ACTA ARITHMETICA Online First version Infinitely many cubic points for X0+ (N ) over Q by Francesc Bars (Barcelona) and Tarun Dalal (Sangareddy) 1. Introduction. A non-singular smooth curve C over a number field K of genus gC > 1 always has a finite set C(K) of K-rational points by a cele- brated result of Faltings (here we fix once and for all K, an algebraic closure S K). We denote the set of all points of degree of at most dSfor C by Γd (C, K) = [L:K]≤d C(L) and of exact degree d by Γ ′ (C, K) = d [L:K]=d C(L), where L ⊆ K runs over the finite extensions of K. A point P ∈ C is said to be a point of degree d over K if [K(P ) : K] = d. The set Γd (C, M ) is infinite for a certain finite extension M/K if C admits a degree at most d map, all defined over M , to a projective line or an elliptic curve with positive M -rank. The converse is true for d = 2 [HaSi], d = 3 [AbHa] and d = 4 under certain restrictions [AbHa, DeFa]. If we fix the number field M in the above results (i.e. an arithmetic statement for Γd (C, M ) with M fixed), we need a precise understanding over M of the set Wd (C) = {v ∈ Picd (C) | h0 (C, Lv ) > 0} where Picd is the usual d-Picard group and Lv the line bundle of degree d on C associated to v. If Wd (C) contains no translates of abelian subvarieties with positive M -rank of Picd (C) then Γd′ (C, M ) is finite (under the assumption that C admits no maps of degree at most d to a projective line over M ). For d = 2 the arithmetic statement for Γd (C, K) follows from [AbHa] (for a sketch of the proof and the precise statement see [Ba, Theorem 2.14]). For d = 3, Daeyeol Jeon [Jeo21] introduced an arithmetic statement and its proof following [AbHa] and [DeFa]. In particular, if gC ≥ 3 and C has no degree 3 or 2 map to a projective line and no degree 2 map to an elliptic curve over K then the set of exact cubic points of C over K, Γ3′ (C, K), is 2020 Mathematics Subject Classification: Primary 11G18; Secondary 11G30, 14G05, 14H10, 14H25. Key words and phrases: cubic points, modular curves, Petri model. Received 14 July 2022. Published online *. DOI: 10.4064/aa220714-10-11 [1] © Instytut Matematyczny PAN, *** 2 F. Bars and T. Dalal infinite if and only if C admits a degree 3 map to an elliptic curve over K with positive K-rank. Observe that if gC ≤ 1 (with C(K) ̸= ∅ for gC = 1), then C has a degree 3 map over K to the projective line, thus Γ3′ (C, K) is always infinite. Thus for curves C with C(K) ̸= ∅ we restrict to gC ≥ 2 in order to study the finiteness of Γ3′ (C, K). Let N be an integer greater than 1 and consider the modular curve X0 (N ) whose non-cusp points correspond to isomorphism classes of isogenies between elliptic curves ϕ : E → E ′ of degree N with cyclic kernel. The rational and quadratic points of X0 (N ) have been studied by many authors. In particular, Jeon [Jeo21] determined the finite set of modular curves X0 (N ) where Γ3′ (X0 (N ), Q) is infinite. Next, the Fricke involution wN on X0 (N ) arises from taking the dual isogeny ϕ̂ : E ′ → E. We define the modular curve X0+ (N ) to be the quotient of X0 (N ) by the group of two elements generated by wN . There is a model for X0+ (N ) over Q, and the study of Q-rational points and quadratic points on those curves attracted the attention of Momose [Mo] and Galbraith [Ga02] and many others. In this paper, we deal with determining whether there are infinitely many cubic points on X0+ (N ) for genus ≥ 2. The values of N for which X0+ (N ) has genus 0 and 1 are listed in Theorem 3 and recall that X0+ (N )(Q) ̸= ∅, because it has a rational cusp. The novelty of the paper compared to previous works on degree 2 and 3 maps to an elliptic curve E with positive Q-rank is considering the cover Q(X0 (N ))/Q(E) by taking into account the action of an Atkin–Lehner in- volution. The main result of the article is the following. Theorem 1. Suppose gX + (N ) ≥ 2. Then Γ3′ (X0+ (N ), Q) is infinite if and 0 only if gX + (N ) = 2 or N is in the following list: 0 gX + (N ) N 0 3 58, 76, 86, 96, 97, 99, 100, 109, 113, 127, 128, 139, 149, 151, 169, 179, 239 4 88, 92, 93, 115, 116, 129, 137, 155, 159, 215 5 122, 146, 181, 185, 227 6 124, 163, 164, 269 7 196, 243 10 236 All computation sources used in the paper are available at https://github. com/Tarundalalmath/X_0-N-with-infinitely-many-cubic-points except the ones for counting points over finite fields, where we use modified versions for X0+ (N ) of the ones already available at different links in [BaGo]. Infinitely many cubic points for X0+ (N ) over Q 3 Having the result of this paper on cubic points for C = X0+ (N ), Theo- rem 1, or [Jeo21] for C = X0 (N ), one can try to determine the whole set of cubic points for such C’s when Γ3′ (C, Q) is finite. This problem could be attacked if the Chabauty method given by Siksek [Si09] (or [BoGaGo]) could apply. 