Cubic and quartic points on modular curves using generalised symmetric Chabauty
aa r X i v : . [ m a t h . N T ] F e b CUBIC AND QUARTIC POINTS ON MODULAR CURVES USINGGENERALISED SYMMETRIC CHABAUTY
JOSHA BOX, STEVAN GAJOVI´C AND PIP GOODMAN
Abstract.
Answering a question of Zureick-Brown, we determine the cubic points on the mod-ular curves X ( N ) for N ∈ { , , , , , } as well as the quartic points on X (65). To doso, we develop a “partially relative” symmetric Chabauty method. Our results generalise currentsymmetric Chabauty theorems, and improve upon them by lowering the involved prime bound.For our curves a number of novelties occur. We prove a “higher order” Chabauty theorem todeal with these cases. Finally, to study the isolated quartic points on X (65), we rigorouslycompute the full rational Mordell–Weil group of its Jacobian. Introduction
This work started at the 2020 Arizona Winter School with a question from David Zureick–Brown: Is it possible to determine the finitely many cubic points on X (65) despite the infinitudeof quadratic points? In this article we answer this question affirmatively by developing a “partiallyrelative” symmetric Chabauty method (Theorem 2.6). In fact, this result allows us to prove thefollowing: Theorem 1.1.
The set of cubic points for each of the curves X (53) , X (57) , X (61) , X (65) , X (67) and X (73) is finite and listed in Section 5. The quartic points on X (65) form an infinite set. This infinite setconsists of inverse images of quadratic points on the quotient curve X +0 (65) and a finite numberof isolated points. The isolated points are listed in § The points introduced in Theorem 1.1 are given relative to models for X ( N ) and X +0 ( N ) ( N as above) listed in §
5. Here, we also provide generators for the Mordell–Weil groups J +0 ( N )( Q )and J (65)( Q ).In addition to the “partially relative” aspect of Theorem 2.6, it allows for multiple maps, ramifiedpoints and makes improvements in the relevant prime bounds. Being able to handle ramified pointsis crucial for determining the cubic points on X (67) (see Example 2.12). Despite the virtues ofTheorem 2.6, there remain points on X (57) and X (73) which cause further difficulty (see Example2.23). We treat these novelties by appealing to a “higher order” Chabauty theorem: Theorem 2.24.For a curve X over a number field K , denote by X ( d ) its d th symmetric power. The K -rationalpoints on X ( d ) are exactly the K -rational effective degree d divisors on X , and the Gal( K/K )-orbit of each point P ∈ X defined over a degree d extension of K gives rise to such a divisor. Bystudying the K -rational points on X ( d ) we can thus study the L -rational points on X for all degree d extensions L/K simultaneously.An important tool to study X ( d ) is the Abel–Jacobi map. Given a K -rational degree d divisor D on X , we define it as ι : X ( d ) → J ( X ) , D [ D − D ] . So when does
X/K have infinitely many points of degree d over K ? Certainly when there is adegree d map ρ : X → P , as the inverse image of P ( K ) provides such an infinite set. Moregenerally, when there is a map ρ : X → C of degree e to a curve C such that C ( f ) ( K ) is infinite, e · f ≤ d and there exists P ∈ X ( d − ef ) ( K ): then such an infinite set is P + ρ ∗ C ( f ) ( K ) ⊂ X ( d ) ( K ) , and ι maps this set into a translate of the abelian subvariety ρ ∗ J ( C ) of J ( X ).Define P ∈ X ( d ) ( K ) to be an isolated point when it is neither in the inverse image of P ( K )under a degree d map X → P , nor does ι ( P ) lie in a translate of a positive rank abelian subvarietyof J ( X ) contained in ι ( X ( d ) ). Recently, Bourdon, Ejder, Liu, Odumodu and Viray [BEL + (i) X has infinitely many points of degree d over K if and only if X ( d ) ( K ) contains a non-isolated point, and(ii) there are finitely many isolated points in X ( d ) ( K ).This result provides a road map for studying X ( d ) ( K ): first describe each infinite set, then de-termine the finite set of isolated points. This is what we have done for the modular curves listedabove, using a generalisation of Chabauty’s method.Chabauty’s method is, classically, a method for effectively computing the rational