the curve is defined over Q. Lemma 4.6 discards E90b as elliptic quotient. Applying Proposition 5.6 we obtain that there does not exist bielliptic quotient. Similarly because there is no matrix A ∈ GL2 (Q) as above that satisfies (7.1), thus E30a is not a bielliptic quotient, and similarly we discard E45a because no matrix A as above that satisfies equation (7.2). After we observe that J0 (90)hw45 i ∼Q(√−3) J0 (90)hw5 i because E15a ∼Q(√−3) E45a. Recall that X0 (90)/w5 and X0 (90)/w45 are isomorphic by use of V3 . See all computation details in name files related to above quotient modular curves in the folder https://github.com/FrancescBars/Mathematica-files-on-Quotient-Modular-Curves (42) Lemma 7.6. The curve X0 (84)/w4 has S2 w14 S2 as a bielliptic involution. X0+ (84) is also bielliptic. But X0 (84)/w3 and X0 (84)/w12 are not bielliptic. Proof. This follows from Proposition 4.21 and [Bar99] resp. from [Jeo18]. For the other two curves we use Proposition 5.6. The Jacobian decomposition over Q is J0 (84)hw3 i ∼ (E14a)2 × (E42a)2 × E84b J0 (84)hw12 i ∼ (E14a)2 × (E21a) × (E42a) × (E84a) From the Jacobian decomposition of J0 (84)hw3 i and J0 (84)hw12 i all endomorphisms are de- fined over Q (no quadratic twist in the elliptic curves involved and is the same decomposition in the algebraic closure of the rationals), see Proposition 5.1. The factors with power 1 are discarded because dim L2,i < dim L, thus are not bielliptic quotients.  The bielliptic quotient a1 b 1 E14a is not possible because there does not exist any matrix A = ∈ GL2 (Q) satis- a2 b 2 fying (7.1)for all Q2 ∈ L2 . Similarly there does not exist such a matrix for E42a for J0 (84)hw3 i satisfying (7.2) for all Q2 ∈ L2 . Thus by Proposition 5.6, they are not bielliptic. See all computation details in https://github.com/FrancescBars/Mathematica-files-on-Quotient-Modular-Curves 8 Quotient modular curves of level N with X0∗(N ) hyper- elliptic 8.1 Quotient modular curves with X0∗ (N ) hyperelliptic and N having two prime divisors In this subsection we discuss the candidates for which X0∗ (N ) is hyperelliptic and N is only divisible by two different primes. So we treat the 15 values N = 88, 104, 112, 116, 117, 135, 147, 153, 184, 284, 136, 171, 207, 176, 279. We emphasize that the curves X0+ (N ) have already been treated in [Jeo18] and are not listed in our Theorem 1.1. So we largely ignore them and only mention a few bielliptic ones among them in passing. 17 Lemma 8.1. The curves X0 (284)/wd are not bielliptic. Proof. This follows easily from Lemma 4.5 by counting F9 -rational points. Lemma 8.2. The curves X0 (184)/w8 , X0 (207)/w9 , X(279)/w9 and X0 (279)/w31 are not biel- liptic. Proof. These curves have an involution whose quotient is the hyperelliptic curve X0∗ (N ). By the Hurwitz formula the number of fixed points is 2g(X0 (N )/wd ) − 4g(X0∗ (N )) + 2 > 8 (see the tables in the Appendix). So by Lemma 4.3 they are not bielliptic. We quickly decide some more curves. Lemma 8.3. (a) The curves X0 (88)/w11 , X0 (112)/w7 and X0 (184)/w23 are bielliptic. (b) On the other hand, X0 (116)/w4 , X0 (153)/w17 and X0 (207)/w23 are not bielliptic. Proof. (a) These curves map of degree 2 to the elliptic curves X0 (44)/w11 , X0 (56)/w7 and ∗ X0 (92)/w23 , respectively (see the table for gN = 0 in the Appendix). So by Lemma 4.16 a bielliptic involution is given by w8 S2 w8 (resp. w16 S2 w16 for X0 (112)/w7 ). Moreover, S2 is a conjugate bielliptic involution. (b) The curve X0 (116)/w4 is by Proposition 4.21 isomorphic to X0 (58), and hence not bielliptic by [Bar99]. Also, X0 (153)/w17 and X0 (207)/w23 are by Proposition 4.22 isomorphic to X0+ (153) resp. X0+ (207) and so not bielliptic by [Jeo18]. Lemma 8.4. Let N ∈ {104, 117, 136, 171, 176} and i = 2 resp. 3 depending on whether N is even or odd. Then Vi wN is a bielliptic involution of X0 (N )/$1 and of X0+ (N ), whereas X0 (N )/$2 is not bielliptic. Here we are using the notation $1 and $2 from Section 2 (xii), which allows us to simultaneously describe the Atkin-Lehner involutions for different N . Proof. Thanks to Lemmas 4.18 respectively 4.19 we can calculate the number of fixed points of every involution Vi wd . We only carry this out for N = 176, for which we obtain the following table v #(v, X0 (176)) v #(v, X0 (176)) id - V2 0 w16 0 V2 w16 4 w11 0 V2 w11 12 w176 12 V2 w176 24 With Lemma 4.14 we can thus check that X0 (176)/w11 has no bielliptic involution in hB(N ), V2 i whereas V2 w176 is a bielliptic involution for the other two curves. To finish the proof we note that by Lemma 4.4 a bielliptic involution v on X0 (176)/w11 would induce the hyperelliptic involution on X0∗ (176), which according to the table above is V2 . So v could only be V2 or V2 w16 . Lemma 8.5. The curve X0 (112)/w16 is not bielliptic. Proof. By exactly the same arguments as in the previous lemma we can show that hB(112), V2 i contains no bielliptic involution for this curve. However, now the problem is that we don’t know the hyperelliptic involution of X0∗ (112). (Actually, V2 does not help because it is a bielliptic involution of X0∗ (112).) But Proposition 4.8 guarantees that the bielliptic involution