2. General considerations. Given a complete curve C over K, the gonality of C is defined as Gon(C) := min {deg(φ) | φ : C → P1 defined over K}. By [Jeo21, Lemma 1.2] (and arguments there) we have Lemma 2. Suppose Gon(C) ≥ 4, P ∈ C(K) and C does not have a degree ≤ 2 map to an elliptic curve. If the set Γ3′ (C, K) is infinite then C admits a K-rational map of degree 3 to an elliptic curve with positive K-rank. The modular curves X0+ (N ) to which Lemma 2 is not applicable are listed in the next result, corresponding to the works [FuHa,Jeo18,HaSh99b] (the list with gX + (N ) ≤ 1 is well-known and follows easily from [BaGo, Appendix]). 0 Theorem 3. (i) The modular curve X0+ (N ) has gX + (N ) = 0 if and only if N is one of 0 the following: 1–21, 23–27, 29, 31, 32, 35, 36, 39, 41, 47, 49, 50, 59, 71. (ii) X0+ (N ) is an elliptic curve (equivalently gX + (N ) = 1) if and only if N 0 is one of the following: 22, 28, 30, 33, 34, 37, 38, 40, 43, 44, 45, 48, 51, 53–56, 61, 63–65, 75, 79, 81, 83, 89, 95, 101, 119, 131. (iii) (Furumoto–Hasegawa) X0+ (N ) is hyperelliptic if and only if N is one of the following: 42, 46, 52, 57, 60, 62, 66–69, 72–74, 77, 80, 85, 87, 91, 92, 94, 98, 103, 104, 107, 111, 121, 125, 143, 167, 191. (iv) (Jeon) X0+ (N ) is bielliptic,i.e. has a degree 2 map to an elliptic curve, if and only if N is one of the following: 42, 52, 57, 58, 60, 66, 68, 70, 72, 74, 76–78, 80, 82, 84–86, 88, 90, 91, 96, 98–100, 104, 105, 108, 110, 111, 117, 118, 120, 121, 123, 124, 128, 135, 136, 141–145, 155, 159, 171, 176, 188. (v) (Hasegawa–Shimura) Gon(X0+ (N )) = 3 if and only if N is one of the following: 4 F. Bars and T. Dalal 58, 70, 76, 82, 84, 86, 88, 90, 93, 96, 97, 99, 100, 108, 109, 113, 115, 116, 117,122, 127, 128, 129, 135, 137, 139, 146, 147, 149, 151, 155, 159, 161, 162, 164, 169, 173, 179, 181, 199, 215, 227, 239, 251, 311. We say that a pair (N, E), where N is a natural number and E is an elliptic curve over Q with positive Q-rank, is admissible if there is a degree 3 map over Q of the form X0+ (N ) → E. The following lemma gives a criterion to rule out the pairs which are not admissible. Lemma 4. If (N, E) is an admissible pair, then: (i) E has conductor M with M | N and for any prime p ∤ N we have + |X 0 (N )(Fpn )| ≤ 3|E(Fpn )| and |X 0 (N )(Fpn )| ≤ 6|E(Fpn )|, ∀n ∈ N; (ii) if the conductor of E is N , then the degree of the strong Weil paramet- rization of E divides 6; (iii) for any prime p ∤ N we have p−1 ψ(N ) + 2ω(N ) ≤ 6(p + 1)2 , 12 Q ω(N ) is the number of prime divisors of N and ψ = where N q|N, q prime (1 + 1/q) is the ψ-Dedekind function; (iv) for any Atkin–Lehner involution wr of X0 (N ) with r ̸= N we have gX + (N ) ≤ 3 + 2 · gX + (N )/wr + 2. 0 0 Proof. Let (N, E) be admissible. Then there is a Q-rational degree 3 mapping f : X0+ (N ) → E and consequently we have a Q-rational degree 6 mapping g : X0 (N ) → E. Hence cond(E) | N . (i) Let p ∤ N be a prime. Since p ∤ N , the curves X0+ (N ), X0 (N ) and E have good reduction at p and the mappings f, g induce the Fp -rational + + mappings f : X 0 (N ) → E and g : X 0 (N ) → E, where X 0 (N ), X 0 (N ) and E denote the mod p reductions of X0+ (N ), X0 (N ) and E respectively. + Hence we have |X 0 (N )(Fpn )| ≤ 3|E(Fpn )| and |X 0 (N )(Fpn )| ≤ 6|E(Fpn )|, for all n ∈ N. (ii) If cond(E) = N , and E ′ denotes the strong Weil curve with strong Weil parametrization φ : X0 (N ) → E ′ , then there exists an isogeny ψ : E ′ → E such that g = ψ ◦ φ, hence the degree of the strong Weil parametrization divides 6. (iii) For any prime p ∤ N we know that |X 0 (N )(Fp2 )| ≥ p−1 12 ψ(N ) + 2ω(N ) (cf. [HaSh99a, Lemma 3.1]) and |E(Fp2 )| ≤ (p + 1)2 . Hence we have p−1 ω(N ) ≤ 6(p + 1)2 . 12 ψ(N ) + 2 (iv) We know f : X0+ (N ) → E is a degree 3 mapping. If wr is an Atkin– Lehner involution on X0 (N ) with r ̸= N , then we have a degree 2 mapping X0+ (N ) → X0+ (N )/wr . The result follows from Castelnuovo’s inequality. Infinitely many cubic points for X0+ (N ) over Q 5 As an immediate application of Lemma 4(iii) we obtain the following: Corollary 5. For N > 623, the pair (N, E) is not admissible. Proof. The proof is similar to that of [HaSh99a, Lemma 3.2]. We will show that for N ≥ 623, there exists a prime p ∤ N such that 12 ψ(N ) > (6(p + 1)2 − 2w(N ) ). p−1 • If 2 ∤ N and N > 623, then choosing p = 2 we have ψ(N ) ≥ N + 1 > 624 = 12(6 · (2 + 1)2 − 2) ≥ 12(6 · (2 + 1)2 − 2w(n) ). • If 2 | N , 3 ∤ N and N > 376, then choosing p = 3 we have ψ(N ) ≥ 3N 2 > 564 = 12 2 (6.16 − 2) > 12 2 (6.16 − 2w(N ) ). • If 2 · 3 | N , 5 ∤ N and N > 321, then choosing p = 5 we have 3 4 12 ψ(N ) ≥ N · · > (6.36 − 2). 