points on acurve in the special case when the rank r of its Mordell–Weil group is strictly smaller than itsgenus g . The effectiveness is due to Coleman [Col85a], who realised that Chabauty’s finitenessproof [Cha41] could be made effective using the machinery of locally analytic p -adic functions.While nowadays many focus on weakening the r < g condition in Chabauty’s method vianon-abelian generalisations (see e.g. [Kim05] and [BBB + ]), the reach of Chabauty’s method isstill being increased. After partial results of Klassen [Kla93], Siksek [Sik09] extended Coleman’sideas to obtain an effective method for computing the K -rational points (when finite) on the d thsymmetric powers of curves over any number field K , provided the stronger condition r < g − ( d − + ]. In the latter article, the Chabauty-like method forsymmetric powers was used to determine X (22) (3) ( Q ), X (25) (3) ( Q ) and the image of X (65) (3) ( Q )in X (65) (3) ( Q ). Using our method, we can now describe the complete set X (65) (3) ( Q ) and listthe finitely many cubic points on X (65).Furthermore, Siksek [Sik09] developed a relative version of his symmetric Chabauty method,which can be used to determine the isolated points on X ( d ) if the infinite set consists entirelyof pullbacks. To be precise, given a map ρ : X → C of curves over K of degree d , Siksek’srelative Chabauty method can determine the remainder X ( d ) ( K ) \ ρ ∗ C ( K ) if that consists entirelyof isolated points. This method was employed by the first named author [Box21] to describe thequadratic points on X ( N ) for N ∈ { , , , , , , } .The curve X (65) admits a degree 2 map ρ : X (65) → X (65) / h w i =: X +0 (65), where w is the Atkin–Lehner involution. This quotient X +0 (65) is an elliptic curve of rank 1, and ρ ∗ ( X +0 (65)( Q )) is an infinite set of degree 2 points. In [Box21], Siksek’s relative Chabauty methodwas used to show that the only isolated degree 2 points are sums of two cusps. For degree 3,however, we obtain for each cusp c ∈ X (65)( Q ) such an infinite set c + ρ ∗ ( X +0 (65)( Q )) ⊂ X (65) (3) ( Q ) , and Siksek’s method cannot be applied anymore to study the isolated points of degree 3, as theinfinite sets are not pullbacks.It is exactly this problem that we solved, by generalising Siksek’s ideas to obtain a “partiallyrelative symmetric Chabauty method” that has the potential to determine the isolated degree d points on a curve X if the infinite sets are of the form P + ρ ∗ C ( ℓ )1 ( K ) + . . . + ρ ∗ n C ( ℓ n ) n ( K ) , where P ∈ X ( e ) ( K ) and ρ i : X → C i are maps of degree d i such that e + ℓ d + . . . + ℓ n d n = d .The result that makes this possible is Theorem 2.6. This theorem has already contributed in [Box]to the proof that all elliptic curves over quartic fields not containing √ X ( N ) for N ∈ { , , , , , } extends a string of papersstudying the modular curves X ( N ) of genus g ∈ { , . . . , } . Bruin and Najman [BN15] determinedthe isolated quadratic points on the hyperelliptic X ( N ) of these genera with finite Mordell–Weilgroup. Subsequently, ¨Ozman and Siksek [OS19] determined the finitely many quadratic pointson the non-hyperelliptic X ( N ) of genus g ∈ { , . . . , } with finite Mordell–Weil group, the firstnamed author [Box21] described the quadratic points on those with infinite Mordell–Weil group.The curves for which we determine the cubic points are exactly those X ( N ) of genus g ∈{ , . . . , } that have infinite Mordell–Weil group and for which the Chabauty condition on the OINTS ON MODULAR CURVES USING GENERALISED SYMMETRIC CHABAUTY 3 rank holds true. Only two of these curves also satisfy the rank condition for quartic points: X (65)and X (57). However, a genus 5 curve admits infinitely many degree four maps to P (see § X (57). For X (65) on the other hand,we are in luck. All degree four maps to P defined over Q factor through the elliptic curve X +0 (65).Applying our partial relative Chabauty methods with respect to this quotient, we determine all ofthe isolated quartic points on X (65). The Magma [BCP97] code to verify all computations madein this paper can be found at https://github.com/joshabox/cubicpoints/ . Acknowledgements.