of X0 (112)/w16 would have to be contained in hw7 , V2 i. 18 Lemma 8.6. The genus 7 curve X0 (116)/w29 is not bielliptic. Proof. The involutions w4 and S2 , both defined over Q, generate a group H ∼ = S3 of automor- phisms of X = X0 (116)/w29 . Each of the three (conjugate) involutions has 8 fixed points on X. If v is a bielliptic involution of X, then H induces an isomorphic group H e of automorphisms on X/v. Under the action of v on the 8 fixed points of an involution in H these can at worst fall together in pairs. So each involution in H e has at least 4 fixed points on X/v. By the Hurwitz formula the curve (X/v)/H e thus has genus 0 and the automorphisms of order 3 in H e have no fixed points on X/v. So the genus 1 curve X/v has a fixed point free automorphism of order 3, defined over Q, and hence it must have a Q-rational 3-torsion point. But the only elliptic curves over Q of level M properly dividing 116 have M = 58 and no 3-torsion by [Cre17]. For M = 116 the modular degrees of the optimal elliptic curves 116a1, 116b1, 116c1 are 120, 8 and 15, and hence too big by Lemma 4.6. Lemma 8.7. The remaining six curves, that is the quotient curves X0 (88)/w8 , X0 (135)/w27 , X0 (135)/w5 , X0 (147)/w3 , X0 (147)/w49 , X0 (153)/w9 , are not bielliptic. Proof. Ordering by genus, the Jacobian decompositions over Q are as follows: J0 (88)hw8 i ∼ (E11a)2 × E44a × E88a J0 (147)hw3 i ∼ E49a × E147a × E147c × Af ; dim Af = 2, f ∈ New(147) J0 (135)hw5 i ∼ E27a × (E45a)2 × E135a × Af ; dim Af = 2, f ∈ New(135) J0 (147)hw49 i ∼ E21a × E147b × Af × Ag ; dim(Af ) = dim(Ag ) = 2; f, g ∈ New(147) J0 (135)hw27 i ∼ (E15a)2 × E45a × E135a × E135b × Af ; dim Af = 2; f ∈ New(135) J0 (153)hw9 i ∼ (E17a)2 × E51a × Af × E153a × E153b; dim Af = 2; f ∈ New(51) For the genus 4 cases there are no quadratic twists, thus all endomorphisms are defined over Q by Proposition 5.1. By Remark 5.8 we consider L2 and L3 , see further details in [BG20, §6]. We compute a nonzero polynomial Q2 ∈ L2 which generates L2 : 48t2 + x2 + xy − 7y 2 − 16z 2 . For E44a and E88a corresponding to z, t we have that Q2 (ω1 , ω2 , ω3 , ω4 ) = Q2 (ω1 , ω2 , −ω3 , ω4 ) and similar for ω4 , but Q3 (. . . , ωi , . . .) − Q3 (. . . , −ωi , . . .) ∈ / L2 (8.1) xi where Q3 ∈ L3 that is not a multiple of L2 , thus not bielliptic. For E11a, we make a change of variables w1 , w2 taking Q2 (x, y, z, t) = 48t2 − 176x2 + y 2 − 16z 2 but this does not satisfy the condition on L3 of equation (8.1) thus is not bielliptic. See the computational details of this example and the remaining ones in files related with such quotient modular curves in the folder https://github.com/FrancescBars/Mathematica-files-on-Quotient-Modular-Curves We only recall that the bielliptic involution is defined over Q for genus ≥ 6 (in such cases it is unique) thus we only need to worry if a bielliptic involution exists and is not defined over Q for the genus 5 curve X0 (147)/w3 . The dimension two factor does not have any elliptic quotient by use of Proposition 5.3, and there only appears an inert twist E49a ∼Q(√−7) E49a, therefore √ √ the bielliptic involution, if it exists, is defined over Q( −7). But because over Q( −7) we have the same Jacobian decomposition, we conclude as we did over Q that there is no bielliptic involution. Thus for all remaining situations, any dimension one factor with power 1 in the Jacobian we only need to observe that dim L2,E < dim L2 and for higher power, a similar argument used in equation (7.1) in the previous section, with a correct choice of the variables, shows that no bielliptic involution appears by Proposition 5.6. 19 8.2 Quotient modular curves with X0∗ (N ) hyperelliptic and N having 3 prime divisors In this part we deal with the cases where X0∗ (N ) is hyperelliptic and N is divisible by 3 different primes. So N is one of ∗ ∗ 168, 180, 198, 204, 276, 380 (all with gN = 2) or 252, 315 (both with gN = 3). Proposition 8.8. Let N be one of the eight numbers just listed. Then none of the curves X0 (N )/wd is bielliptic. Proof. From [Jeo18] we know that none of the eight curves X0+ (N ) is bielliptic. So we can assume d 6= N . We consider the covering X0 (N )/wd → X0 (N )/hwd , wN i. Since wN always has fixed points, by Lemma 4.4 a bielliptic involution of X0 (N )/wd would induce a hyperelliptic involution on X0 (N )/hwd , wN i. But by the tables in the Appendix none of these curves is hyperelliptic. Now we investigate the curves X0 (N )/W with |W | = 4. Lemma 8.9. V3 w7 is a bielliptic involution of the curves X0 (252)/hw4 , w9 i, X0 (252)/hw9 , w7 i and X0 (252)/hw4 , w63 i. Moreover V3 is a second bielliptic involution of X0 (252)/hw4 , w63 i. By contrast, X0 (252)/hw4 , w7 i, X0 (252)/hw9 , w28 i, X0 (252)/hw7 , w36 i and X0 (252)/hw36 , w28 i are not bielliptic. Proof. Using Lemma 4.19 we obtain the following table with number of fixed points. v #(v, X0 (252)) v #(v, X0 (252)) id - V3 0 w4 8 V3 w4 0 w9 0 V3 w9 0 w7 0 V3 w7 24 w36 0 V3 w36 0 w28 0 V3 w28 8 w63 24 V3 w63 24 w252 8 V3 w252 8 With Lemma 4.14 we can then easily check three things, namely that V3 w7 is a bielliptic involution for the first three curves (and V3 as well for X0 (252)/hw4 , w63 i), that the other four curves have no bielliptic involution in hB(N ), V3 i, and that V3 induces the hyperelliptic involution on X0∗ (N ). Combining the second and third fact proves that the four curves are not bielliptic, because a bielliptic involution would by Lemma 4.4 induce the hyperelliptic involution on X0∗ (N ), and hence would be in hB(N ), V3 i. Lemma 8.10. V2 w168 is a bielliptic involution of the curves X0 (168)/hw8 , w3 i, X0 (168)/hw8 , w7 i, X0 (168)/hw3 , w56 i and X0 (168)/hw7 , w24 i. Moreover, V2 w8 is a second bielliptic involution of X0 (168)/hw3 , w56 i. The remaining curves X0 (168)/hw3 , w7 i, X0 (168)/hw8 , w21 i and X0 (168)/hw24 , w56 i are not bielliptic. Proof. Apart from using the involutions V2 wd the proof is completely analogous to that of Lemma 8.9, except for the case of X0 (168)/hw24 , w56 i, in which case we cannot apply Lemma 4.4 because the genus is 4. But being hyperelliptic this curve therefore cannot also be bielliptic by the Castelnuovo inequality (Lemma 4.1). 20 Unfortunately the method we just used does not really work for the six other values of N , as for them we don’t know the hyperelliptic involution of X0∗ (N ). For example, V3 induces a bielliptic involution on X0∗ (180), X0∗ (198) and X0∗ (315). Lemma 8.11. None of the curves X0 (380)/W with |W | = 4 is bielliptic. Proof. Assume that such a curve is bielliptic and let E be the elliptic curve it covers. Then E (the reduction of E modulo 3) would by Lemma 4.5 have the maximally possible number of 16 F9 -rational points. So E must necessarily be supersingular. This is equivalent to j(E) = 0. This in turn is equivalent to the condition that for the coefficients ai of the global minimal model of E the expression a21 + a2 is divisible by 3. With this one can quickly exclude almost all candidates for E in [Cre17]. Only the isogeny classes 190a and 380a remain. But the curves in 190a have 7 rational points over F3 and hence the same number over F9 . And for 380a the degree of the strong Weil uniformization is 24 for the elliptic curve 380a1 and 240 for 380b1, so both too big by Lemma 4.6. In the previous lemma the four curves of genus bigger than 10 could also have been quickly excluded using the following method. Lemma 8.12. The curves X0 (276)/W with W any one of hw4 , w3 i, hw4 , w69 i, hw3 , w92 i, hw12 , w92 i, as well as X0 (204)/hw4 , w3 i and X0 (315)/hw9 , w7 i are all not bielliptic. Proof. For all these curves the map to the (hyperelliptic) curve X0∗ (N ) is given by an involution with more than 8 fixed points. So by Lemma 4.3 they are not bielliptic. Lemma 8.13. None of the curves X0 (315)/W with |W | = 4 is bielliptic. Proof. The curve X0 (315)/hw9 , w7 i has already been excluded in the last lemma. So let X be one of the other six curves and assume it is bielliptic with corresponding elliptic curve E. A necessary condition for this is that for the reductions modulo 2 we have |X(F2k )| ≤ 2|E(F2k )| for all k. With MAGMA one can calculate |X(F2k )| for small values of k. It turns out that |X(F4 )| = 18 for all six curves. Hence E must have the maximally possible number of 9 rational points over F4 . So it also only has 9 rational points over F16 . This excludes the curves X0 (315)/hw9 , w5 i and X0 (315)/hw9 , w35 i because they both have |X(F16 )| = 26. Moreover, |E(F4 )| = 9 implies j(E) = 0. This is equivalent to the coefficient a1 of the global minimal model of E being even. So with one glance at [Cre17] one can exclude that E has conductor 15, 21, 45, 63 or 105. Only conductor 35 or 315 is possible. But for 315 the degree of the strong Weil uniformization is too big. And an elliptic curve with conductor 35 does not appear in the Jacobian of X0 (315)/hw7 , w45 i and X0 (315)/hw45 , w63 i. J(X0 (315)/hw7 , w45 i) ∼Q (E15a) × (E21a)2 × A35,x2 +z−4 × (E63a) × A315,x2 +2x−1 J(X0 (315)/hw45 , w63 i) ∼Q (E15a) × (E21a) × (E105a) × (E315a) × A315,x2 +2x−1 × A315,x2 −5 . The other two curves of genus 8 are isomorphic to the above two by Lemma 4.22, so also not bielliptic. From Proposition 4.21 we obtain the following isomorphisms over Q. 21 Lemma 8.14. X0 (180)/hw4 , w9 i ∼ = X0 (90)/w9 , so by Lemma 7.5 and Proposition 4.21 it has (90) S2 V3 w10 S2 as a bielliptic involution. The following curves are not bielliptic by Section 7 resp. [BG20] resp. [BGK20]. X0 (180)/hw4 , w5 i ∼ = X0 (90)/w5 ; X0 (180)/hw4 , w45 i ∼ = X0 (90)/w45 ; ∼ X0 (198)/hw9 , w11 i = X0∗ (396); X0 (204)/hw4 , w17 i ∼ = X0 (102)/w17 ; X0 (204)/hw4 , w51 i ∼ = X0 (102)/w51 ; ∼ X0 (276)/hw4 , w23 i = X0 (138)/w23 . Lemma 8.15. The curve X0 (204)/hw3 , w68 i is not bielliptic. Proof. By [HS06] this curve is trigonal. If it also were bielliptic, then by the Castelnuovo inequality its genus could be at most 4. But it has genus 5. Lemma 8.16. The remaining next 15 curves are not bielliptic. X0 (180)/hw9 , w5 i, X0 (180)/hw9 , w20 i, X0 (180)/hw5 , w36 i, X0 (180)/hw36 , w20 i, X0 (198)/hw2 , w9 i, X0 (198)/hw2 , w11 i, X0 (198)/hw2 , w99 i, X0 (198)/hw9 , w22 i, X0 (198)/hw11 , w18 i, X0 (198)/hw18 , w22 i, X0 (204)/hw3 , w17 i, X0 (204)/hw17 , w12 i, X0 (204)/hw12 , w51 i, X0 (276)/hw3 , w23 i, X0 (276)/hw23 , w12 i Proof. First we consider the genus 5 curves. For that we need to study quadratic twists because the possible bielliptic involution (or automorphisms) could be not defined over Q. J0 (180)hw9 ,w20 i ∼Q E15a × E30a × E36a × E90a × E90b J0 (180)hw5 ,w36 i ∼Q E20a × E45a × (E30a)2 × E90b J0 (180)hw36 ,w20 i ∼Q E15a × E30a × E90b × E90c × E180a J0 (198)hw2 ,w99 i ∼Q E11a × E33a × E66a × E99a × E198a J0 (198)hw11 ,w18 i ∼Q E66a × E66b × E99a × E99b × E99d J0 (204)hw12 ,w51 i ∼Q E17a × E34a × E102a × E102b × E204b J0 (276)hw23 ,w12 i ∼Q E69a × E92a × E138a × E138b × E138c We find the following quadratic twists (we are only interested in a fixed Jacobian, thus we do not compare here quadratic twist for modular forms between different Jacobians): E90a ∼Q(√−3) E90b, E30a ∼Q(√−3) E90c and E36a ∼Q(√−3) E36a. √ Anyway, we know by V3 the following modular curves are isomorphic over Q( −3): X0 (180)/hw5 , w36 i ∼ = X0 (180)/hw36 , w20 i, X0 (198)/hw2 , w99 i ∼ = X0 (198)/hw11 , w18 i and X0 (198)/hw2 , w11 i ∼ = X0 (198)/hw18 , w22 i. These isomorphism induce isogenous Jacobians, by the following twists: E15a ∼Q(√−3) E45a, E180a ∼Q(√−3) E20a, E11a ∼Q(√−3) E99d, E33a ∼Q(√−3) E99b, E198a ∼Q(√−3) E66b, E66a ∼Q(√−3) E198b, E66c ∼Q(√−3) E198e, and √ E99a ∼Q(√−3) E99c. Thus, over Q( −3): J0 (180)hw9 ,w20 i ∼Q(√−3) E15a × E30a × E36a × (E90b)2 J0 (180)hw36 ,w20 i ∼Q(√−3) E15a × (E30a)2 × E90b × E180a In the above situation dim L2 = 3 and for each elliptic curve with power one we obtain dim L2,E < 3, thus they are not bielliptic quotients. For the E 2 -factors √ with E an elliptic curve we prove that there is no A ∈ GL2 (K) where K = Q or Q( −3) respectively of the Jacobian decomposition satisfying Proposition 5.6. Thus none of these curves is bielliptic. See the complete computation details on each curve and for the following ones in this proposition in name files related with such quotient modular curves in the folder 22 https://github.com/FrancescBars/Mathematica-files-on-Quotient-Modular-Curves Now consider the genus ≥ 6 curves. It is enough to study them over the rationals (we are not interested where the general endomorphism are defined for the problem on biellipticity). The Jacobian decomposition is as follow: J0 (180)hw9 ,w5 i ∼Q (E20a)2 × (E30a)2 × E36a × (E90b)2 J0 (198)hw2 ,w11 i ∼Q (E66a)2 × E99a × E99b × E99d × E198e J0 (198)hw18 ,w22 i ∼Q E11a × E33a × E66a × E66c × E99c × E198b J0 (198)hw2 ,w9 i ∼Q (E11a)2 × E33a × E66a × E99a × E99c × E198d J0 (198)hw9 ,w22 i ∼Q (E11a)2 × E33a × E66a × E66c × E99a × E99c J0 (204)hw3 ,w17 i ∼Q (E34a)2 × Af,68 × (E102a)2 × E(204a) J0 (204)hw17 ,w12 i ∼Q (E34a)2 × E51a × Af3 ,68 × E102a × E102c J0 (276)hw3 ,w23 i ∼Q E92a × (E138a)2 × (E138c)2 × Af4 ,276 where dim(Af,68 ) = dim(Af3 ,68 ) = dim(Af4 ,276 ) = 2 and the number next to fi is the level where it appears as newform. By Theorem 5.3 they do not give any elliptic quotient. Moreover, for each non-repeated factor of dimension one in the Jacobian we have dim(L2,E ) < dim(L) and for the terms E 2 with E an elliptic curve, there is no matrix A ∈ GL2 (Q) satisfying Proposition 5.6. All in all we obtain that none is a bielliptic quotient. 9 Quotient modular curves of level N with X0∗(N ) not subhyperelliptic Proposition 9.1. Let N be non-square free such that X0∗ (N ) is not subhyperelliptic. Then the only bielliptic curve X0 (N )/WN where WN is a proper subgroup of B(N ) is X0 (144)/hw144 i. Proof. Let X0 (N )/WN be bielliptic and X0∗ (N ) not subhyperelliptic. Then X0∗ (N ) must be bielliptic by Lemma 3.1. So we are dealing with a subset of the levels in case (v) of Theorem 3.2. Furthermore, by Lemma 4.4 (b) the covering X0 (N )/WN → X0∗ (N ) must be totally unramified. Next we point out that every Atkin-Lehner involution wd that has a fixed point on X0 (N ) must be contained in WN . Otherwise it would induce an involution with fixed points on X0 (N )/WN , contradicting the fact that the map to X0∗ (N ) is totally unramified. In particular, the full Atkin-Lehner involution wN , which by [Ogg74, p.454] always has fixed points, must be contained in WN . Now we are ready to settle N = 420 = 22 · 3 · 5 · 7, the only case in this paper where N has more than by 3 different primes divisors. As just said, w420 ∈ W420 . And as w4 fixes a cusp by [Ogg74, Proposition 3], also w4 ∈ W420 . Moreover, w20 , w35 , w84 and w140 also all have fixed points by [Ogg74, p.453]. So they also all must be contained in W420 , leading to the contradiction W420 = B(420). Next we treat the cases where N is divisible by 3 different primes. Thus, N is one of the following: ∗ N gN 234, 240, 252, 294, 312, 315, 348, 476 3 228, 260, 264, 280, 300, 306, 342 4 364, 444, 495 5 558 7 23 Obviously it suffices to prove that there are no such bielliptic curves with |WN | = 4. As necessarily wN ∈ WN , there are at most three possible choices for WN , namely h$1 , $2 ∗ $3 i, h$2 , $1 ∗ $3 i and h$3 , $1 ∗ $2 i. Moreover, the covering X0 (N )/WN → X0∗ (N ) is unrami- ∗ fied if and only if X0 (N )/WN has genus 2gN − 1. By the tables in the Appendix the only curves surviving this test are X0 (260)/hw4 , w65 i and X0 (300)/hw4 , w75 i, both of genus 7. But they are also not bielliptic. By Proposition 4.21 they are isomorphic to X0 (130)/hw65 i and X0 (150)/hw75 i, respectively. For the second curve we have shown in Lemma 7.1 that it is not bielliptic. The exact same proof works for N = 130, or alternatively (since 130 is square-free) we could invoke [BGK20]. Finally, if N has only two different prime divisors it belongs to one of the following sets: ∗ N gN 144, 152, 164, 189, 196, 236, 245, 248 3 148, 160, 172, 200, 224, 225, 242, 275 4 As discussed earlier, then necessarily WN = hwN i. But by [Jeo18, Theorem 1.1] X0+ (144) is the only such curve that is bielliptic. 