This is the submitted version of the article: Bars Cortina, Francesc; González Rovira, Josep. «The automorphism group of the modular curve X0* (N) with square-free level». Transactions of the American Mathematical Society, Vol. 374 (April 2021). DOI 10.1090/tran/8391 This version is available at https://ddd.uab.cat/record/243414 under the terms of the license The automorphism group of the modular curve X0∗(N ) with square-free level Francesc Bars∗and Josep González † Abstract We determine the automorphism group of the modular curve X0∗ (N ), obtained as the quotient of the modular curve X0 (N ) by the group of its Atkin-Lehner involutions, for all square-free values of N . 1 Introduction In [KM88], Kenku and Momose determined the automorphism group of all modular curves X0 (N ) with genus > 1, except for N = 63, which was solved by Elkies in [Elk90]. Subsequently, Harrison detected a mistake in these results concerning to the curve X0 (108). In [Har14], it is proved that this curve has an extra involution which does not arise from the normalizer of Γ0 (108) in SL2 (R), as it happens for the curves X0 (37) and X0 (63). For the modular curve X0+ (N ) = X0 (N )/hwN i, where wN denotes the Fricke involution, Baker and Hasegawa determined the automorphism group when N is a prime in [BH03] and, later, in [Gon16] it was determined the automorphism group when N is the square of a prime. In this paper we focus our attention to the modular curves X0∗ (N ) = X0 (N )/B(N ), where B(N ) is the group of the Atkin-Lehner involutions of the modular curve X0 (N ), when N is square-free. The interest in these modular curves is due to their moduli interpretation. For a number field K, the non cuspidal K-rational points of X0∗ (N ) parametrize a class of K-curves. More precisely, these points parametrize elliptic curves E/K having the property that for every Galois conjugation σ ∈ Gal(K/K) there is an isogeny between E and E σ of degree a divisor of N . When X0∗ (N ) has genus < 3, these parametrizations are described in [GL98] and [BGX21]. We point out that for a square-free N 6= 37, B(N ) is the automorphism group of X0 (N ) when its genus is greater than one and, thus, a non trivial automorphism of X0∗ (N ) does not come from the action of an automorphism of X0 (N ). More generally, let us denote by A(N ) the subgroup of SL2 (R) generated by Γ0 (N ) and B(N ). In [Lan01], it is proved that A(N ) is its normalizer in SL2 (R). Hence, a non trivial automorphism of X0∗ (N ) is exceptional in the sense that does not come from a linear fractional transformation on the complex upper-half plane. In [BH03][Corollary 2.6], it is proved that for a square-free integer N such that the genus of X0 (N ) is greater than 1, the group Aut X0∗ (N ) is elementary 2-abelian and every automorphism ∗ of X0∗ (N ) is defined over Q. If the curve X0∗ (N ) has a non trivial involution and the genus of the quotient curve is zero, then it is hyperelliptic. When the genus of the quotient curve is one, then the curve X0∗ (N ) is bielliptic. ∗ First author is supported by MTM2016-75980-P † The second author is partially supported by DGI grant MTM2015-66180-R. 1 In [HH96], it is proved that X0∗ (N ) is hyperelliptic if, and only if, the curve has genus two and, moreover, all these values of N are determined. In [BG19], all values of N for which X0∗ (N ) is bielliptic are determined and for each of these curves its automorphism group is determined. All these results are summarized in the next theorem. 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, (i) The modular curve X0∗ (N ) is hyperelliptic if, and only if, it has genus two. In this case, the automorphism group has order 2 if, and only if, N is in the following list 67, 73, 85, 93, 103, 107, 115, 133, 134, 146, 154, 161, 165, 167, 170, 177, 186, 191, 205, 206, 209, 213, 221, 230, 266, 285, 286, 287, 299, 357. (ii) The automorphism group of a modular curve X0∗ (N ) of genus 3 is non trivial if, and only if, the curve is bielliptic. (iii) 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 the genus of the curve is greater than two, otherwise it is the Klein group. The goal of this article is to complete the values of N such that the group Aut(X0∗ (N )) is non trivial and describe this group for all these values. Among the curves X0∗ (N ) with genus > 1, there are exactly 37 that are hyperelliptic and 12 that are bielliptic and non hyperelliptic. In [BG19], it is proved that the curve X0∗ (366) of genus 4 has automorphism group of order 2 and the genus of the quotient curve by the non trivial involution is 2. Hence, it is reasonable to expect that there are a few curves X0∗ (N ) of genus > 3 that are not bielliptic and its automorphism group is non trivial. The main result of this paper, that is presented in the following theorem, gives a precise answer to this question. Theorem 2. Let N be a square-free integer such that the curve X0∗ (N ) has genus > 3 and it is not bielliptic, i.e. N 6= 370. Then, the group Aut(X0∗ (N )) is not trivial, if and only if, N = 366, 645. In both cases, the order of this group is 2 and the genus of the quotient curve by the non trivial involution is 2. The paper is organized as follows. In section 2, we fix the notation and recall some general facts. In Section 3, we show the main tools that we will use to determine the group Aut(X0∗ (N )) for a fixed value of N . Sections 4 and 5 are devoted to prove Theorem 2 for odd and even levels respectively. The key point in these two last sections is the determination of a finite set of positive integers containing all levels N such that X0∗ (N ) has a non trivial automorphism group. In section 4, for the odd levels, this result follows from Proposition 3. This proposition is based on an idea already used in [KM88][Lemma 2.7], [BH03][Lemma 3.3] and [Gon16][Lemma 6], but always proved with different arguments because the involved modular curves in these statements are different. In section 5, the determination of a finite set for the even levels is obtained from Proposition 4. This proposition presents an unknown inequality involving the genera of the curves X0∗ (N ) and X0∗ (2N ) for all odd square-free values of N . 