The authors would like to warmly thank David Zureick-Brown forinitiating this project, and both Jackson Morrow and David for their help during the projectsessions of the 2020 Arizona Winter School. We are also grateful to the organisers of the schoolfor providing such excellent conditions in Tucson, and the first and last named authors thank theschool’s donors for their financial support.We thank Samir Siksek for many valuable comments and suggestions, which have improvedthis work greatly. Likewise, we thank Tim Dokchitser, Steffen M¨uller, Filip Najman, and LazarRadiˇcevi´c for their help.2.
A general symmetric Chabauty theorem
In this section we present a common generalisation of Theorems 3.2 and 4.3 in [Sik09], whichare Chabauty-type theorems for computing K -rational points on symmetric powers of curves.Before this, we give an overview of Siksek’s Chabauty method.2.1. Uniformisers and differentials.
Let K be a finite extension of Q p with ring of integers R and residue field k . Consider a curve X/K together with a minimal proper regular model X /R for X . We write e X for the special fibre of X , and similarly denote reductions of objects associated to X with a tilde. Denote by O S the sheaf of functions on a scheme S . For a sheaf F on S , we denoteby F s its stalk at s . When A is a discrete valuation ring (DVR), we denote by b A its completion.Finally, when s ∈ S is such that O S,s is a DVR, U ⊂ S is an open subset containing s and F is an O S -module, we denote by loc s the map F ( U ) → F s . For x ∈ X ( K ) and y ∈ e X ( k ), we denote by D ( x ) and D ( y ) the points in X ( K ) reducing to e x and y respectively.When x ∈ X is non-singular, a uniformiser t ∈ b O X,x is called a local coordinate at x ∈ X .When t also reduces to a uniformiser in the reduction b O e X, e x , we say that it is a well-behavedlocal coordinate or well-behaved uniformiser . This just means that the maximal ideal of b O X , e x isgenerated by t and p as an R -module. The following facts can be found for example in [LT02].When t is a well-behaved local coordinate at x , we can evaluate t at points in the residue disc D ( x )of x , yielding a bijection between D ( x ) and the maximal ideal of R .Denote by Ω X/K and Ω X /R (sometimes abbreviated to Ω) the sheafs of regular differentials on X and X respectively. A choice of uniformiser t ∈ b O X,x gives rise to the identifications b O X,x = K J t K and b Ω X,x = K J t K d t . If t is moreover well-behaved, we have b O X , e x = R J t K and b Ω X , e x = R J t K d t .Now consider ω ∈ H ( X, Ω X/K ). Then H ( X , Ω X /R ) is a lattice in H ( X, Ω X/K ), so aftermultiplication by a constant in R , we may assume that ω ∈ H ( X , Ω X /R ). In sum, given awell-behaved uniformiser s at a point Q ∈ X ( K ), we can writeloc Q ( ω ) = ∞ X n =0 a n s n d s with a n ∈ R for all n. (1)Consider any point P ∈ X ( K ). Lemma 2.1.
The map ι : X → J ( X ) , P [ P − P ] induces an isomorphism ι ∗ : H ( J, Ω J/K ) ≃ H ( X, Ω X/K ) of global differential forms independent of the choice of P .Proof. See e.g. [Sik09, Proposition 2.1]. (cid:3)
We shall thus use ι ∗ to pass between these two spaces. JOSHA BOX, STEVAN GAJOVI´C AND PIP GOODMAN
Now suppose that we have a map ρ : X → C between two curves over K . We obtain apushforward map ρ ∗ : J ( X ) → J ( C ) and pullback map ρ ∗ : J ( C ) → J ( X ), leading to adecomposition up to isogeny J ( X ) ∼ J ( C ) × A, where A ⊂ J ( X ) is an abelian subvariety. Denote by π A the map J ( X ) → A . We also obtain apush-forward, or “trace map”, Tr : H ( X, Ω X/K ) → H ( C, Ω C/K ) . Lemma 2.2.