10 Quadratic points In this section we prove our second main result, Theorem 1.3. Let X be a curve of genus at least 2 which has a Q-rational point. Then by [Bar18, The- orem 2.14] X has infinitely many quadratic points if and only if X is hyperelliptic or X has an involution v, defined over Q, such that X/v is an elliptic curve E with positive rank over Q. Now we specialize to X being of the form X(N )/WN . The hyperelliptic ones have already been determined in [FH99]. On the other hand, if the bielliptic involution v is defined over Q, then the conductor of the elliptic curve E will be a divisor M of N . So our main tool will be [Cre17]. ∗ We start with the cases where gN = 1 and N has 3 different prime divisors. Among these, for N = 84, 90, 120, 126, 132, 140, 150 there are no divisors M for which there exist elliptic curves of positive rank. For N = 156 and 220 the only possibility is M = N . So in that case the map from X0 (N ) to E of degree 2|WN | ≤ 8 must factor through the strong Weil parametriza- tion. But by [Cre17] for the elliptic curves of positive rank the degree of that parametrization is 12 for N = 156 and 36 for N = 220. We also mention as a warning that if M is a proper divisor of N it seems that one cannot expect any help from the strong Weil parametrization from X0 (M ) to E. For example, X0 (22) has two bielliptic involutions defined over Q, namely w2 and w22 . As there are no elliptic curves with conductor 22, the quotient curves must be isogenous to the elliptic curve X0 (11). But the bielliptic maps do not factor through the canonical map to X0 (11), which has degree 3. ∗ By the same token, for the cases where gN = 1 and N has only 2 different prime divisors, we can exclude all levels N except 99 and 124. Indeed Jac(X0∗ (99)) ∼Q E99a, which has rank 1, furnishing the curves X0 (99)/hwd i with infinitely many quadratic points. For N = 124 again the degree of the strong Weil parametriza- tion (here 6) is too small. 24 By exactly the same method one can exclude all remaining bielliptic curves in the table of Theorem 1.1 except for the following two constellations: N = 171 with M = 57 and N = 176 with M = 88. So assume that there is a map of degree 2, defined over Q, from X0 (171)/hw9 i to the elliptic curve E with conductor 57 and positive rank over Q. For the reduction modulo 2 one easily verifies #E(F2 ) = 5. And with the general formula #E(Fq2 ) = #E(Fq )(2q + 2 − #E(Fq )) we see that E does not acquire more points over F4 . This leads to a contradiction in Lemma 4.5. The exact same proof works for N = 176 if we reduce the elliptic curve with conductor 88 modulo 3 and count the F9 -rational points. Finally we provide some more information on the two curves that are bielliptic but not over Q. Lemma 10.1. The genus 5 curve X √0 (126)/w63 has exactly two bielliptic involutions, namely V3 and V3 w9 , both√defined over Q( −3). This curve has only finitely many points that are quadratic over Q( −3). The same holds for the curve X0 (252)/hw4 , w63 i, which is isomorphic to X0 (126)/w √ 63 by Proposition 4.21. Its two bielliptic involutions are V3 and V3 w7 , both defined over Q( −3). Proof. In Lemma 7.3 we exhibited the two bielliptic involutions of X0 (126)/w63 . Since the curve X0 (126)/hw63 , w2 i has genus 2, we can apply Lemma 4.11. It tells us that if X0 (126)/w63 had further bielliptic involutions, it would have one defined over Q. So one checks with Petri that X0 (126)/w63 has no bielliptic involution over Q. More computationally, one could also show √ from the splitting of the Jacobian that all automorphisms of X0 (126)/w63 are defined over Q( −3) and then use Petri to determine the bielliptic involutions over that field. Finally one checks that √ the elliptic quotients of X√0 (126)/w63 by V3 resp. V3 w9 , namely the base change of E14a to Q( −3) has rank 0 over Q( −3). Acknowledgements We are very happy to thank Professor Josep González for his different comments and sugges- tions. We are grateful to unknown referees for detailed and constructive reports and for many suggestions that improved the presentation of the paper. A Computations of the genus We list the genus of X0 (N )/WN with WN ≤ B(N ) such that N is non-square-free and not a power of a prime, and X0∗ (N ) is of genus ≤ 1, or is hyperelliptic or is bielliptic. These levels N are listed in Theorem 3.2. For such list we omit: N = 420 (a product involving four primes, see the results of §9 to discard N = 420), and N = 12, 18, 20, 24 because gN ≤ 1. Recall the definition of $i from Section 2 (xii). 