2 2 Notation and general facts Let N > 1 be a square-free integer. We fix, once and for all, the following notation. (i) We denote by B(N ) the group of the Atkin-Lehner involutions of X0 (N ). If N 0 |N , B(N 0 ) can be also identified as the subgroup of B(N ) formed by the Atkin-Lehner involutions wd such that d|N 0 . (ii) The integer ω(N ) is the number of primes dividing N . In particular, |B(N )| = 2ω(N ) . ∗ (iii) The integers gN and gN are the genus of X0 (N ) and X0∗ (N ) respectively. (iv) We denote by NewN the set of normalized newforms in S2 (Γ0 (N )) and New∗N is the subset of NewN formed by the newforms invariant under the action of the group of the Atkin-Lehner involutions B(N ). (v) S2 (N ) = S2 (Γ0 (N )), S2∗ (N ) is the vector space S2 (N )B(N ) , J0 (N ) = Jac(X0 (N )) and J0∗ (N ) = Jac(X0∗ (N )). (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|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 expresses that 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) GQ denotes the absolute Galois group Gal(Q/Q) once an algebraic closure Q of Q has been fixed. A summary on modular abelian varieties can be found in [BGGP05][Section 3]. We recall some known facts that will be used. The Q-vector space Ω1X0 (N )/Q is the subspace of elements in S2 (N )dq/q with rational q-expansion, i.e. S2 (N )dq/q ∩ Q[[q]]. For N square-free integer ∗ and gN ≥ 1, it is known that Y Y J0∗ (N ) ∼ Q Af . 1M |N f ∈New∗M i=1 and, thus, ∗ 2 gN X |X0∗ (Fpn )| = pn + 1 − αin . i=1 On the other hand, the basis of the regular differentials formed by all Galois conjugates of hi (q) dq/q allows us to compute equations for X0∗ (N ) when gN ∗ > 1. Similarly, if X is a curve defined over Q of genus g > 0 for which there is a non constant morphism X0∗ (N ) → X defined over Q, then there is a subset S of ∪1 1. 3 Preliminary results In this section, we present some tools that will be applied to decide whether the group Aut(X0∗ (N )) is trivial or not, and to determine this group when it is non trivial. For a curve X defined over Q of genus ≥ 2 and of good reduction at a prime p, one has that AutQ (X) ,→ AutFp (X ⊗ Fp ) (cf. [Liu02][Proposition 3.10.38]). We can apply [Gon17][Theorem 2.1] to discard the existence of involutions defined over Q. More precisely, we will use the following criterion (see [Gon17][Remark 2.2] taking N = 2). Lemma 1. Let X be a curve defined over Fp of genus g > 2. Consider the sequence X Pp (n) := mod [( µ(n/d)|X(Fpd )|)/n, 2] d|n where mod [r, 2] denotes 0 or 1 depending on P whether r is even or not, and µ is the Moebius function. If there is an integer k > 0 such that kn≥0 (2n+1)Pp (2n+1) > 2g +2, then AutFp (X) contains no involution. 4 By Petri’s Theorem, we know 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 . Q As said in the above section, J0∗ (N ) ∼ Ah1 × · · · × Ahn for some normalized eigenforms h1 , · · · , hk ∈ S2∗ (N ). These abelian varieties are simple and pairwise nonisogenous over Q and, any involution u of the curve leaves stable each Ahi acting on Ω1Agi as the product by −1 or the identity. Choose a basis {ω1 , · · · , ωgN∗ } of Ω1X ∗ (N )/Q obtained as the ordered union of bases of all 0 Ω1Ah /Q . An involution u of X0∗ (N ) induces a linear map u∗ : Ω1X ∗ (N )/Q → Ω1X ∗ (N )/Q sending i 0 0 ∗ (ω1 , · · · , ωgN∗ ) 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 ≥ 2 . (3.1) The genus of the quotient curve is the cardinality of the set I = {i : εi = 1} and {ωj }j∈I is a basis of the pullback of the regular differentials of the quotient curve. 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, that will be the main tool to determine the group Aut(X0∗ (N )) for a fixed level N . ∗ Lemma 2. Assume X0∗ (N ) is nonhyperelliptic, i.e. gN > 2. Let ω1 , · · · , ωgN∗ be a basis of 1 ΩX ∗ (N )/Q as above. Then, 0 (i) Let u be a non trivial involution of X0∗ (N ). Consider the curve Xu = X0∗ (N )/hui. Denote by gu the genus of Xu . Then there exists an integer k < n and factors Ah1 , · · · , Ahk such Q that Jac(Xu ) ∼ Ah1 × · · · × Ahk and, thus, satisfying Q(−x1 , −x2 , · · · , −xgu , xgu +1 , · · · , xgN∗ −1 , xgN∗ ) ∈ Li for all Q ∈ Li and for all i ≥ 2 . (3.2) ∗ (ii) If gu := dim Ah1 × · · · × Ahk < gN and the condition (3.2) is satisfied, then there exists ∗ an involution v of X0 (N ) such that Q Q Jac(X/hvi) ∼ Ah1 × · · · × Ahk or Jac(X/hvi) ∼ Ahk+1 × · · · × Ahn In order to make easy the computation of the condition (3.2) for L2 , we introduce the vector Lns subspace P 2 of polynomials in L2 that do not contain square monomials, i.e. polynomials of the form 1≤i 2|Xu (F`m )| for some integer m ≥ 1, then we can discard this possibility. Another way is the following. If g > 2 and we know that Xu is not hyperelliptic, we check if the regular differentials ω1 , · · · , ωg satisfy Petri’s Theorem. That is, if g > 3 and the dimension of the corresponding vector space L2 is not (g − 3)(g − 2)/2 or g = 3 and dim L4 6= 1, then we can discard this