We have (i) H ( X, Ω X/K ) = ρ ∗ H ( C, Ω C/K ) ⊕ Ker(Tr) , and (ii) ι ∗ π ∗ A H ( A, Ω A/K ) = Ker(Tr) .Proof.
Part (i) follows from surjectivity of the trace map. For part (ii) it suffices to comparedimensions and to check that ι ∗ π ∗ A H ( A, Ω A/K ) ⊂ Ker(Tr). (cid:3)
Coleman integration.
We consider the notation from the previous subsection. Again let K be a finite extension of Q p . Also let B/K be any abelian variety. In [Col85b, Section II], Colemandefines a pairing, now called
Coleman integration : H ( B, Ω B/K ) × B ( K ) → K, ( ω, P ) Z P ω. (2)We note that Coleman defines this integration in a much more general setting than this, but weshall only be concerned with the above case of abelian varieties and differentials of the first kind. Proposition 2.3.
The integration pairing (2) is (i) locally analytic in P ∈ B ( K ) , (ii) Z -linear on the right, (iii) K -linear on the left, (iv) its left-hand kernel is zero, (v) its right-hand kernel is B ( K ) tors , and (vi) if g : B → B ′ is a morphism of abelian varieties over K , then Z P g ∗ ω = Z g ( P ) ω for all ω ∈ H ( B ′ , Ω B ′ /K ) and P ∈ B ( K ) .Proof. Parts (i) and (ii) are [Col85b, Theorem 2.8]. Part (iii) follows from part (i) and [Col85b,Proposition 2.4 (i), (iii)]. Part (iv) follows from [Col85b, Theorem 2.8 (ii)], and Part (v) is [Col85b,Theorem 2.11]. Finally, part (vi) is [Col85b, Theorem 2.7]. (cid:3)
Lemma 2.4.
Suppose that G ⊂ B ( K ) is a subgroup of rank r . Then there is a vector space V ⊂ H ( A, Ω B/K ) of dimension at least dim( B ) − r such that for all ω ∈ V , we have Z P ω = 0 for all P ∈ G, where the closure is inside the p -adic topology on B ( K ) .Proof. Suppose that D , . . . , D r generate G up to torsion. Then G ⊗ K = KD + . . . + KD r , ofdimension ≤ r . Now we find V because (2) extends to an exact pairing between H ( B, Ω B/K ) and B ( K ) ⊗ K by Proposition 2.3. (cid:3) When B = J ( X ) /K and G = J ( X )( K ), we call this space V the space of vanishing differentials .Moreover, we call V ∩
Ker(Tr) the space of vanishing differentials with trace zero . This is thepullback along π A of the space of vanishing differentials on A .More notation: by pulling back along ι , we obtain integrals H ( X, Ω X/K ) × X ( K ) → K, ( ω, Q, P ) Z PQ ω := Z [ P − Q ] ( ι ∗ ) − ( ω ) . Proposition 2.5. (i)
For
P, Q ∈ X ( K ) such that P ∈ D ( Q ) , and a well-behaved uniformiser s at Q , we have for each ω ∈ H ( X, Ω X/K ) Z PQ ω = ∞ X n =0 a n n + 1 s ( P ) n +1 , OINTS ON MODULAR CURVES USING GENERALISED SYMMETRIC CHABAUTY 5 where loc Q ( ω ) = P n a n s n as in (1). We call such an integral between points in the sameresidue class a tiny integral . (ii) If ρ : X → C is a non-constant morphism of curves over K with good reduction, then Z ρ ∗ D ω = Z D Tr( ω ) for all ω ∈ H ( X, Ω X/K ) and every degree 0 divisor D on C defined over K .Proof. Part (i) follows from Proposition 2.3 (i) together with the Fundamental Theorem of Calculusproved in [Col85b, Proposition 2.4 (ii)]. Part (ii) is [Sik09, Lemma 2.2]. (cid:3)
An overview of the symmetric Chabauty–Coleman method.