2 3 4 • If 2 · 3 · 5 | N , 7 ∤ N and N > 319, choosing p = 7 we have 3 4 6 12 ψ(N ) ≥ N · 2 · 3 · 5 > 6 (6.64 − 2). • If 2 · 3 · 5 · 7 | N , choose p to be the smallest prime not dividing N . After applying Lemma 4 (see Appendix B for a list of N ’s that we can discard in each item), we are reduced to a finite set of N ’s. To deal with the remaining admissible pairs, the next two lemmas will be helpful. Lemma 6. Let E/Q be an elliptic curve of conductor N and let φ : X0 (N ) → E be the strong Weil parametrization of degree k defined over Q. Suppose that wN acts as +1 on the modular form fE associated to E. Then φ factors through X0+ (N ) (and k is even). Proof. Consider the mapping φ : X0 (N ) → E. Following [CaEm, p. 424] (or [De, §2]), the fact that wN fE = fE implies φ ◦ wN = φ + P , where P is a torsion point of E given by P = φ(0) − φ(∞), where 0, ∞ are the corresponding cusps on X0 (N ) with φ(∞) = OE (recall that OE denotes the zero point of E). Because the sign of the functional equation of fE is −1, the Q-rank of E is odd (cf. [MaSD, §3.1]); this implies that P = OE (see [CaEm]), so φ factors through the quotient X0 (N )/⟨wN ⟩ and wN acts as the identity on E. Lemma 7. Consider a degree k map φ : X → E defined over Q where X is a quotient modular curve X0 (N )/WN with WN a proper subgroup of B(N ) (B(N ) is the subgroup of Aut(X0 (N )) generated by all Atkin–Lehner involutions). Assume that cond(E) = M (M | N ). Let d ∈ N with d || M , (d, N/d) = 1 and wd ∈/ WN be such that wd acts as +1 on the modular form fE associated to E. 6 F. Bars and T. Dalal (i) If E has no non-trivial 2-torsion over Q, then φ factors through X/⟨wd ⟩ and k is even. (ii) If E has non-trivial 2-torsion over Q and k is odd, then we obtain a degree k map φ′ : X/⟨wd ⟩ → E ′ by taking the wd -invariant to φ, where E ′ is an elliptic curve isogenous to E. Proof. Let E(C) ∼ = C/Λ. The mapping φ can be considered as a mapping τ in the complex field, φ̃ : Γ \H → C/Λ, defined by τ 7→ i∞ const · f (τ ′ ) dτ ′ , where Γ := ⟨Γ0 (N ), WN ⟩ and f ∈ d ⟨WN ⟩ L d|N/M QfE (q ) ∈ S2 (Γ0 (N )) (cf. [Go]). Since wd acts on fE as +1, it also acts on f as +1. Moreover, φ̃(wd τ ) − φ̃(τ ) = P is independent of τ . Thus φ ◦ wd = φ + P . Since wd is an involution, we obtain 2P ∈ Λ, and P is a 2-torsion point of E(C) (which could be the trivial zero point of E, i.e. belonging to Λ). Therefore we have the following commutative diagram (proj is the usual projection map): φ X(C) C/Λ proj proj φwd X/⟨wd ⟩(C) C/⟨Λ, P ⟩ Thus if E has no non-trivial 2-torsion over Q, then P is the trivial zero of E and φ factors through X/⟨wd ⟩. On the other hand, if E has non-trivial 2-torsion over Q and k is odd, then from the above commutative diagram we see that P is a non-trivial 2-torsion point of E and φ induces a Q-rational degree k mapping φ′ : X/⟨wd ⟩ → E ′ where E ′ (C) ∼ = C/⟨Λ, P ⟩ and E ′ is isogenous to E. As an immediate corollary of Lemma 7 we obtain Corollary 8. Let N be natural number which is not a power of a prime number. Take a pair (N, E) with conductor of E equal to M with M | N and M ̸= N . Let d be a natural number with d || M , (d, N/d) = 1 such that wd acts as +1 on the modular form fE associated to E. Suppose that E has no non-trivial 2-torsion over Q. Then (N, E) is not admissible. Proof. If (N, E) is admissible, then we have a degree 3 mapping φ : X0+ (N ) → E. Since wd acts as +1 on fE and E has no non-trivial 2-torsion over Q, by Lemma 7 the map φ factors through X0+ (N )/⟨wd ⟩. This is a contradiction since φ has degree 3. 