possibility. The result in the next proposition allows us to use the knowledge of the group of auto- morphisms of lower levels, that can be a source for getting bounds for the even levels N such that the curves X0∗ (N ) may have non trivial automorphisms. Previously, we will present the following lemma. Lemma 3. Let X be a curve defined over Q of genus greater than 1 and let p be a prime of good reduction of X. Assume that there is a non trivial automorphism w̃ ∈ AutFp X/Fp that is not a hyperelliptic involution. If there exists an automorphism v ∈ EndQ (Jac X) such that its reduction modulo p coincides with the induced automorphism w̃∗ ∈ EndFp Jac(X/Fp ), then there exists an automorphism w ∈ AutQ X such that the induced automorphism w∗ ∈ EndQ (Jac X) satisfies w∗ = ±v. Proof. If w̃ is not a hyperelliptic involution, then v 6= − Id. The automorphism w e respects the canonical polarization of the curve X/Fp . Since the reduction of the endomorphism ring is injective, v also respects the canonical polarization of the curve X/Q. Therefore, by Torelli’s Theorem, there exists an automorphism w ∈ AutQ X such that the induced morphism w∗ ∈ EndQ (Jac X) is v or −v.  Proposition 1. Let N > 1 be a square-free integer and let p be a prime dividing N such that ∗ gN/p > 1. If there exists a non trivial involution u of X0∗ (N ) such that the action of u on J0 (N/p) is non trivial, then the group Aut(X0∗ (N/p)) is non trivial or the curve X0∗ (N/p)/Fp ∗ is hyperelliptic. Proof. Let φ and φp be the morphisms from X0 (N ) to X0 (N/p) induced by the automorphisms of the complex upper half-plane given by z 7→ z and z 7→ pz respectively. Consider the morphism ν = φ∗ + p φ∗p : J0 (N/p) → J0 (N ), that is defined over Q. For every cusp form h ∈ S2∗ (N/p) one has that ν(h) ∈ S2∗ (N ) and, moreover, ν is an injective linear map from S2∗ (N/p) into S2∗ (N ) because ν sends a basis of S2∗ (N/p) to a set of linearly independents cusp forms (cf. (2.1)). Let π and π 0 be the following natural projections π : X0 (N/p) → X0∗ (N/p), π 0 : X0 (N ) → X0∗ (N ). The kernel of the morphism ν 0 := π∗0 ◦ ν ◦ π ∗ : J0∗ (N/p) −→ J0∗ (N ) is finite since J0∗ (N/p) and the abelian variety A := ν 0 (J0∗ (N/p)) have the same dimension. Moreover, for any Hecke operator TM with gcd(M, N ) = 1, one has ν · TM = TM · ν (cf. [Li75]) and ν 0 has the same property as ν. Since (End J0∗ (N )) ⊗ Q is isomorphic to the product of totally real number fields, by applying the arguments used in [Rib75][Proposition3.2], we obtain that for any abelian subvariety B of J0∗ (N ) the algebra (End B) ⊗ Q is generated by the Hecke operators TM with gcd(M, N ) = 1. 6 Let u be a non trivial involution of X0∗ (N ). This involution induces an involution u∗ of J0∗ (N ) which leaves stable A because A and J0∗ (N )/A are defined over Q and have no isogenous quotients. Hence, u∗ induces an involution of A, that we still denominate by u∗ . The dual isogeny νb0 : A → J0∗ (N/p) also satisfies the condition νb0 ◦ TM = TM ◦ νb0 for all Hecke operators TM with gcd(M, N ) = 1. Since u∗ ∈ End(A) lies in the algebra generated by the Hecke operators TM with gcd(M, N ) = 1, the involution u∗ leaves stable ker νb0 and, thus, provides an involution v of the quotient A/ ker νb0 = J0∗ (N/p). Now, assume that u∗ ∈ End A is non trivial. By using Hecke operators, we get that v is non trivial and satisfies u∗ · ν 0 = ν 0 ◦ v. On the one hand, the normalization of X0∗ (N )/Fp is the curve X0∗ (N/p)/Fp (cf. [Has97][Section 5]) and u induces an involution u e of X0∗ (N/p)/Fp . On the other hand, the reduction of v modulo p coincides with the endomorphism of Jac(X0∗ (N/p)/Fp ) induced by the involution u e. Now, the statement follows from Lemma 3.  Corollary 1. Assume that there exist a non trivial involution u of X0∗ (N ) and a prime p|N ∗ with gN/p > 2. If the group Aut(X0∗ (N/p)) is trivial and X0∗ (N/p)/Fp is not hyperelliptic, ∗ then gu ≥ gN/p and ν(S2∗ (N/p))dq/q is contained in the pullback of Ω1Jac(Xu ) . In particular, ∗ ∗ ∗ gN/p ≤ (gN + 1)/2 since by Hurwitz theorem one has gu ≤ (gN + 1)/2. Note that a necessary condition for X0∗ (N/p)/Fp being hyperelliptic is that the following inequalities are satisfied |X0∗ (N/p)(Fpn )| ≤ 2pn + 2 for all integers n > 0. The hyperelliptic curves have a different behaviour depending on whether they are defined over fields of characterisitic 2 or not. Since [BGGP05][Lemma 2.5] can be applied to hyperelliptic curves defined over fields of characteristic different from 2 and X0 (N ) admits a regular model defined over Z[1/N ], we have a similar result to [BGGP05][Lemma 6] for modular curves over Fp with p 6= 2. More precisely, in our case: Lemma 4. Assume p is an odd prime not dividing M . Let X/Q be a curve of genus g > 2 for which there is a no constant morphism π : X0∗ (M ) → X and let S be the Z-module S2∗ (M ) ∩ Z[1/N ][[q]]. The curve X/Fp is hyperelliptic if, and only if, there is a basis f1 , · · · , fg of S whose reductions mod p satisfy  q i + O(q i ) if the cusp ∞ is not a Weierstrass point of X/Fp , fi (q)/q (mod p) = 2i−1 q + O(q 2i−1 ) otherwise, (3.4) and for any such a basis, the functions on X/Fp defined by fg−1 qdx/dq x= (mod p) , y= (mod p) , fg fg satisfy y 2 = P (x) for an unique square-free polynomial P (X) ∈ Fp [X] which has degree 2g + 1 or 2g + 2 depending on whether the cusp ∞ is a Weierstrass point or not of X/Fp . For the case p = 2, we use the following result. ∗ Lemma 5. Let N > 1 be an odd square-free integer such that gN > 2. If X0∗ (N )/F2 is hyperelliptic, then N is in the set {183, 185, 187, 203, 335, 345, 385}. Proof. Put n = ω(N ). By Ogg ( [Ogg74][Theorem 3]), if X0∗ (N )/F2 is hyperelliptic, then ψ(N ) + 2n ≤ 2n+1 |P1 (F4 )| = 5 · 2n+1 . 