We refer the reader to[Wet97] and [MP12] for a clear overview of the Chabauty–Coleman method in the classical case.Here we give an overview of Chabauty and Coleman’s original ideas in the symmetric power setting,after which we explain what needs to be changed in the relative case. For simplicity we work over Q , but everything generalises to number fields.We consider an integer d , a prime p and a curve X/ Q p of genus g X and minimal proper regularmodel X / Z p . To determine X ( d ) ( Q ), it suffices to determine each of the residue discs separately.So consider e Q ∈ X ( d ) ( F p ) and its inverse image under the reduction map, the residue disc D ( e Q ) ⊂ X ( d ) ( Q p ). We define the Abel–Jacobi map via D ∈ X ( d ) ( Q ): ι : X ( d ) → J ( X ) , D [ D − D ] . Chabauty’s idea was to consider the diagram(3) D ( e Q ) ∩ X ( d ) ( Q ) J ( X )( Q ) D ( e Q ) J ( X )( Q p ) ιι and instead determine ι ( D ( e Q )) ∩ J ( X )( Q ) , (4)where the closure is inside the p -adic topology on J ( X )( Q p ). This set contains ι ( D ( e Q ) ∩ X ( d ) ( Q )).Let r be the rank of J ( X )( Q ). By Lemma 2.4, we obtain a space V ⊂ H ( X, Ω X/K ) of dimensiondim( V ) ≥ g X − r such that for all ω ∈ V , we have Z D ω = 0 for all D ∈ J ( X )( Q ) . In particular, when the
Chabauty condition r < g X − ( d − d such linearly independent differentials, and dimensions suggest that Z p ( e Q ) := ( P ∈ D ( e Q ) | Z ι ( P ) ω = 0 for all ω ∈ V ) is finite. In the case d = 1, Chabauty [Cha41] proved that J ( Q ) ∩ ι ( D ( e Q )) is indeed finite when (5)is satisfied. For d > X ( d ) ( Q ) can still be infinite even when (5) holds, e.g. due to the existenceof a map ρ : X → C of degree ≤ d . For now, however, let us assume that X satisfies (5), and thatwe have no reason to believe that X ( d ) ( Q ) ∩ D ( e Q ) is infinite regardless.It was Coleman’s idea [Col85a] to introduce integration of differentials and compute Z p ( e Q )instead. Such zero sets can be computed for d = 1 by evaluating Coleman integrals between Q p -rational points. In Sage on hyperelliptic curves, the computation of Coleman integrals over Q p hasbeen implemented by Balakrishnan, Bradshaw, and Kedlaya [BBK10], [Bal15], while Balakrishnanand Tuitman [BT20] wrote a Magma implementation for all plane curves. For d >
1, however, onewould need to evaluate Coleman integrals between points
P, Q ∈ X ( K ) for extensions K/ Q p ofdegree >
1, and this has not yet been done.Instead, therefore, we shall restrict our attention to tiny (i.e. computable) integrals only: thissuffices when combining the information from multiple primes p using the Mordell–Weil sieve (seeSection 3). The drawback here is that information on J ( X )( Q ) is needed. Thanks to the sieve, weneed to consider only residue discs D ( e Q ) containing a known point Q ∈ X ( d ) ( Q ). For each such JOSHA BOX, STEVAN GAJOVI´C AND PIP GOODMAN point Q , (assuming that X ( d ) ( Q ) is finite) there is a prime p such that X ( d ) ( Q ) ∩ D ( e Q ) = {Q} .To compute X ( d ) ( Q ), it then suffices to have a criterion to decide whether Z p ( e Q ) = {Q} .Given the known point Q , we can choose D = Q to define ι . Then for P ∈ D ( e Q ), theintegral R ι ( P ) ω is a sum of d tiny integrals. By studying the power series obtained from these tinyintegrals via Proposition 2.5 (i), Siksek [Sik09, Theorem 3.2] found a criterion for deciding whether Z p ( e Q ) = {Q} .2.4. An overview of our relative Chabauty–Coleman method.