25 A.1 N a product of two primes N P rimef actor gX0 (N ) gX0 (N )/<$1 > gX0 (N )/<$2 > gX0 (N )/<$1 ∗$2 > = gX + (N ) 0 gX0∗ (N ) = 0 28 22 ∗ 7 2 1 0 1 44 22 ∗ 11 4 2 1 1 45 32 ∗ 5 3 1 1 1 50 2 ∗ 52 2 1 1 0 54 2 ∗ 33 4 2 1 1 56 23 ∗ 7 5 3 1 1 92 22 ∗ 23 10 5 1 4 gX0∗ (N ) = 1 40 23 ∗ 5 3 2 2 1 48 24 ∗ 3 3 2 2 1 52 22 ∗ 13 5 2 3 2 63 32 ∗ 7 5 3 3 1 68 22 ∗ 17 7 3 4 2 72 23 ∗ 32 5 2 3 2 75 3 ∗ 52 5 3 3 1 76 22 ∗ 19 8 4 3 3 80 24 ∗ 5 7 3 4 2 96 25 ∗ 3 9 3 5 3 98 2 ∗ 72 7 4 3 2 99 32 ∗ 11 9 5 3 3 100 22 ∗ 52 7 2 4 3 108 22 ∗ 33 10 4 4 4 124 22 ∗ 31 14 7 3 6 188 22 ∗ 47 22 11 4 9 26 N P rimef actor gX0 (N ) gX0 (N )/<$1 > gX0 (N )/<$2 > gX0 (N )/<$1 ∗$2 > = gX + (N ) 0 gX0∗ (N ) = 2 88 23 ∗ 11 9 4 5 4 104 23 ∗ 13 11 6 6 3 112 24 ∗ 7 11 6 4 5 116 22 ∗ 29 13 6 7 4 117 32 ∗ 13 11 5 6 4 135 33 ∗ 5 13 7 6 4 147 3 ∗ 72 11 5 6 4 153 32 ∗ 17 15 7 6 6 184 23 ∗ 23 21 11 5 9 284 22 ∗ 71 34 17 7 14 gX0∗ (N ) = 3 136 23 ∗ 17 15 7 8 6 144 24 ∗ 32 13 7 7 5 152 23 ∗ 19 17 8 9 6 164 22 ∗ 41 19 9 10 6 171 32 ∗ 19 17 9 9 5 189 33 ∗ 7 19 8 10 7 196 22 ∗ 72 17 7 9 7 207 32 ∗ 23 21 11 8 8 236 22 ∗ 59 28 14 10 10 245 5 ∗ 72 21 10 9 8 248 23 ∗ 31 29 15 9 11 gX0∗ (N ) = 4 148 22 ∗ 37 17 8 9 8 160 25 ∗ 5 17 9 9 7 172 22 ∗ 43 20 10 9 9 176 24 ∗ 11 19 10 10 7 200 23 ∗ 52 19 10 10 7 224 25 ∗ 7 25 13 11 9 225 32 ∗ 52 19 9 10 8 242 2 ∗ 112 22 11 10 9 275 52 ∗ 11 25 13 11 9 gX0∗ (N ) = 5 279 32 ∗ 31 29 15 15 9 In the previous tables N means that X0∗ (N ) is an hyperelliptic curve, and k with k an integer means that the quotient modular curve with such genus is an hyperelliptic curve. 27 A.2 N a product of three primes N 60 84 90 120 126 132 140 150 156 F actor(N ) 22 ∗ 3 ∗ 5 22 ∗ 3 ∗ 7 2 ∗ 32 ∗ 5 23 ∗ 3 ∗ 5 2 ∗ 32 ∗ 7 22 ∗ 3 ∗ 11 22 ∗ 5 ∗ 7 2 ∗ 3 ∗ 52 22 ∗ 3 ∗ 13 WN = {id} 7 11 11 17 17 19 19 19 23 h$1 i 3 5 6 9 9 9 9 10 11 h$2 i 4 5 5 9 9 10 10 10 11 h$3 i 4 6 5 9 9 7 10 10 12 h$1 $2 i 4 5 6 7 9 10 8 9 11 h$1 $3 i 2 6 6 9 7 7 10 7 12 h$2 $3 i 1 6 5 5 5 10 7 7 6 h$1 $2 $3 i 3 4 4 7 7 8 7 8 10 h$1 , $2 i 2 2 3 4 5 5 4 5 5 h$1 , $3 i 1 3 3 5 4 2 5 4 6 h$2 , $3 i 1 3 2 3 3 4 4 4 3 h$1 , $2 $3 i 0 2 2 2 2 4 2 3 2 h$2 , $1 $3 i 1 2 2 4 3 3 4 3 5 h$3 , $1 $2 i 2 2 2 3 4 3 3 4 5 h$1 $2 , $1 $3 i 0 3 3 2 2 4 3 2 3 B(N ) 0 1 1 1 1 1 1 1 1 N 220 168 180 198 204 276 380 234 240 F actor(N ) 22 ∗ 5 ∗ 11 23 ∗ 3 ∗ 7 22 ∗ 32 ∗ 5 2 ∗ 32 ∗ 11 22 ∗ 3 ∗ 17 22 ∗ 3 ∗ 23 22 ∗ 5 ∗ 19 2 ∗ 32 ∗ 13 24 ∗ 3 ∗ 5 WN = {id} 31 25 25 29 31 43 55 35 37 h$1 i 15 13 11 14 15 21 27 18 19 h$2 i 16 13 13 15 16 22 28 17 19 h$3 i 13 13 13 12 16 13 25 18 19 h$1 $2 i 16 11 11 14 16 22 28 18 19 h$1 $3 i 13 9 11 15 12 19 25 15 15 h$2 $3 i 10 13 13 12 13 22 16 16 15 h$1 $2 $3 i 14 11 11 13 13 18 24 15 17 h$1 , $2 i 8 6 5 7 8 11 14 9 10 h$1 , $3 i 5 5 5 6 6 5 11 8 8 h$2 , $3 i 4 7 7 5 7 7 7 8 8 h$1 , $2 $3 i 4 6 5 5 5 9 6 7 7 h$2 , $1 $3 i 6 4 5 7 5 8 11 6 7 h$3 , $1 $2 i 6 5 5 5 7 5 11 8 9 h$1 $2 , $1 $3 i 4 4 5 6 5 10 7 7 6 B(N ) 1 2 2 2 2 2 2 3 3 N 252 294 312 315 348 476 228 260 264 F actor(N ) 22 ∗ 32 ∗ 7 2 ∗ 3 ∗ 72 23 ∗ 3 ∗ 13 32 ∗ 5 ∗ 7 22 ∗ 3 ∗ 29 22 ∗ 7 ∗ 17 22 ∗ 3 ∗ 19 22 ∗ 5 ∗ 13 23 ∗ 3 ∗ 11 WN = {id} 37 41 49 41 55 67 35 37 41 h$1 i 17 21 25 21 27 33 17 17 19 h$2 i 19 21 25 19 28 34 17 19 21 h$3 i 19 21 25 21 28 34 18 19 21 h$1 $2 i 19 21 25 19 28 34 17 19 19 h$1 $3 i 19 17 19 21 22 30 18 19 21 h$2 $3 i 13 17 17 17 19 19 18 19 21 h$1 $2 $3 i 17 17 23 17 25 29 16 15 17 h$1 , $2 i 9 10 13 9 14 17 8 9 9 h$1 , $3 i 9 9 10 11 11 15 9 9 10 h$2 , $3 i 7 9 9 8 10 10 9 10 11 h$1 , $2 $3 i 5 8 8 7 8 7 8 7 8 h$2 , $1 $3 i 9 7 9 8 10 13 8 8 9 h$3 , $1 $2 i 9 9 12 8 13 15 8 8 8 h$1 $2 , $1 $3 i 7 7 6 8 7 8 9 10 10 B(N ) 3 3 3 3 3 3 4 4 4 N 280 300 306 342 364 444 495 558 F actor(N ) 23 ∗ 5 ∗ 7 22 ∗ 3 ∗ 52 2 ∗ 32 ∗ 17 2 ∗ 32 ∗ 19 22 ∗ 7 ∗ 13 22 ∗ 3 ∗ 37 32 ∗ 5 ∗ 11 2 ∗ 32 ∗ 31 WN = {id} 41 43 47 53 51 71 65 89 h$1 i 21 19 23 26 25 35 33 45 h$2 i 21 22 23 27 26 35 33 45 h$3 i 21 22 22 27 26 36 29 45 h$1 $2 i 19 22 23 26 26 35 33 45 h$1 $3 i 17 22 24 24 24 36 29 41 h$2 $3 i 21 19 22 21 23 24 33 33 h$1 $2 $3 i 19 19 20 24 23 32 25 41 h$1 , $2 i 10 10 11 13 13 17 17 23 h$1 , $3 i 9 10 11 12 12 18 13 21 h$2 , $3 i 11 10 10 11 12 12 15 17 h$1 , $2 $3 i 10 7 9 9 10 10 13 15 h$2 , $1 $3 i 8 10 10 11 11 16 11 19 h$3 , $1 $2 i 9 10 9 12 12 16 11 21 h$1 $2 , $1 $3 i 8 10 11 9 11 12 15 15 B(N ) 4 4 4 4 5 5 5 7 The double vertical lines separate the different values of g0∗ . A red entry for the genus means that the curve is hyperelliptic. 