12 7 ∗ Hence, ψ(N ) ≤ 2n 108 and, after discarding the values N such that gN ≤ 2, we obtain the following 59 values of N : 97, 109, 113, 127, 137, 139, 149, 151, 157, 163, 173, 179, 181, 183, 185, 187, 193, 197, 199, 201, 203, 211, 217, 219, 235, 237, 247, 249, 253, 259, 265, 267, 273, 291, 295, 301, 303, 305, 309, 319, 321, 323, 329, 335, 341, 345, 355, 371, 377, 385, 391, 399, 429, 435, 455, 465, 483, 561, 595 . We can discard the levels N such that AN,m := |X0∗ (N )(F2m )| − 2(2m + 1) > 0. Since (N, m) AN,m (N, m) AN,m (N, m) AN,m (N, m) AN,m (97, 1) 1 (193, 1) 2 (267, 2) 4 (341, 2) 5 (109, 2) 1 (197, 2) 5 (273, 1) 1 (355, 2) 3 (113, 2) 1 (199, 2) 2 (291, 1) 3 (371, 2) 5 (137, 2) 2 (201, 1) 1 (295, 2) 2 (377, 2) 4 (139, 2) 1 (211, 2) 5 (301, 2) 3 (391, 1) 1 (149, 2) 2 (219, 1) 1 (303, 2) 3 (399, 1) 1 (151, 2) 1 (235, 1) 1 (305, 2) 4 (429, 2) 1 (157, 1) 2 (237, 1) 1 (309, 1) 2 (435, 1) 1 (163, 1) 2 (249, 2) 4 (319, 2) 1 (465, 2) 2 (173, 2 4 (253, 1) 1 (321, 2) 5 (483, 2) 5 (179, 2) 2 (259, 2) 2 (323, 2) 2 (561, 1) 1 (181, 2) 4 (265, 1) 2 (329, 2) 2 (592, 2) 3 By Lemma 1, we also can discard the values N for which we know that X0∗ (N )/F2 does not have any involution: 127, 217, 247.  Note that for a prime p we know that the automorphism group is trivial when gp∗ > 2 ∗ (cf. [BH03][Theorem 1.1]). For gN = 3, the group Aut(X0∗ (N )) is non trivial if, and only if, ∗ X0 (N ) is bielliptic (cf. [BG19]Lemma 13]). For this reason in the next sections, we exclude the ∗ cases gN ≤ 3 that can be found in Table 4 in [BG20][ Appendix]. 4 Odd levels ∗ We know that when gN > 1, if there is a non trivial involution u of X0∗ (N ), then the cusp ∞ is not a fixed point of u (cf. [BH03][Lemma 3.2]). The following result, similar to Lemma 6 in [Gon16], is the key result that allow us to determine a finite set of odd square-free integers containing all odd square-free integers N such that the group Aut(X0∗ (N )) is non trivial. ∗ Proposition 2. Assume that the square-free integer N is odd, gN > 1 and there is a non ∗ ∗ trivial involution u of X0 (N ). Then, the Q-gonality of X0 (N ) is ≤ 6 and u has at most 12 fixed points. Proof. Take Q = u(∞) and let P ∈ X0 (N ) be such that π(P ) = u(∞), where π is the natural projection of X0 (N ) onto X0∗ (N ). Since Q is not a cusp, there is an elliptic curve E defined over Q and a N -cyclic subgroup CN of E(Q) such that P = (E, CN ). The other pre-images of Q under π are the points wd (P ) = (E/Cd , (E[d] + CN/d )/Cd ) ∀d|N , where Cd denotes the d-cyclic subgroup of C, that is Cd = CN ∩ E[d]. 8 For any noncuspidal S ∈ X0∗ (N ), we consider the divisor DS := (uT2 − T2 u) (∞ − S) , where T2 denotes the Hecke operator viewed as a correspondence of the curve X0∗ (N ). We claim that DS is nonzero but linearly equivalent to zero. The endomorphism algebra End(J0∗ (N )) ⊗ Q is commutative. Hence, u T2 = T2 u and DS is a principal divisor. Next, we will prove that DS is nonzero. If DS is a zero divisor, then uT2 (∞) must be equal to T2 u(∞) because T2 (∞) = 3∞ and ∞ is not in the support of T2 (S). To prove that DS is a nonzero divisor, we only need to prove that the condition 3(Q) = T2 (Q) cannot occur for a noncuspidal point Q ∈ X0∗ (N ). Let Gi , 1 ≤ i ≤ 3, be the three 2-cyclic subgroups of E[2]. Since 3 X T2 (Q) = π((E/Gi , (CN + Gi )/Gi )) , i=1 the condition 3(Q) = T2 (Q) implies that each elliptic curve E/Gi is isomorphic to E/Cd for some d|N . Therefore, E has an endomorphism whose kernel is a 2d-cyclic subgroup and, thus, E has CM by a quadratic order O of discriminant D. The conductor of the discriminant D cannot be even because 2d is a norm of O and 2d 6≡ 0 (mod 4). Since π(P ) = π(wd (P )), this property holds for all elliptic curves E/Cd and, thus, also for all elliptic curves E/Gi . Now, we claim that for every elliptic curve E with CM by the order of discriminant D with odd conductor, there is at least a 2-subgroup G of E[2] such that the discriminant of the order End(E/G) has even conductor. This fact implies that T2 (Q) 6= 3Q and, thus, DS is nonzero. Indeed, let [a, b, c] := ax2 + bxy + cy 2 ∈ Z[x, y] be a primitive quadratic form of discriminant D (with odd conductor). The primitive quadratic forms attached to the three elliptic curves E/Gi are Q1 = [4a, 2b, c]/ gcd(c, 2), Q2 = [a, 2b, 4c]/ gcd(a, 2), Q3 = [4a, 2b−4a, a−b+c]/ gcd(2, a−b+c) . If the discriminants of Q1 and Q2 are equal to D, then a and c must be even. Since [a, b, c] is primitive, b must be odd and this fact leads to the contradiction that the discriminant of Q3 is 4D, with even conductor. By taking S = u(∞), DS is defined over Q and, thus, the Q-gonality is at most 6. Finally, since u∗ (DS ) 6= DS for some noncuspidal point S ∈ X0∗ (N )(C), any non trivial automorphism of X0∗ (N ) has at most 12 fixed points (cf. Lemma 3.5 of [BH03]).  Corollary 2. When N is odd, if gu is the genus of the curve X0∗ (N )/hui for a non trivial involution u of X0∗ (N ), then ∗ gN −5 g∗ + 1 ≤ gu ≤ N . 2 2 Moreover, if J0∗ (N ) has a simple factor of dimension larger than (gN ∗ + 5)/2, then Aut(X0∗ (N )) is trivial. Proof. The inequalities follows from the Riemann-Hurwitz formula applied to the projection X0∗ (N ) → X0∗ (N )/hui. For the last assertion see [BH03][Corollary 3.9]  Lemma 6. Assume that the Q-gonality of X0∗ (N ) is at most 6. If N is odd, then ψ(N ) ≤ 2ω(N ) 348 . (4.1) 9 ψ(N ) Proof. redAgain by Ogg, we know that 2ω(N ) + ≤ |X0 (N )(F4 )|. The statement follows 12 from the fact that |X0 (N )(F4 )| ≤ 2ω(N ) · 6 · |P1 (F4 )|.  