3. The curve X0+ (N ) with N not listed in Theorem 3. Here by Lemma 2 it is enough to determine the admissible pairs (N, E). After apply- ing Lemma 4 (see Appendix B for a list of N ’s that we can discard in each item), we are reduced to the following finite set of candidates for admissible pairs. Infinitely many cubic points for X0+ (N ) over Q 7 (N, E) AL-action on E (N, E) AL-action on E (106, 53a) w53 = + (195, 65a) w5 = +, w13 = + (114, 57a) w3 = +, w19 = + (196, 196a) w196 = + (130, 65a) w5 = +, w13 = + (202, 101a) w101 = + (158, 79a) w79 = + (231, 77a) w7 = +, w11 = + (163, 163a) w163 = + (236, 118a) w2 = +, w59 = + (166, 83a) w83 = + (236, 236a) w236 = + (172, 43a) w43 = + (237, 79a) w79 = + (174, 58a) w2 = +, w29 = + (243, 243a) w243 = + (178, 89a) w89 = + (249, 83a) w83 = + (182, 91a) w7 = +, w13 = + (258, 43a) w43 = + (182, 91b) w7 = −, w13 = − (258, 129a) w3 = +, w43 = + (183, 61a) w61 = + (267, 89a) w89 = + (185, 37a) w37 = + (269, 269a) w269 = + (185, 185c) w185 = + When (N, E) is in the table above with cond(E) = N , the strong Weil parametrization X0 (N ) → E has degree 6. Thus we conclude by Lemma 6 that (N, E) is an admissible pairing. More precisely, we have Corollary 9. For N = 163, 185, 196, 236, 243, 269 the modular curve X0+ (N ) has infinitely many cubic points over Q. To deal with the remaining cases we use Lemma 7. Corollary 10. For N = 106, 114, 158, 166, 172, 174, 178, 182, 183, 202, 231, 237, 249, 258, 267, the set Γ3′ (X0+ (N ), Q) is finite. Proof. Let N be as in the statement and (N, E) be a pair appearing in the above table. Then cond(E) | N , cond(E) ̸= N and E has no non-trivial 2-torsion over Q. By Corollary 8, we conclude that the pair (N, E) is not admissible (for (182, 91b) use the Atkin–Lehner operator w91 ). The result follows. Proposition 11. The modular curves X0+ (130) and X0+ (195) each have finitely many cubic points over Q. Proof. We need to check the pairs (130, 65a) and (195, 65a). Considering φ : X0+ (130) → 65a of degree 3, we know that w5 and w13 act as +1. Since the degree of Q(X0+ (130))/Q(X0∗ (130)) is coprime to 3 (recall that X0∗ (N ) := X0 (N )/B(N ) where B(N ) is the subgroup of Aut(X0 (N )) generated by all Atkin–Lehner involutions), by applying Lemma 7 twice with w5 and w13 we obtain a degree 3 morphism (moreover an isogeny) X0∗ (130) → E ′ between elliptic curves, where E ′ is isogenous to 65a (note that X ∗ (130) has genus 1 and its Cremona level is 65a). This is a contradiction since the elliptic curve 65a has no non-trivial 3-torsion over Q, and also no 3-isogeny over Q by [Cr]. 8 F. Bars and T. Dalal Thus (130, 65a) is not admissible. A similar argument holds for the pair (195, 65a): recall that X0∗ (195) has genus 1 and its Cremona level is 65a. 4. The curve X0+ (N ) with N listed in Theorem 3. Recall that X0+ (N )(Q)̸= ∅. Thus Γ3′ (X0+ (N ), Q) is an infinite set when gX + (N ) ≤ 1. 0 We assume, once and for all, gX + (N ) ≥ 2. 0 4.1. The levels N with X0+ (N ) hyperelliptic. We deal with the following levels N : gX + (N ) N 0 2 42, 46, 52, 57, 62, 67, 68, 69, 72, 73, 74, 77, 80, 87, 91, 98, 103, 107, 111, 121, 125, 143, 167, 191 3 60, 66, 85, 104 4 92, 94 For such hyperelliptic curves, we pick the model given by Hasegawa [Ha] if gX + (N ) = 2, and by Furumoto and Hasegawa [FuHa] when gX + (N ) ≥ 3. 0 0 Theorem 12 ([JKS04, Lemma 2.1]). Let X be a curve of genus 2 over a perfect field k. If X has at least three k-rational points, then there exists a map X → P1 of degree 3 which is defined over k. As an immediate consequence of the last theorem, we have Proposition 13. X0+ (N ) has infinitely many cubic points over Q for N ∈ {42, 46, 52, 57, 67, 68, 69, 72, 73, 74, 77, 80, 91, 103, 107, 111, 121, 125, 143, 167, 191}. Proof. Using MAGMA it can be easily checked that in this case the genus 2 hyperelliptic curve X0+ (N ) has at least three Q-rational points. The remaining values of N with gX + (N ) = 2 are N = 62, 87, 98. 0 Proposition 14. For N ∈ {62, 87}, the set Γ3′ (X0+ (N ), Q) is infinite. Proof. Consider N = 62. An affine model of X0+ (62) is given by Y : y 2 = x6 − 8x5 + 26x4 − 42x3 + 29x2 + 2x − 11. Then Y has two Q-rational points ((1 : 1 : 0) and (1 : −1 : 0)) which are the “points at infinity”, and the hyperelliptic involution permutes them. Therefore, from [Jeo21, Lemma 2.2] we conclude that there is a Q-rational degree 3 mapping X0+ (62) → P1 , and consequently X0+ (62) has infinitely many cubic points over Q. A similar argument works for N = 87 with the model Y : y 2 = x6 − 4x5 + 12x4 − 22x3 + 32x2 − 28x + 17. Lemma 15. The genus 2 curve X0+ (98) has infinitely many cubic points over Q. Infinitely many cubic points for X0+ (N ) over Q 9 Proof. An (affine) model of X0+ (98) is given by y 2 = 4x5 − 15x4 + 30x3 − 35x2 + 24x − 8. Suppose D is a degree 3 effective Q-rational divisor on a curve of genus 2. By the Riemann–Roch theorem we have dim L(D) = 2. Observe that with y = 1 in the model we get 0 = (x2 − x + 1) x3 − 11 2 15 9  4 x + 4 x− 4 . Let t1 , t2 , t3 be the roots of the equation x3 − 11 4 x + 4 x − 4 . Then Pi := 2 15 9 (ti , 1) ∈ X0+ (98)(K) for 1 ≤ i ≤ 3 (where K is a cubic extension of Q defined by the polynomial t3 − 11 4 t + 4 t − 4 ). Furthermore, the divisor 2 15 9 [P1 + P2 + P3 ] is a Q-rational effective divisor of degree 3. By Riemann– Roch we have dim L([P1 + P2 + P3 ]) = 2. Therefore, there exists a Q-rational function f with exactly three poles and consequently there is a degree 3 mapping X0+ (98) → P1 defined over Q. The result follows. Consider X0+ (N ) hyperelliptic with gX + (N ) ≥ 3. By [Jeo21, §2.3], in order 0 for X0+ (N ) to have infinitely many cubic points, W3 (X0+ (N )) must contain an elliptic curve with positive Q-rank. Thus, by Cremona tables [Cr] we obtain (because there is no elliptic curve with Q-rank ≥ 1 for levels dividing N ): Corollary 16. For N ∈ {60, 66, 85, 94, 104}, the set Γ3′ (X0+ (N ), Q) is finite. Proposition 17. X0+ (92) has infinitely many cubic points over Q. Proof. The strong Weil modular parametrization ϕ : X0 (92) → 92b has degree 6 and 92b has Q-rank 1, and w92 acts as +1 on 92b; therefore, we have a Q-rational degree 3 map X0+ (92) → 92b by Lemma 6. 4.2. Trigonal curves X0+ (N ). Suppose that Gon(X0+ (N )) = 3. The levels N are: gX + (N ) N 0 3 58, 76, 86, 96, 97, 99, 100, 109, 113, 127, 128, 139, 149, 151, 169, 179, 239 4 70, 82, 84, 88, 90, 93, 108, 115, 116, 117, 129, 135, 137, 147, 155, 159, 161, 173, 199, 215, 251, 311 5 122, 146, 181, 227 6 164 If gX + (N ) = 3, then the projection from a Q-rational cusp defines a 0 degree 3 map X0+ (N ) → P1 over Q (cf. [HaSh99a, p. 136]). On the other hand, it is known that every curve C/K of genus ≥ 5 with Gon(C) = 3 has a degree 3 map X → P1 over K (cf. [NS, Theorem 2.1], [HaSh99a, Corollary 1.7]). 10 F. Bars and T. Dalal Thus, we restrict to Gon(X0+ (N )) = 3 and gX + (N ) = 4. 0 It is well known that a non-hyperelliptic curve of genus 4 lies either on a quadratic cone or on a ruled surface (cf. [HaSh99a, p. 136]), and by Petri’s theorem a model of the curve can be computed in P3 as the intersection of a degree 2 and a degree 3 homogeneous equations. Following [HaSh99a, pp. 131, 136] it can be checked that for N = 159 the curve X0+ (N ) lies on a quadratic cone over Q and for N = 88, 93, 115, 116, 129, 137, 155, 215 the curve X0+ (N ) lies on a ruled surface over Q. On the other hand, for N = 70, 82, 84, 90, 108, 117, 135, 147, 161, 173, 199, 251, 311 the curve X0+ (N ) lies on a ruled surface either over a quadratic extension of Q or over a bi- quadratic extension of Q. Hence in these last levels the trigonal maps are not defined over Q. For example, consider X0+ (70); the quadratic surface is given by xz − y 2 + 8yw − z 2 − 10zw − 9w2 , which after a suitable coordinate change can be converted into the equation √ √ x2 − y 2 − z 2 + 7w2 = (x + y)(x − y) − (z + 7 w)(z − 7 w), √ and this surface is isomorphic to the ruled surface uv − st over Q( 7). See details in Appendix A for all X0+ (N ) trigonal with gX + (N ) = 4. 0 From the discussion so far we have Theorem 18. Assume that gX + (N ) ≥ 3. Then X0+ (N ) is trigonal over 0 Q if and only if N is in the following list: 58, 76, 86, 88, 93, 96, 97, 99, 100, 109, 113, 115, 116, 122, 127, 128, 129, 137, 139, 146, 149, 151, 155, 159, 164, 169, 179, 181, 215, 227, 239. In particular, for such N , the set Γ3′ (X0+ (N ), Q) is infinite. Assume now that Gon(X0+ (N )) = 3, but X0+ (N ) does not admit a de- gree 3 map to the projective line P1 over Q. Hence in these cases X0+ (N ) contains infinitely many cubic points over Q when W3 (X0+ (N )) contains a translation of the elliptic curve E with positive Q-rank [Jeo21, p. 352]. Proposition 19. For N = 70, 82, 84, 90, 108, 117, 135, 147, 161, 173, 199, 251, 311, the curve X0+ (N ) has finitely many cubic points over Q. Proof. For N = 70, 84, 90, 108, 147, 161, 173, 199, 251, 311 there is no el- liptic curve E of positive Q-rank with cond(E) | N . Hence in these cases, X0+ (N ) contains finitely many cubic points over Q. For N = 82, 117 and 135, X0+ (N ) is bielliptic and there are elliptic curves of positive Q-rank with cond(E) | N . By arguments in [Jeo21, p. 353], if there is no Q-rational degree 3 mapping X0+ (N ) → E where E is an elliptic curve of positive Q-rank and cond(E) | N , then W3 (X0+ (N )) has no translation of an elliptic curve with positive Q-rank. Infinitely many cubic points for X0+ (N ) over Q 11 In these cases only the pairs (82, 82a), (117, 117a) or (135, 135a) could ap- pear. If any of these pairs (N, E) is admissible (i.e. there is a Q-rational degre 6 mapping X0 (N ) → E), then the degree of the strong Weil parametrization of E should divide 6. For 82a, 117a and 135a the degrees of the strong Weil parametrization are 4, 8 and 16 respectively. Thus no such pairs are admissible. The result follows. 