28 B Table of bielliptic quotient curves We consider (N, WN ) with N non-square free of genus gW ≥ 2 which is a bielliptic curve. We assume that WN 6= B(N ) and WN 6= hwN i, and nontrivial. Let F be the field where are defined all automorphism elements of End(J0 (N )WN ), which is Q or a quadratic field K. Such a field is determined from the Q-isogeny decomposition of the Jacobian of (N, WN ) by using mainly Propositions 5.1 and 5.3. We indicate the elliptic factors through K-isogeny if the field K is not Q (when gWN ≤ 5). By (w, E) a couple where w indicates a bielliptic involution as an element of Aut(X0 (N )/WN ) mod WN , and E denotes the corresponding bielliptic quotient modulo Q-isogeny (not Q-isomorphism!) and (w, E) a couple as before but defined all over K in particular E as K-isogeny. We add · · · in the tables to indicate that the modular curve X0 (N )/WN could have more bielliptic involutions. For all bielliptic (N, WN ) that are not hyperelliptic, such that no Atkin-Lehner appears as a bielliptic involution (or gWN ≥ 6 or that are listed in Theorem 1.1), we determine all bielliptic involutions by use of computations related to Petri’s theorem, (in the situations where Petri’s theorem applies). This procedure can be done also for the rest of quotient curves where we can apply the Petri methodology. One may consult all the computations done in MATHEMATICA in: https://github.com/FrancescBars/Mathematica-files-on-Quotient-Modular-Curves For (N, WN ) hyperelliptic and bielliptic, with hyperelliptic involution u, the bielliptic involu- tions can be computed by MAGMA or SAGE by using the explicit equation given by in [Has95] and [FH99]. In such situations we do not make explicit the bielliptic involutions except if one corresponds to an Atkin-Lehner involution. 29 (N, WN ) (w, E) Q − Jacobiandecomp. (40, hw8 i) (w5 , E20a ∼ X0∗ (40)), (u ◦ w5 , E40a) E20a × E40a (40, hw5 i) (w8 , E20a), (V2 , E20a), (S2 , E20a), (w8 S2 w8 , E20a) (E20a)2 (22) (22) (44, hw4 i) (S2 w2 S2 , E11a), (S2 w22 S2 , E11a) (E11a)2 (48, hw16 i) (w3 , 24a ∼ X0∗ (48)), (u ◦ w3 , 48a) E24a × E48a E24a ∼Q(√−1) E48a (48, hw3 i) (w16 , E24a), (V2 , E24a), (S2 , E24a), (S2 w16 S3 , E24a) (E24a)2 (52, hw4 i) (w13 , E26b ∼ X0∗ (52)), (u ◦ w13 , E26a) E26a × E26b (60, hw20 i) (w3 , E15a), (w3 ◦ u = w4 , E30a) E15a × E30a (30) (30) (60, hw4 , w3 i) (S2 w10 S2 , E15a), (S2 w2 S2 , E30a) E15a × E30a (72, hw8 i) (w9 , E36a ∼ X0∗ (72)), (, u ◦ w9 , E72a) E36a × E72a (84, hw4 , w3 i) (w7 , E42a ∼ X0∗ (84)), (u ◦ w7 , E14a) E14a × E42a (84, hw4 , w21 i) (w7 , E42a), (u ◦ w7 , E14a) E14a × E42a (84, hw3 , w28 i) (w7 , E42a), (u ◦ w7 , E14a) E14a × E42a (84, hw7 , w12 i) (w3 , E42a ∼ X0∗ (84)), (u ◦ w3 , E21a) E21a × E42a (90, hw9 , w5 i) (w9 , E30a ∼ X0∗ (90)), (u ◦ w9 , E90b) E30a × E90b (90, hw2 , w45 i) (w9 , E30a), (u ◦ w9 , E15a) E15a × E30a (90, hw9 , w10 i) (w5 , E30a), (u ◦ w5 , E15a) E15a × E30a (90, hw5 , w18 i) (w9 , E30a), (u ◦ w9 , E45a) E30a × E45a (100, hw4 i) (w25 , E50b ∼ X0∗ (100)), (w25 ◦ u, E50a) E50a × E50b E50a ∼Q(√5) E50b (120, hw8 , w15 i (w3 , E20a ∼ X0∗ (120)), (u ◦ w3 , E40a) E20a × E40a (120, hw15 , w40 i (w3 , E20a), (u ◦ w3 , E120a) E20a × E120a (126, hw2 , w63 i (w9 , E21a ∼ X0∗ (126)), (u ◦ w9 , E14a) E14a × E21a (126, hw14 , w18 i (w9 , E21a), (u ◦ w9 , E126a) E21a × E126a (132, hw4 , w11 i) (w3 , E66b ∼ X0∗ (132)), (u ◦ w3 , E66c) E66b × E66c (140, hw4 , w35 i) (w5 , E70a ∼ X0∗ (140)), (u ◦ w5 , E14a) E14a × E70a (150, hw6 , w50 i) (w2 , E15a ∼ X0∗ (150)), (u ◦ w2 , E150a) E15a × E150a (156, hw4 , w39 i) (w13 , E26b ∼ X0∗ (156)), (u ◦ w13 , E26a) E26a × E26b Table 1. Case gWN = 2 For genus 3 table below we list first the three hyperelliptic ones. We remind that the automorphism group can be computed in MAGMA and here we do not make it explicit if is not an Atkin-Lehner involution. For example, two of the bielliptic involutions of X0 (63)/w9 that√ are not defined over Q are simply √ listed as (∗∗, Ẽ). Their elliptic quotient Ẽ is Q( −3)-isogenous to Y 2 = Ẽ), (∗, √ +1 + 6 −3X − 27X − (26 + 6 −3)X 3 . 2 30 (N, WN ) (w, E) Q − Jacobiandecomp. (56, hw8 i) (V2 w8 , E14a) (E14a)2 × E56b (30) (30) (60, hw4 i) (S2 w5 S2 , E30a), (S2 w6 S2 , E15a), (E15a)2 × E30a (30) (S2 w30 S2 , E15a). (63, hw9 i) (w7 , E21a ∼ X0∗ (63)) E21a × Af,63 , (∗∗, Ẽ), (∗, Ẽ) Af,63 ∼Q(√−3) Ẽ 2 (52, hw7 i) (w8 , E26b ∼ X0∗ (52)), ... (E26b)2 × E52a (63, hw7 i) (w9 , E21a), ... (E21a)2 × E63a, E21a ∼Q(√−3) E63a (68, hw4 i) (w17 , E34a ∼ X0∗ (68)), ... (E17a)2 × E34a (72, hw9 i) (w8 , E36a ∼ X0∗ (72)), ... E24a × (E36a)2 (75, hw3 i) (w25 , E15a ∼ X0∗ (75)), ... (E15a)2 × E75a (75, hw25 i) (w3 , E15a), ... E15a × E75b × E75c, E15a ∼Q(√5) E75b (76, hw19 i) (w4 , E38b ∼ X0∗ (76)).... (E38b)2 × E76a (80, hw16 i) (w5 , E20a ∼ X0∗ (80)), ... E20a × E40a × E80a E40a ∼Q(√−1) E80a (84, hw4 , w7 i) (w3 , E42a ∼ X0∗ (84)), ... (E21a)2 × E42a (84, hw3 , w7 i) (w4 , E42a), ... (E42a)2 × E84b (84, hw12 , w21 i) (w3 , E42a), ... E14a × E42a × E84a (90, hw2 , w9 i) (w5 , E30a ∼ X0∗ (90)... E15a × E30a × E90a (90, hw2 , w5 i) (w9 , E30a)... (E30a)2 × E45a (90, hw10 , w18 i) (w5 , E30a)... E15a × E30a × E90c E30a ∼Q(√−3) E90c (96, hw32 i) (w3 , E24a ∼ X0∗ (96)), ... E24a × E48a × E96a E24a ∼Q(√−1) E48a (98, hw49 i) (w2 , E14a ∼ X0∗ (98)), ... E14a × Af,98 (99, hw11 i) (w9 , E99a ∼ X0∗ (99)), ... E99a × E99b × E99d (120, hw3 , w5 i) (w8 , E20a ∼ X0∗ (120)), ... (E20a)2 × E24a (120, hw5 , w24 i) (w8 , E20a), ... (E20a)2 × E30a (124, hw31 i) (w4 , E62a ∼ X0∗ (124)), ... (E62a)2 × E124b (126, hw9 , w7 i) (w2 , E21a ∼ X0∗ (126)), ... (E21a)2 × E42a (126, hw9 , w14 i) (w2 , E21a ∼ X0∗ (126)), ... (E21a) × Af,63 (132, hw3 , w44 i) (w7 , E66b ∼ X0∗ (132)), ... E11a × E33a × E66b (132, hw11 , w12 i) (w7 , E66b), ... E44a × E66a × E66b (140, hw7 , w20 i) (w4 , E70a ∼ X0∗ (140)), ... Af,35 × E70a (140, hw35 , w20 i) (w4 , E70a), ... E14a × E70a × E140a (150, hw2 , w75 i) (w25 , E15a ∼ X0∗ (150)), ... E15a × E30a × E50a (150, hw3 , w50 i) (w25 , E15a), ... (E15a)2 × E75a (156, hw3 , w13 i) (w4 , E26b ∼ X0∗ (156)), ... (E26b)2 × E52a (156, hw39 , w12 i) (w4 , E26b), ... E26a × E26b × E156b Table 2. Case gWN = 3 31 (N, WN ) (w, E) Q − Jacobiandecomp. (60, hw3 i) (w5 , E20a), (w20 , E15a), .... (E15a)3 × E20a (60, hw5 i) (w4 , E30a), (w3 , E20a), ... (E20a)2 × (E30a)2 (68, hw17 i) (w4 , E34a ∼ X0∗ (68)), ... 