There are 471 values of N (square-free and odd, N ≥ 3) satisfying the condition (4.1), ∗ whose maximum value is 3003. Excluding all values that are prime or gN ≤ 3, we obtain 293 values. By applying Lemma 1 to X0∗ (N )/Fp , we can discard 248 values of N . More precisely, with p = 2 the values 247, 253, 259, 267, 291, 301, 305, 319, 323, 327, 339, 355, 365, 377, 381, 391, 393, 395, 403, 407, 411, 413, 417, 427, 451, 453, 469, 471, 473, 481, 485, 489, 493, 501, 505, 511, 519, 527, 533, 535, 537, 543, 545, 553, 559, 565, 573, 579, 581, 589, 591, 595, 597, 611, 627, 629, 633, 635, 651, 655, 667, 669, 671, 679, 681, 685, 687, 695, 697, 699, 703, 707, 713, 717, 721, 723, 731, 737, 741, 745, 749, 755, 759, 763, 771, 777, 779, 781, 785, 789, 791, 793, 795, 799, 803, 805, 807, 813, 815, 817, 831, 835, 843, 849, 851, 865, 869, 871, 879, 885, 889, 893, 895, 897, 899, 901, 903, 905, 913, 915, 917, 921, 923, 933, 935, 939, 943, 949, 951, 955, 959, 965, 969, 973, 979, 985, 993, 995, 1001, 1003, 1005, 1007, 1011, 1015, 1023, 1027, 1037, 1041, 1043, 1045, 1057, 1065, 1067, 1073, 1081, 1085, 1095, 1099, 1105, 1111, 1113, 1115, 1121, 1131, 1133, 1135, 1139, 1141, 1145, 1147, 1157, 1159, 1169, 1173, 1177, 1185, 1189, 1199, 1207, 1209, 1211, 1219, 1221, 1235, 1239, 1241, 1243, 1245, 1247, 1261, 1265, 1271, 1273, 1281, 1295, 1311, 1353, 1407, 1419, 1435, 1443, 1455, 1463, 1479, 1491, 1505, 1515, 1533, 1545, 1547, 1581, 1595, 1599, 1605, 1635, 1645, 1653, 1659, 1677, 1695, 1705, 1729, 1743, 1749, 1767, 1771, 1785, 1833, 1855, 1885, 1887, 1955, 1995, 2015, 2035, 2093, 2145, 2415, 2805, 3003 , with p = 3 the values 445, 1495, 1615, with p = 5 the values 623, with p = 7 the values 583, 753, 1551 and for p = 11 the value 1335. Corollary 2 allows us to exclude the values 235, 237, 273, 341, 385, 415, 435, 497, 515, 517, 649, 767, 715, 989, 1079, 1309 . So, we only have to consider 29 values for N , which we present together with the splitting ∗ of the corresponding jacobians collected by the genus gN . From now on, the splitting r Y J0∗ (N ) ∼ Afi , with fi ∈ New∗Mi and dim Af i = ni Q i=1 ∗ Pr will be presented as n1M1 + · · · + nrMr . Obviously, gN = i=1 ni and, for a divisor M of N , ∗ P one has gM = Mi |N ni . 10 N J0∗ (N ) N J0∗ (N ) 201 267 + 1201 + 1201 265 153 + 1265 + 2265 + 2265 219 273 + 1219 + 1219 447 3149 + 3447 321 2107 + 2321 561 3187 + 3561 609 3203 + 3609 335 267 + 2335 615 1123 + 2205 + 1615 + 2615 345 2115 + 2345 309 2103 + 5309 399 157 + 2133 + 1399 437 2437 + 5437 483 2161 + 2483 861 1123 + 2287 + 4861 371 153 + 1371 + 3371 665 2133 + 6665 465 293 + 1155 + 2465 1155 177 + 2 165 + 3 385 + 1 1155 + 11155 551 2551 + 3551 689 153 + 1689 + 2689 + 2689 + 3689 705 1141 + 5235 + 1705 + 2705 555 137 + 1185 + 1185 + 2555 987 1141 + 3329 + 2987 + 3987 645 143 + 1129 + 1215 + 2645 1365 165 + 191 + 3273 + 1455 + 31365 663 2221 + 3663 957 4319 + 7957 1055 3211 + 3211 + 31055 + 61055 ∗ Proposition 3. Let N > 1 be an odd square-free integer such that gN > 3. The group ∗ Aut(X0 (N )) is non trivial if, and only if, N = 645. In this case, the order of this group is 2 and an equation for the quotient curve by the non trivial involution is given by Y 2 = X 6 + 8X 4 + 20X 2 + 12X + 4 . Proof. In order to apply Petri’s theorem to the curve X0∗ (N ), we will take a basis of Ω1X ∗ (N )/Q 0 as in Lemma 2, following the order showed in the splitting tables. ∗ For gN = 4, dim L2 = 1. In all cases in the above table, dim Lns 2 = 0. The genus of a quotient curve by an involution must be 2. We have to check the condition (3.2) for all pairs xi , and xj such that ωi and ωj are a basis of the regular differentials corresponding to an abelian quotient of J0∗ (N ) of dimension 2. A nonzero polynomial Q ∈ L2 neither satisfies Q(−x1 , −x2 , x3 , x4 ) = Q for N 6= 399 nor Q(x1 , −x2 , −x3 , x4 ) = Q for N = 399. Therefore, for all these cases the curves X0∗ (N ) have trivial automorphism group. ∗ For gN = 5, dim L2 = 3. The genus of a quotient curve by an involution must be 2 or 3. It is suffices to check the condition (3.2) for all pairs xi , and xj such that ωi and ωj is a basis of the regular differentials corresponding to an abelian quotient of J0∗ (N ) of dimension 2. For the values in the above table N 6= 645, one has dim Lns 2 = 0 and the polynomials of L2 are not simultaneously even in the variables xi , xj . Hence, for all these cases the curves X0∗ (N ) have trivial automorphism group. For N = 645, we know that X0∗ (645) is not trigonal (cf. [HS00]) and, thus, L2 defines the curve. The set ∪M |645 New∗M is New∗43 = {f1 }, New∗129 = {f2 }, New∗215 = {f3 }, √ √ New∗645 = {f4 = q + 2q 2 + · · · , f5 = q − 2q 2 + · · · } . Taking the following basis of Ω1X ∗ (645)/Q : 0 X X X ω1 = ( d f1 (q d ))dq/q , ω2 = ( d f2 (q d ))dq/q , ω3 = ( d f3 (q d ))dq/q , d|15 d|5 d|3 11 f4 + f5 f5 − f4 ω4 = dq/q and ω5 = √ dq/q , 2 2 2 we get that L2 = {Q = a Q1 + b Q2 + c Q3 : a, b, c ∈ Q}, where Q1 = 6x21 + 5x1 x2 + 7x1 x3 − 11x2 x3 − 9x23 + 2x24 + 48x4 x5 + 16x25 , Q2 = 2x1 x2 + 3x22 − 2x1 x3 + 4x2 x3 − 3x23 − 4x24 + 16x25 , Q3 = x2 x4 − x3 x4 − 2x1 x5 + x2 x5 + x3 x5 . Now, Lns 2 = hQ3 i. For all possible choices (i, j) ∈ {(1, 2), (1, 3), (2, 3), (4, 5)}, we only have the possibility (i, j) = (4, 5) satisfying condition (3.3), i.e. Q − Q(x1 , x2 , x3 , −x4 , −x5 ) ∈ Lns2 for all Q ∈ L2 . Therefore, there is a unique non trivial involution u sending (ω1 , ω2 , ω3 , ω4 , ω5 ) to ±(ω1 , ω2 , ω3 , −ω4 , −ω5 ). The number of eigenvalues equal to 1 is the genus of Xu = X0∗ (N )/hui and the corresponding