4.3. X0+ (N ) bielliptic and not hyperelliptic and not trigonal. Sup- pose X0+ (N ) is bielliptic but neither hyperelliptic nor trigonal. Following [Jeo21, p. 353], if Γ3′ (X0+ (N ), Q) is infinite, then W3 (X0+ (N )) contains a translation of an elliptic curve E with positive Q-rank, equivalently (N, E) is an admissible pair. The levels that remain to study are 78, 105, 110, 118, 120, 123, 124, 136, 141, 142, 144, 145, 171, 176, 188. Proposition 20. Suppose X0+ (N ) is bielliptic and not hyperelliptic and not trigonal. Then the only admissible pair is (124, 124a); in particular, for all such curves, Γ3′ (X0+ (N ), Q) is infinite if and only if N = 124. Proof. For N = 78, 105, 110, 120, 144, 188 there is no possible (N, E) be- cause there is no elliptic curve satisfying (iii) in Lemma 4 by Cremona tables [Cr]. For N = 118, 123, 124, 136, 141, 142, 145 the only possible admissible pairs (N, E) have cond(E) = N . If they were admissible, we get a degree 6 map from X0 (N ) → E and the degree of the strong Weil parametriza- tion of E (see Cremona tables [Cr] for such degrees) should divide 6, and no such case happens except (124, 124a), for which by Lemma 6 the Weil parametrization of degree 6 factors through X0+ (124) because w124 in 124a acts as +1. Finally, take N = 171, 176; the pairs to study are (171, 171b), (171, 57a) and (176, 88a). The pair (171, 171b) we discard as before, because the strong Weil parametrization for 171b is 8. We can apply Corollary 8 with w19 and w11 respectively to deduce that (171, 57a) and (176, 88a) are not admissible. Appendix A. A model for trigonal X0+ (N ) with gX + (N ) = 4. For 0 a detailed discussion on how to construct the models we refer the reader to [Si] and [Ga]. Curve Petri model X0+ (70) x2 w − 7xw2 − y 3 + 3y 2 z + 2y 2 w − 3yz 2 − 16yzw + 28yw2 + z 3 + 11z 2 w − 19zw2 − 27w3 , xz − y 2 + 8yw − z 2 − 10zw − 9w2 12 F. Bars and T. Dalal Curve Petri model X0+ (82) x2 w − 2xyw − 5xw2 − yz 2 + 5yzw + yw2 + 2z 3 − 12z 2 w + 23zw2 − 9w3 , xz − 3xw − y 2 + 2yz − 4z 2 + 10zw − 4w2 X0+ (84) x2 w − 2xyw − 5xw2 − y 2 z − y 2 w + 3yz 2 + 6yzw + 5yw2 − 2z 3 − 6z 2 w + 4zw2 + 4w3 , xz − xw − y 2 + 2yz + yw − 3z 2 + w2 X0+ (88) x2 z − xy 2 − xyz − 2xz 2 + y 3 + 6y 2 z − 9y 2 w − 8yz 2 + 33yw2 + 5z 3 + 6z 2 w − 12zw2 − 30w3 , xw − yz + yw + z 2 − zw − 5w2 X0+ (90) x2 w − 2xyw − 3xw2 − y 2 z − y 2 w + 3yz 2 + 6yzw + 3yw2 − 2z 3 − 5z 2 w + zw2 , xz − xw − y 2 + 2yz + yw − 3z 2 X0+ (93) x2 z − xy 2 − xyz − 2xz 2 + y 3 + 7y 2 z − 11y 2 w − 10yz 2 + 7yzw + 29yw2 + 6z 3 + 2z 2 w − 16zw2 − 21w3 , xw − yz + yw + z 2 − 2zw − 3w2 X0+ (108) x2 w − 3xw2 − y 3 + 2y 2 z − 8yzw + 12yw2 − 2z 3 + 12z 2 w − 22zw2 + 5w3 , xz − y 2 + 4yw − 6zw − w2 X0+ (115) x2 z − xy 2 − xyz − 2xz 2 + y 3 + 5y 2 z − 9y 2 w − 4yz 2 − 6yzw + 29yw2 + 2z 3 + 5z 2 w − 22w3 , xw − yz + yw + z 2 − 4w2 X0+ (116) x2 z − xy 2 − 2xz 2 + 4y 2 z + 2y 2 w − 6yz 2 − 8yzw + 3yw2 + 4z 3 + 9z 2 w − 4zw2 − 4w3 , xw − yz + z 2 − 3w2 X0+ (117) x2 w − xyw − 5xw2 − y 2 z + y 2 w + yz 2 + yzw + yw2 − z 3 + 2zw2 + 4w3 , xz − y 2 + yz + yw − 3z 2 + 2zw − 4w2 X0+ (129) x2 z − xy 2 − 2xz 2 + 5y 2 z − 7yz 2 − 3yzw + 3yw2 + 4z 3 + 3z 2 w − 3zw2 − w3 , xw − yz + z 2 − zw − w2 X0+ (135) x2 w − 2xyw − 3xw2 − y 3 + 3y 2 z + 2y 2 w − 3yz 2 + 2yw2 + z 3 + w3 , xz − 2xw − y 2 + 2yz + 3yw − 2z 2 − zw X0+ (137) x2 z − xy 2 − xz 2 + 3y 2 z + 2y 2 w − 6yz 2 − yzw − 3yw2 + 3z 3 + 2z 2 w − zw2 + 2w3 , xw − yz + z 2 − zw − w2 X0+ (147) x2 w − xyw − 6xw2 − y 2 z + yz 2 + 2yw2 − z 3 + z 2 w + 3zw2 + 7w3 , xz − xw − y 2 + yz − 2z 2 + zw + w2 X0+ (155) x2 z − xy 2 − xyz − xz 2 + y 3 + 3y 2 z − 5y 2 w − 2yz 2 + 2yzw + 7yw2 + z 3 − 2zw2 − 3w3 , xw − yz + yw − 2w2 Infinitely many cubic points for X0+ (N ) over Q 13 Curve Petri model X0+ (159) x2 z − xy 2 + xyz − 3xz 2 + 2y 2 z + y 2 w − 8yzw + 3yw2 + 7z 2 w − zw2 − 2w3 , xw − yw − z 2 + 2zw − 2w2 X0+ (161) x2 w − 5xw2 − y 2 z + yz 2 + 2yw2 − 3z 2 w + 9zw2 − 4w3 , xz − xw − y 2 + 3yw − z 2 + zw − 3w2 X0+ (173) x2 w − xyw + 6xw2 − 2y 2 w − yz 2 + 4yzw + 6yw2 + 4z 2 w − 17zw2 − 6w3 , xz + 2xw − y 2 + yz + 3yw − 6zw − 3w2 X0+ (199) x2 w + 2xyw + xw2 − y 3 − y 2 z + 2y 2 w + yz 2 − 5yzw + 3zw2 − 5w3 , xz + 2xw − y 2 − 2yz + 3yw − 4w2 X0+ (215) x2 z − xy 2 − xyz − xz 2 + y 3 + 2y 2 z − 3y 2 w − 2yzw + 5yw2 + z 3 − z 2 w + zw2 − 2w3 , xw − yz + yw + zw − 2w2 X0+ (251) x2 w − 5xw2 − y 2 z − y 2 w + yz 2 + yw2 + z 2 w − zw2 + 4w3 , xz − 2xw − y 2 + yw + w2 X0+ (311) x2 w − xyw − y 3 + y 2 z + 2y 2 w − yz 2 − 2yzw − yw2 + z 2 w, xz − xw − y 2 + yz + 2yw − z 2 − 2zw Curve Quadratic surface √ X0+ (70) Diagonal form: x2 − y 2 − z 2 + 7w2 , lies on a ruled surface over Q( 7) X0+ (82) Diagonal form: 3x2 − 12y 2 − 4z 2 − w2 , √ lies on a ruled surface over Q( −1) √ X0+ (84) Diagonal form: 2x2 − 6y 2 − 3z 2 + w2 , lies on a ruled surface over Q( 3) X0+ (88) Diagonal form: 5x2 + 5y 2 − 5z 2 − 5w2 , lies on a ruled surface over Q X0+ (90) Diagonal form: 2x2 − 6y 2 − 3z 2 − 3w2 , √ √ lies on a ruled surface over Q( 3, −1) X0+ (93) Diagonal form: 4x2 + 3y 2 − 4z 2 − 3w2 , lies on a ruled surface over Q √ X0+ (108) Diagonal form: −x2 − y 2 + z 2 + 3w2 , lies on a ruled surface over Q( 3) X0+ (115) Diagonal form: 3x2 + 4y 2 − 3z 2 − 4w2 , lies on a ruled surface over Q X0+ (116) Diagonal form: 3x2 − y 2 + z 2 − 3w2 , lies on a ruled surface over Q X0+ (117) Diagonal form: 11x2 − 33y 2 − 3z 2 − 15w2 , √ √ lies on a ruled surface over Q( 3, −5) X0+ (129) Diagonal form: x2 − 5y 2 + 5z 2 − w2 , lies on a ruled surface over Q √ X0+ (135) Diagonal form: x2 − 2y 2 − 2z 2 + 9w2 , lies on a ruled surface over Q( 2) X0+ (137) Diagonal form: x2 − 5y 2 + 5z 2 − w2 , lies on a ruled surface over Q √ X0+ (147) Diagonal form: 7x2 − 14y 2 − 2z 2 + w2 , lies on a ruled surface over Q( 2) X0+ (155) Diagonal form: 2x2 + 2y 2 − 2z 2 − 2w2 , lies on a ruled surface over Q X0+ (159) Diagonal form: 2y 2 − z 2 − 2w2 , lies on a quadratic cone over Q 14 F. Bars and T. Dalal Curve Quadratic surface √ X0+ (161) Diagonal form: x2 − y 2 − z 2 − 3w2 , lies on a ruled surface over Q( −3) X0+ (173) Diagonal form: −x2 − 3y 2 + 3z 2 + 37w2 , √ lies on a ruled surface over Q( 37) √ X0+ (199) Diagonal form: −x2 − y 2 + z 2 + 33w2 , lies on a ruled surface over Q( 33) X0+ (215) Diagonal form: x2 + 2y 2 − z 2 − 2w2 , lies on a ruled surface over Q √ X0+ (251) Diagonal form: −x2 − y 2 + z 2 + 5w2 , lies on a ruled surface over Q( 5) X0+ (311) Diagonal form: 3x2 − 3y 2 − z 2 − 3w2 , √ lies on a ruled surface over Q( −3) Appendix B. The sieves to reduce to a finite set of N to consider. Here we consider the levels N that do not appear in Theorem 3. Using Ogg’s classical argument as in the proof of [HaSh99b, Lemma 3.2] one finds that if N ≥ 624, there is no Q-rational degree 6 mapping X0 (N ) → E for any E, and consequently no degree 3 map X0+ (N ) → E over Q for N ≥ 624. Now by Lemma 4(i) we can discard the existence of such a degree 3 map for the following N : 252, 260, 264, 272, 276, 280, 288, 290, 294, 296, 300, 304, 306, 308, 310, 312, 315, 316, 318, 320, 322, 324, 328, 330, 332, 336, 340, 342, 344, 345, 348, 350, 352, 354, 356, 357, 360, 364, 366, 368, 370, 372, 374–376, 378, 380, 382, 384, 385, 386, 388, 390, 392, 394, 396, 398–400, 402, 404–406, 408, 410, 412, 414, 416, 418, 420, 422-426, 428-430, 432, 434–436, 438, 440–442, 444, 446, 448, 450, 452–456, 458–460, 462, 464–466, 468, 470–472, 474–478, 480, 482–486, 488–490, 492, 494–498, 500–502, 504–508, 510–520, 522, 524–528, 530–540, 542–546, 548–556, 558–562, 564–623. By Lemma 4(iii) we can discard all pairs (N, E) for the following N : 126, 132, 133, 134, 140, 150, 157, 165, 168, 177, 180, 186, 187, 193, 194, 206, 211, 213, 217, 221, 223, 230, 233, 240, 241, 247, 250, 253, 255, 257, 261, 263, 266, 268, 271, 279, 281, 283, 287, 292, 293, 295, 299, 307, 313, 317, 319, 321, 323, 329, 334, 337, 341, 343, 349, 353, 355, 358, 365, 367, 379, 383, 391, 397, 401, 403, 409, 411, 413, 417, 419, 421, 439, 447, 449, 457, 461, 463, 479, 487, 491, 499, 509, 521, 523, 529, 541, 547. By the use of (iii) and (v) in Lemma 4 we can discard N in the list: 102, 112, 138, 152, 153, 156, 160, 170, 175, 189, 190, 192, 197, 200, 201, 203, 205, 207, 208, 209, 210, 214, 216, 218, 219, 220, 225, 226, 229, 235, 238, 245, 254, 274, 275, 277, 278, 289, 291, 298, 302, 309, 314, 327, 331, 335, 338, 339, 346, 347, 359, 361, 362, 373, 377, 381, 389, 431, 433, 437, 443, 451, 467, 469, 493, 503, 557, 563. For N in the table below, using Lemma 4(v) we can eliminate all (N, E) with cond(E) = N ; the remaining pairs (N, E) where cond(E) | N and cond(E) ̸= N (rankQ (E) ≥ 1) can be eliminated by Lemma 4(ii), i.e. by computing Fpr -points on X0 (N ) with p ∤ N in the first two columns and the Infinitely many cubic points for X0+ (N ) over Q 15 last one for X0+ (N ) instead of X0 (N ). Thus we can discard all the levels N