2 (E34a) × Af,x2 −2x−2 , dim(Af ) = 2 (76, hw4 i) (w19 , E38b ∼ X0∗ (76)), ... (E19a)2 × E38a × E38b (80, hw5 i) (w16 , E20a ∼ X0∗ (80)), ... (E20a)3 × E80b E20a ∼Q(√−1) E80b (98, hw2 i) (w49 , E14a ∼ X0∗ (98)), ... (E14a)2 × E49a × E98a E14a ∼Q(√−7) E98a ∗ (100, hw25 i) (w4 , E50b ∼ X0 (100)), ... E20a × (E50b)2 × E100a E20a ∼Q(√5) E100a (108, hw4 i) (w27 , E54b ∼ X0∗ (108)), ... (E27a)2 × E54a × E54b E54a ∼Q(√−3) E54b (108, hw27 i) (w4 , E54b), ... E36a × (E54b)2 × E108a (112, hw7 i) (S2 , E56a), (w16 S2 w16 , E56a) (E56a)2 × E112a × E112c ∗ (120, hw8 , w3 i) (w5 , E20a ∼ X0 (120))... (E15a)2 × E20a × E40a (120, hw3 , w40 i) (w5 , E20a)... (E15a)2 × E20a × E24a ∗ (126, hw2 , w7 i) (w9 , 21a ∼ X0 (126))... (E21a)2 × E63a × E126b E21a ∼Q(√−3) E126b (126, hw7 , w18 i) (w9 , E21a)... (E21a)2 × E42a × E63a E21a ∼Q(√−3) E63a ∗ (132, hw3 , w11 i) (w4 , E66b ∼ X0 (132)), ... E44a × (E66b)2 × E132b (132, hw4 , w33 i) (w3 , E66b), ... (E11a)2 × E66b × E66c (132, hw12 , w33 i) (w4 , E66b), ... E11a × E66b × E66c × E132a ∗ (140, hw4 , w5 i) (w7 , E70a ∼ X0 (140)), ... E14a × (E35a)2 × E70a (140, hw5 , w7 i) (w4 , E70a), ... E20a × (E70a)2 × E140b (140, hw5 , w28 i) (w4 , E70a), ... E14a × E20a × E35a × E70a ∗ (150, hw2 , w25 i) (w3 , E15a ∼ X0 (150)), ... E15a × E30a × E75b × E75c E15a ∼Q(√5) E75b (150, hw3 , w25 i) (w2 , E15a), ... (E15a)2 × E50b × E150c (150, hw25 , w6 i) (w2 , E15a), ... E15a × E50b × E75b × E75c E15a ∼Q(√5) E75b (168, hw3 , w56 i) (V2 w168 , E14a), (V2 w8 , E24a) E14a × E24a × E42a × E84b ∗ (188, hw47 i) (w4 , E94a ∼ X0 (188))... (E94a)2 × Af,x2 −x−3 dim(Af ) = 2 ∗ (220, hw5 , w11 i) (w4 , E110b ∼ X0 (220)), ... E20a × E44a × (E110b)2 (220, hw4 , w55 i) (w11 , E110b), ... (E11a)2 × E110c × E110b (220, hw44 , w55 i) (w11 , E110b), ... E11a × E110a × E110b × E220b Table 3, case gWN = 4 32 (N, WN ) (w, E) Q − Jacobiandecomp. (42) (84, hw4 i) (S2 w14 S2 , E21a) (E14a)2 × (E21a)2 × E42a (88, hw11 i) (S2 , E44a), (w8 S2 w8 , E44a) (E44a)2 × E88a × Af,88,x2 −x+4 (90, hw9 i) (V3 w10 , E15a) (E15a)2 × E30a × E90a × E90b E90a ∼Q(√−3) E90b (96, hw3 i) (w32 , E24a = X0∗ (96))... (E24a)3 × E32a × E96b (99, hw9 i) (w11 , E99a = X0∗ (99))... (E11a)2 × E33a × E99a × E99c E99a ∼Q(√−3) E99c (117, hw9 i) (V3 w117 , E39a) E39a × Af1 × Af2 Af2 ∼Q(√−3) Ẽ 2 (120, hw15 i) (V2 w40 , E24a), (w8 S2 w8 , E20a), (S2 , E20a) (E20a)2 × E24a × E40a × E120a (120, hw8 , w5 i) (w3 , E20a = X0∗ (120))... (E20a)2 × (E30a)2 × E120b (126, hw63 i) (V3 , E14a), (V3 w9 , E14a) E14a × (E21a)2 × E42a × E126a E14a ∼Q(√−3) E126a (126, hw2 , w9 i) (w7 , E21a = X0∗ (126)), ... (E14a)2 × E21a × Af,63,x2 −3 Af ∼Q(√−3) Ẽ 2 (132, hw4 , w3 i) (w11 , E33a = X0∗ (132))... (E11a)2 × (E33a)2 × E66b (140, hw4 , w7 i) (w5 , E70a = X0∗ (140))... A2f,35 × E70a (150, hw2 , w3 i) (w25 , E15a = X0∗ (150))... (E15a)2 × E50a × E75a × E150b (156, hw4 , w3 i) (w13 , E26b = X0∗ (156))... (E26a) × (E26b) × (E39a)2 × E78a (156, hw3 , w56 i) (w4 , E26b)... E26a × E26b × E39a × E78a × E156a (156, hw13 , w12 i) (w4 , E26b)... (E26b)2 × Af × E52a (168, hw8 , w7 i) (V2 w168 , E21a) (E21a)2 × E42a × E84b × E168a (168, hw7 , w24 i) (V2 w168 , E21a) (E21a)2 × E42a × E56a × E84b (90) (180, hw4 , w9 i) (S2 V3 w10 S2 , E15a) (E15a)2 × E30a × E90a × E90b E90a ∼Q(√−3) E90b (184, hw23 i) (S2 , E92a), (w8 S2 w8 , E92a) (E92a92 × E184b × Af,184,x2 +x−4 (220, hw4 , w11 i) (w5 , E110b = X0∗ (220))... (E55a)2 × E110b × Af (252, hw4 , w63 i) (V3 , E14a), (V3 w7 , E14a) E14a × (E21a)2 × E42a × E126a E14a ∼Q(√−3) E126a Table 4, Case gWN = 5 gWN (N, WN ) (w, E) Q − Jacobiandecomp. 6 (104, hw8 i) (V2 w104 , E26a) (E26a)2 × E26b × E52a × Af,104 (156, hw4 , w13 i) (w3 , E26b = X0∗ (156)) (E26b)2 × A2f,39 2 (168, hw8 , w3 i) (V2 w168 , E14a) (E14a) × E42a × E56b × E84b × E168b (220, hw5 , w44 i) (w4 , E110b = X0∗ (220)) E11a × E20a × Af × E110b × E110c (220, hw11 , w20 i) (w4 , E110b) E44a × E55a × E110b × Af × E220a 7 (120, hw24 i) (V2 w40 , E15a) (E15a)2 × (E20a)2 × E30a × E40a × E120a (124, hw4 i) (w31 , E62a = X0∗ (124) (Af1 ,31 )2 × E62a × Af3 ,62 (136, hw8 i) (V2 w136 , E17a) (E17a)2 × E34a × Af3 ,64 × Af4 ,136 (252, hw9 , w7 i) (V3 w7 , E36a) (E21a)3 × E36a × (E42a)2 × E84b 8 (220, hw4 , w5 i) (w11 , E110b = X0∗ (220)) (E11a)2 × A2f × E110b × E110c 9 (126, hw9 i) (V3 w7 , E14a) (E14a)2 × (E21a)2 × E42a × (Af,63 )2 2 (171, hw9 i) (V3 w171 , E19a) (E19a) × E57a × E57b × E57c × Af,171 , dim(Af ) = 4 (252, w9 , hw4 i) (V3 w7 , E14a) (E14a)2 × (E21a)2 × E42a × (Af,63 )2 10 (176, hw16 i) (V3 w176 , E11a) (E11a)3 × E44a × E88a × Af1 ,88 × E176a × Af2 ,176 11 (188, hw4 i) (w47 , X0∗ (188) = E94a) A2f1 × E94a × Af3 , dim(Af1 ) = 4 Table 5, gWN ≥ 6 33 References [Acc94] Robert D. 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