eigenvectors among the differentials {ωi , i ≤ 5} is a basis of the pullback of Ω1Xu . To decide the precise sign, we compute the number of fixed points of u. The set of such points in P4 (Q) is the set of points of the form (0, 0, 0, x4 , x5 ) and of the form (x1 , x2 , x3 , 0, 0) satisfying Qi (x1 , · · · , x5 ) = 0 for 1 ≤ i ≤ 3. Since this set has 4 points, all of them of the form x4 = x5 = 0, the genus of Xu is 2 and {ω4 , ω5 } form a basis of the pullback of Ω1Xu . Take X = ω4 /ω5 and Y = dx/ω5 and we obtain Y 2 = X 6 + 8X 4 + 20X 2 + 12X + 4 . ∗ For gN = 6, we get dim Lns 2 = 0. The genus of a quotient curve by an involution must be 2 or 3. For the possible choices of pairs (i, j) or triples (i, j, k), there are polynomials in L2 that are not simultaneously even in the corresponding variables. Hence, all curves X0∗ (N ) of genus 6 have trivial automorphism group. ∗ For gN ≥ 6, in all cases dim Lns 2 = (g − 1)(g − 6)/2. We have to consider all choices ∗ ∗ (i1 , · · · , ir ) such that (gN − 5)/2 ≤ r ≤ (gN + 1)/2 and ωi1 , · · · , ωir is a basis of the pullback of ∗ regular differentials of a quotient of J0 (N ) of dimension r. After computing the vector space L2 , we can claim that in all these cases there are polynomials in L2 not satisfying the condition (3.3). Nevertheless, we note that some of these computations can be simplified by using Proposition and Remark 1. Indeed, for the pairs (N, p) in the set {(1155, 3), (705, 3), (987, 3), (1365, 3), (1365, 5), (957, 3), (1055, 5)} , the curve X0∗ (N/p) has trivial automorphism group and X0∗ (N/p)/Fp is not hyperelliptic. Hence, the choice (i1 , · · · , ir ) should contain the variables corresponding to the basis of the pullback of Ω1J ∗ (N/p) . Next, in the four following examples, we show how we use these results. 0 For instance, if X0∗ (1365) of genus 9 has a nontrivial involution u, by Porposition1, the jacobian of the quotient curve Xu has J0∗ (273) and J0∗ (455) as factors. This fact leads to the contradiction that Jac(Xu ) has a factor of dimension 6 when the genus of Xu is at most 5. For N = 957, take a basis {gi }1≤i≤4 of S2∗ (319) ∩ Q[[q]] and consider the set of the cusp forms hi (q) = gi (q) + 3gi (q 3 ) ∈ S2∗ (957). By observing the splitting of J0∗ (957), the vector space spanned by hi (q)dq/q would be the pullback of the regular differential of the unique possible quotient curve and the vector space of the homogenous polynomial in Q[x1 , · · · , x4 ] of degree 2 vanishing at these differentials would have dimension 1. After a computation, the dimension obtained is 0. So, we can exclude the level 957. Note that for X0∗ (957), dim L2 = 36. For N = 705, take a basis {gi }1≤i≤5 of S2∗ (235) ∩ Q[[q]] and consider the set of the cusp forms hi (q) = gi (q) + 3gi (q 3 ) ∈ S2∗ (957). The vector space spanned by hi (q)dq/q should be the pullback of the regular differential of the unique possible quotient curve and the vector space of 12 the homogenous polynomial in Q[x1 , · · · , x5 ] of degree 2 vanishing at these differentials should have dimension 3. After a computation, the dimension obtained is 0. So, we can exclude the level 705. If X0∗ (1551) has a non trivial involution, then the jacobian of the genus six curve X0∗ (211) is isogenous to the jacobian of the quotient curve. Since |X0∗ (1555)(F2 )|−2|X0∗ (211)(F2 )| = 4 > 0, we can discard N = 1551.  5 Even levels The next proposition is the key result to determine the even levels. Proposition 4. Let N > 1 be an odd square-free integer. Then, ∗ ∗ g2N − 2gN ≤ 2. ∗ ∗ ∗ Moreover, g2N − 2gN < −1 for all N > 1239 and, for the particular case gN > 2, one has that ∗ ∗ (i) g2N = 2 gN − 1 if, and only if, N is in the set {109, 113, 139, 151, 203, 227, 259, 263, 319, 355, 411, 445, 451, 455, 461, 491, 505, 521, 555, 573, 581, 591, 695, 699, 779, 1001, 1131, 1239} . ∗ ∗ (ii) g2N = 2 gN if, and only if, N is in the set {173, 267, 281, 295, 339, 341, 359, 371, 377, 413, 419, 429, 431, 447, 479, 483, 501, 551, 623, 627, 645, 663, 671, 755, 789, 987} . ∗ ∗ (iii) g2N = 2 gN + 1 if, and only if, N is in the set {149, 179, 239, 249, 251, 269, 305, 311, 321, 329, 393, 395, 519, 545, 689, 861, 897} . ∗ ∗ (iv) g2N = 2 gN + 2 if, and only if, N = 303. Proof. Let us denote by P the set of integer primes dividing N and set n := |P|. We have that the genera of X0 (N ) and X0 (2N ) are ψ(N ) ν2 ν3 ψ(N ) ν2 gN = 1 + − − − 2n−1 , g2N = 1 + − − 2n , (5.1) 12 4 3 4 4 where   0 if ∃p ∈ P , p ≡ −1 (mod 4), 0 if ∃p ∈ P , p ≡ −1 (mod 3), ν2 = , ν3 = , 2n otherwise. n−v3 (N ) 2 otherwise. where v3 denotes the 3-adic valuation. The genera of X0∗ (N ) and X0∗ (2N ) are ∗ 1 1 X ∗ 1 1 X gN = 1 + n (gN − 1) − n+1 ν(N, d) , g2N = 1 + n+1 (g2N − 1) − n+2 ν(2N, d) , 2 2 2 2 1 73. (iii) For ω(N ) = 6 when p1 ≥ 7 or p6 > 569. (iv) For ω(N ) = 5 when p1 ≥ 13 or p5 > 3373. (v) For ω(N ) = 4 when p1 ≥ 23 or p4 > 16573. (vi) For ω(N ) = 3 when p1 ≥ 53 or p3 > 37993. (vii) For ω(N ) = 2, when p1 ≥ 269 or p2 > 63737. (viii) For ω(N ) = 1 when p1 > 54277 . ∗ ∗ The statement is obtained after computing g2N − 2gN for the remaining values of N .  Remark 2. The proof presented in the above proposition needs a laborious computation because it involves a lot of possibilities and, thus, many class numbers. In opinion of the authors, there ∗ ∗ has to exist a deeper explanation to justify the inequality stated for g2N − 2 gN . 