appearing in the table below. N E pr N E pr N E pr 148 37a 32 297 99a 52 244 61a 32 154 77a 32 301 43a 52 244 122a 32 184 92b 32 325 65a 32 248 124a 52 198 99a 52 326 163a 32 273 91a 2 204 102a 52 333 37a 52 273 91b 22 212 53a 32 351 117a 22 282 141a 52 212 106a 52 363 121a 52 282 141d 72 224 112a 32 369 123a 22 305 61a 7 228 57a 52 369 123b 72 395 79a 22 232 58a 32 371 53a 32 234 117a 52 387 43a 22 242 121b 52 387 129a 24 246 82a 72 393 131a 52 246 123a 52 407 37a 32 246 123b 72 415 83a 32 256 128a 32 427 61a 32 259 37a 32 445 89a 22 265 53a 32 473 43a 22 270 135a 72 481 37a 22 285 57a 22 286 143a 32 By Lemma 4(iv) we can discard N = 222, 262, 284, 303. Acknowledgements. The second author wishes to thank University Grants Commission, India, for the financial support provided in the form of Research Fellowship to carry out this research work at IIT Hyderabad. The second author would also like to thank the organizers and lecturers of the programme “CMI-HIMR Summer School in Computational Number Theory”. Some part of the paper was written during the second author’s visit to the Universitat Autònoma de Barcelona and the second author is grateful to the Department of Mathematics for their support and hospitality. The first author is supported by PID2020-116542GB-I00. References [AbHa] D. Abramovich and J. Harris, Abelian varieties and curves in Wd (C), Compos. Math. 78 (1991), 227–238. [Ba] F. Bars, On quadratic points of classical modular curves, in: Contemp. Math. 701, Amer. Math. Soc., 2018, 17–34. 16 F. Bars and T. Dalal [BaGo] F. Bars and J. González, Bielliptic modular curves X0∗ (N ), J. Algebra 559. (2020), 726–759. [BoGaGo] J. Box, S. Gajović and P. Goodman, Cubic and quartic points on modular curves using generalized symmetric Chabauty, arXiv:2102.08236v2 (2021). [CaEm] F. Calegari and M. Emerton, Elliptic curves of odd modular degree, Israel J. Math. 169 (2009), 417–444. [Cr] J. Cremona, https://johncremona.github.io/ecdata/. [DeFa] O. Debarre and R. Fahlaoui, Abelian varieties in Wdr (C) and points of bounded degree on algebraic curves, Compos. Math. 88 (1993), 235–249. [De] C. Delaunay, Critical and ramification points of the modular parametrization of an elliptic curve, J. Théor. Nombres Bordeaux 17 (2005), 109–124. [FuHa] M. Furumoto and Y. Hasegawa, Hyperelliptic quotients of modular curves X0 (N ), Tokyo J. Math. 22 (1999), 105–125. [Ga02] S. D. Galbraith, Rational points on X0+ (N ) and quadratic Q-curves, J. Théor. Nombres Bordeaux 14 (2002), 205–219. [Ga] S. D. Galbraith, Equations for modular curves, https://www.math.auckland. ac.nz/~sgal018/thesis.pdf. [Go] J. González, Equations of bielliptic modular curves, JP J. Algebra Number Theory Appl. 27 (2012), 45–60. [HaSi] J. Harris and J. H. Silverman, Bielliptic curves and symmetric products, Proc. Amer. Math. Soc. 112 (1991), 347–356. [Ha] Y. Hasegawa, Table of quotient curves of modular curves X0 (N ) with genus 2, Proc. Japan Acad. Ser. A Math. Sci. 71 (1995), 235–239. [HaSh99a] Y. Hasegawa and M. Shimura, Trigonal modular curves, Acta Arith. 88 (1999), 129–140. [HaSh99b] Y. Hasegawa and M. Shimura, Trigonal modular curves X0+d (N ), Proc. Japan Acad. Ser. A Math. Sci. 75 (1999), 172–175. [JKS04] D. Jeon, C. H. Kim and A. Schweizer, On the torsion of elliptic curves over cubic number fields, Acta Arith. 113 (2004), 291–301. [Jeo18] D. Jeon, Bielliptic modular curves X0+ (N ), J. Number Theory 185 (2018), 319–338. [Jeo21] D. Jeon, Modular curves with infinitely many cubic points, J. Number Theory 219 (2021), 344–355. [MaSD] B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math. 25 (1974), 1–61. [Mo] F. Momose, Rational points on the modular curves X0+ (N ), J. Math. Soc. Japan 39 (1987), 269–286. [NS] K. V. Nguyen and M.-H. Saito, D-gonality of modular curves and bounding torsions, arXiv:alg-geom/9603024 (1996). [Si09] S. Siksek, Chabauty for symmetric powers of curves, Algebra Number Theory 3 (2009), 209–236. [Si] S. Siksek, Explicit Methods for Modular Curves, http://homepages.warwick.ac. uk/staff/S.Siksek/teaching/modcurves/lecturenotes.pdf. Francesc Bars Tarun Dalal Departament Matemàtiques, Edif. C Department of Mathematics Universitat Autònoma de Barcelona Indian Institute of Technology Hyderabad 08193 Bellaterra, Catalonia Kandi, Sangareddy 502285, India E-mail: francesc@mat.uab.cat E-mail: ma17resch11005@iith.ac.in Abstract (will appear on the journal’s web site only) We determine all modular curves X0+ (N ) that admit infinitely many cubic points over the rational field Q.