5.1 Candidates X0∗ (2N ) with non trivial automorphism group ∗ ∗ Combining Propositions 1 and 4, the cases g2N − 2 gN < −1 can be discarded, because of the ∗ ∗ inequality gN > (g2N + 1)/2 (see Corollary 1). Now, we only have to study the automorphism group of X0∗ (2N ) for the odd values N contained in the following five lists: ∗ ∗ (i) The list of values of N with gN ≤ 2 such that g2N > 3: `1 = {101, 107, 131, 161, 167, 177, 191, 205, 209, 213, 221, 285, 287, 299, 357} By Lemma 1 we discard N = 191 at p = 5. For the remaining N , the splitting of the jacobian of X0∗ (2N ) is N J0∗ (2N ) N J0∗ (2N ) 101 1101 + 3202 167 2167 + 2314 131 1131 + 1262 + 2262 177 1118 + 2177 + 1354 107 2107 + 1214 + 1214 213 1142 + 2213 + 1426 161 2161 + 2322 285 157 + 1190 + 1285 + 1570 N J0∗ (2N ) N J0∗ (2N ) 205 182 + 2205 + 2410 287 182 + 2287 + 1574 + 1574 209 2209 + 3418 357 1102 + 1238 + 2357 + 1714 221 2221 + 3442 299 2299 + 4598 15 The genus of a quotient curve of any of these curves X0∗ (2N ) by an involution non ∗ ∗ bielliptic must be equal to 2 when g2N = 4 and equal to 2 or 3 when 5 ≤ g2N ≤ 6. Hence, we can discard N = 101 and for the remaining cases we apply Lemma 2. Finally, we get that all curves X0∗ (2N ) with N ∈ `1 have trivial automorphism group. ∗ (ii) The list containing the values of N with gN > 2 such that X0∗ (N )/F2 could be hyperel- liptic: `2 = {183, 185, 187, 203, 335, 345, 385} . The splitting of the jacobian of X0∗ (2N ) N J0∗ (2N ) N J0∗ (2N ) 183 161 + 1126 + 2183 203 158 + 3203 + 1406 185 137 + 1185 + 1185 + 1370 385 177 + 1154 + 3385 187 3187 + 1374 335 267 + 2335 + 2670 345 2115 + 1138 + 2345 + 1690 The curves X0∗ (2N ) with N ∈ `2 \{183} have trivial automorphism group. Indeed, we can discard N = 187 because J0∗ (374) has no two dimensional quotients and the remaining values of N can be excluded by applying Lemma 2. For X0∗ (366) the automorphism group has order 2 (cf. [BG19]). ∗ (iii) The list containing the values of N with gN > 2 such that Aut(X0∗ (N )) is not trivial (except to 183 because it is in the list `2 ) is: `3 = {249, 303, 455} , with N 249 303 455 J0∗ (2N ) 183 + 1166 + 1249 + 1249 + 3498 1101 + 3202 + 2303 + 2606 165 + 191 + 1455 + 2910 By applying Lemma 2, we get that all these three curves X0∗ (2N ) have trivial automor- phism group. We point out that X0∗ (202)/F3 is not hyperelliptic (see Lemma 4) and the automorphism group of X0∗ (202) is trivial. Hence, by Proposition 1, for X0∗ (606) we only have to consider as possible quotient the curve whose jacobian is isogenous to J0∗ (202). ∗ ∗ ∗ (iv) The list `4 contains the values of N with gN > 2 and −1 ≤ g2N − 2gN ≤ 0, except to N = 203 that is in the list `2 . So, the list `4 is the union of the sets {109, 113, 139, 151, 227, 259, 263, 319, 355, 411, 445, 451, 455, 461, 491, 505, 521, 555, 573, 581, 591, 695, 699, 779, 1001, 1131, 1239} , ∗ ∗ for which g2N − 2gN = −1, and {173, 267, 281, 295, 339, 341, 359, 371, 377, 413, 419, 429, 431, 447, 479, 483, 501, 551, 623, 627, 645, 663, 671, 755, 789, 987} . For all these values, X0∗ (N ) has trivial automorphism group and its reduction modulo 2 is not hyperelliptic (see Lemma 5). By Proposition 1, if X0∗ (2N ) has a non trivial involution, then the jacobian of the quotient curve is isogenous to J0∗ (N ) because gN∗ is the greatest ∗ genus of a quotient curve of X0 (2N ). 16 ∗ ∗ When g2N − 2gN = −1, if X0∗ (2N ) has a non trivial involution, by Hurwitz, this cannot have fixed points. Hence, for all odd primes p - N and all integers k > 0, the number R(2N, pk ) := |X0∗ (2N )(Fpk )| must be even. Since N pk R(2N, pk ) N pk R(2N, pk ) N pk R(2N, pk ) N pk R(2N, pk ) 109 33 21 319 3 9 491 35 75 695 3 9 5 113 3 9 355 3 7 505 3 147 699 5 23 139 33 21 411 55 3323 521 33 29 779 33 21 151 3 17 445 33 29 555 11 17 1001 35 305 3 3 7 227 3 9 451 3 11 573 5 89 1131 5 75993 259 5 17 455 3 7 581 33 27 1239 5 15 263 33 17 461 3 15 591 5 17 we can discard all these values. ∗ ∗ When g2N = 2gN , by applying Lemma 1, we can discard some values of N . With the prime p = 3, the values of N in the set {173, 281, 359, 377, 419, 431, 479, 671, 755} , and with the prime p = 5, the following values {413, 501, 623, 789} . We only have to consider N J0∗ (2N ) N J0∗ (2N ) 295 1118 + 3295 + 2590 551 158 + 2551 + 3551 + 21102 + 21102 429 1143 + 1286 + 2429 + 2858 645 143 + 1129 + 1215 + 1258 267 189 + 2178 + 3267 + 2534 +1430 + 2645 + 31290 341 4341 + 4682 663 1102 + 2221 + 3442 + 3663 + 11326 483 2161 + 2322 + 2483 + 1966 + 1966 447 3149 + 1298 + 3298 + 3447 + 1894 + 1894 339 3113 + 2226 + 2339 + 1678 + 2678 627 157 + 2209 + 3418 + 3627 + 31254 371 153 + 1106 + 1371 + 3371 987 1141 + 2282 + 3329 + 4658 + 2987 +1742 + 3742 +3987 + 31974 It can be checked that the curves X0∗ (322)/F3 , X0∗ (226)/F3 ,X0∗ (430)/F3 , X0∗ (442)/F3 , X0∗ (298)/F3 , X0∗ (418)/F3 , and X0∗ (658)/F3 are not hyperelliptic. By Proposition 1, the jacobian of a quotient curve de X0∗ (2N ) is isogenous to J0∗ (N ). Thus, we can discard the values N ∈ {483, 339, 645, 663, 447, 627, 987} . Although X0∗ (178) has a unique non trivial involution, whose quotient curve is isogenous to an elliptic curve of conductor 89, the curve X0∗ (534) can also be discarded. Indeed, a non trivial involution of X0∗ (534) should induce the identity on the elliptic curve of conductor 89 and, thus on the curve X0∗ (178). But this fact leads to the contradiction that the jacobian of the quotient curve would have as a factor an abelian surface attached to a newform of level 178. For the remaining values, i.e. N ∈ {295, 429, 341, 371, 551} , we apply Lemma 2 and we get that the five curves X0∗ (2N ) have trivial automorphism group. 17 ∗ ∗ ∗ (v) The list of values N with gN > 2 and g2N > 2gN , except to N = 249 and N = 303 because they are in the lists `3 , are the following: `5 = {149, 179, 239, 251, 269, 303, 305, 311, 321, 329, 393, 395, 519, 545, 689, 861, 897} . ∗ ∗ For all these cases, g2N −2gN = 1. By Proposition 1, if X0∗ (2N ) has a non trivial involution, then the pullback of the regular differentials of the quotient curve is the pullback of Ω1J ∗ (N ) 0 or Ω1J ∗ (N )×E for some elliptic curve E in the decomposition of J0∗ (2 N ), whose conductor 0 does not divide N . By applying Lemma 1 to X0∗ (2 N ), we can discard the values of N in the set {149, 179, 239, 251, 269, 311, 393, 519, 545} , taking p = 5 for 2 N = 2 · 393, 2 · 519, 2 · 179, p = 7 for 2 N = 2 · 545 and p = 3 for the remaining cases. The splitting of J0∗ (2N ) for the non discarded cases is as follows N J0∗ (2N ) N J0∗ (2N ) 329 3329 + 4658 861 182 + 1123 + 1246 + 2287 + 1574 + 1574 305 161 + 1122 + 3305 + 4610 +4861 + 11722 + 11722 + 21722 321 2107 + 1214 + 1214 + 2321 + 3642 897 1138 + 2299 + 4598 + 5897 + 31794 395 179 + 1158 + 3395 + 4790 689 153 + 1106 + 1689 + 2689 + 3689 +3689 + 31378 + 61378 It can be checked that the curves X0∗ (214)/F3 , X0∗ (574)/F3 , and X0∗ (598)/F3 are not hyperelliptic. By Proposition 1, the values N ∈ {321, 861, 897} can be discarded. For the remaining values, i.e. N = 329, 305, 395, 689, we apply Lemma 2 and we get that the five curves X0∗ (2N ) have trivial automorphism group. For instance, for N = 689 we can proceed in a easier way. If X0∗ (1378) has a non trivial involution u, then the jacobian of the quotient curve Xu is isogenous to J0∗ (689) or J0∗ (689) × Ag , where g is the only newform in New∗106 . None of these curves could be hyperelliptic because |Xu (F3 )| > 8. Take a basis g1 , · · · , g9 of S2∗ (689) ∩ Q[[q]]. Put hi (q) := gi (q)+2gi (q 2 ) and h10 (q) := g(q)+13g(q 13 ). It can be checked that the dimension of the vector space of homogenous polynomials in 9 variables (resp. 10 variables) vanishing at h1 , · · · , h9 (resp. h1 , · · · , h10 ) has dimension 1. Hence, N = 689 can be excluded. Note that for X0∗ (1378), dim L2 = 153. As a consequence of this analysis and taking into account [BG19][Proposition 24], we get the following result. ∗ Proposition 5. Let N > 1 be an even square-free integer such that gN > 3 and X0∗ (N ) is not ∗ bielliptic. The group Aut(X0 (N )) is non trivial if, and only if, N = 366. In this case, the order of this group is 2 and an equation for the quotient curve by the non trivial involution is given by Y 2 = X 6 − 6X 5 + 23X 4 − 42X 3 + 53X 2 − 24X + 4 . This proposition together Proposition 3 concludes the proof of Theorem 2. Acknowledgements We thank C. Ritzenthaler for his valuable comments and B. Poonen, whose contribution was decisive to prove Proposition 4. 18 References [BGGP05] M. H. Baker, E. González-Jiménez, J. González, and B. Poonen. Finiteness results for modular curves of genus at least 2. Amer. J. Math., 127(6):1325–1387, 2005. [BH03] M. H. Baker and Y. Hasegawa. Automorphisms of X0∗ (p). J. Number Theory, 100(1):72–87, 2003. [BG19] F. Bars and J. González. Bielliptic modular curves X0∗ (N ) with square-free levels. Math. Comp., 88(320):2939–2957, 2019. [BG20] Francesc Bars and Josep González. Bielliptic modular curves X0∗ (N ). J. Algebra, 559:726–759, 2020. [BGX21] F. Bars, J. González, and X. Xarles. Hyperelliptic parametrizations of Q-curves. To be published in Ramanujan Journal., 2021. [Cox13] D. A. Cox. Primes of the form x2 +ny 2 . Pure and Applied Mathematics (Hoboken). John Wiley & Sons, Inc., Hoboken, NJ, second edition, 2013. Fermat, class field theory, and complex multiplication. [Elk90] N. D. Elkies. The automorphism group of the modular curve X0 (63). Compositio Math., 74(2):203–208, 1990. [GL98] Josep González and Joan-C. Lario. Rational and elliptic parametrizations of Q- curves. J. Number Theory, 72(1):13–31, 1998. [Gon16] Josep González. Automorphism group of split Cartan modular curves. Bull. Lond. Math. Soc., 48(4):628–636, 2016. [Gon17] J. González. Constraints on the automorphism group of a curve. J. Théor. Nombres Bordeaux, 29(2):535–548, 2017. [Har14] M. Harrison. A new automorphism of X0 (108). https//arxiv.org/abs/1108.5595., 2014. [Has97] Y. Hasegawa. Hyperelliptic modular curves X0∗ (N ). Acta Arith., 81(4):369–385, 1997. [HH96] Y. Hasegawa and K. Hashimoto. Hyperelliptic modular curves X0∗ (N ) with square- free levels. Acta Arith., 77(2):179–193, 1996. [HS00] Y. Hasegawa and M. Shimura. Trigonal modular curves X0∗ (N ). Proc. Japan Acad. Ser. A Math. Sci., 76(6):83–86, 2000. [Klu77] P. G. Kluit. On the normalizer of Γ0 (N ). In Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), pages 239–246. Lecture Notes in Math., Vol. 601, 1977. [KM88] M. A. Kenku and F. Momose. Automorphism groups of the modular curves X0 (N ). Compositio Math., 65(1):51–80, 1988. [Lan01] M.L. Lang. Normalizers of the congruence subgroups of the Hecke groups G4 and G6 . J. Number Theory, 90(1):31–43, 2001. 19 [Li75] W. C. W. Li. Newforms and functional equations. Math. Ann., 212:285–315, 1975. [Liu02] Q. Liu. Algebraic geometry and arithmetic curves, volume 6 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné, Oxford Science Publications. [Ogg74] A. P. Ogg. Hyperelliptic modular curves. Bull. Soc. Math. France, 102:449–462, 1974. [Rib75] K. A. Ribet. Endomorphisms of semi-stable abelian varieties over number fields. Ann. Math. (2), 101:555–562, 1975. [Ser89] J.-P. Serre. Lectures on the Mordell-Weil theorem. Aspects of Mathematics, E15. Friedr. Vieweg & Sohn, Braunschweig, 1989. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt. 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 20