We continue the notationfrom the previous subsection. We now assume that we do have reason to believe that X ( d ) ( Q ) ∩ D ( e Q ) is infinite, due to the existence of a curve C with minimal proper regular model C / Z p , a map ρ : X → C such that ρ : X → C has degree f , a positive integer e such that e · f ≤ d , and knownpoints P ∈ X ( d − ef ) ( Q ) and Q ∈ C ( e ) ( Q ) such that Q := P + ρ ∗ Q ∈ D ( e Q ). Indeed, we then havea family P + ρ ∗ C ( e ) ( Q ) ⊂ X ( d ) ( Q )intersecting D ( e Q ). Even when X satisfies the Chabauty condition (5), the approach from theprevious subsection fails when either C ( e ) ( Q ) is infinite or when C ( e ) does not satisfy the Chabautycondition r C < g C − ( e −
1) itself, where r C = rk J ( C )( Q ) and g C is the genus of C .As in Section 2.1, there is an abelian variety A ⊂ J ( X ) such that J ( X ) ∼ J ( C ) × A , and wedefine π A : J ( X ) → A . As before, define the Abel–Jacobi map ι using Q . We now replace J ( X )in Diagram (3) by A , and note that π A ◦ ι (cid:16) P + ρ ∗ C ( e ) ( Q ) (cid:17) = { } , so the entire family has been collapsed to a single point on A . It may be easier to determine A ( Q ) ∩ π A ( ι ( D ( e Q ))) than it is to determine J ( X )( Q ) ∩ ι ( D ( e Q )). Again by Lemma 2.2, we find aspace V ⊂ H ( A, Ω A/K ) of dimension dim( V ) ≥ dim( A ) − rank( A ( Q )) such that for all ω ∈ V Z D ω = 0 for each D ∈ A ( Q ) . Let r X be the rank of J ( X )( Q ), r C the rank of J ( C )( Q ), and g X and g C be the genera of X and C respectively. If the Chabauty condition(6) r X − r C < g X − g C − ( d − V ) ≥ d and the dimensions suggest that the common zero set of R ω for ω ∈ V has finite intersection with π A ◦ ι ( D ( e Q )). Define Z p,A ( e Q ) := ( P ∈ D ( e Q ) | Z PQ ω = 0 for all ω ∈ Ker(Tr) ) , where Tr : H ( X, Ω X/K ) → H ( C, Ω C/K ) is the trace map. By Lemma 2.2 (ii) and Proposition2.3 (vi), this is the inverse image to D ( e Q ) of the common zero set of the integrals R ω for ω ∈ V .Analogous to the case of the previous section, we now desire a criterion to decide if Z p,A ( e Q ) =( P + ρ ∗ C ( e ) ( Q )) ∩ D ( e Q ). This is the purpose of Theorem 2.6.2.5. The main theorem.
Consider a point Q on a curve X over a field K , and a regular 1-form ω ∈ H ( X, Ω X/K ). We expand ω around Q in terms of a uniformiser t Q at Q , givingloc Q ( ω ) = P j ≥ a j t jQ . We define v ( ω, t Q , k ) := (cid:0) − a , a , . . . , ( − k − a k − (cid:1) . Theorem 2.6.
Let ρ j : X → C j for j ∈ { , . . . , h } be degree d j maps of curves over a numberfield K , and consider an effective divisor Q = Q + Q + . . . + Q h , where Q ∈ X ( e ) ( K ) and Q j ∈ ρ ∗ j C ( ℓ j ) j ( K ) for j ≥ . Let n = e + d ℓ + . . . + d h ℓ h . Suppose that r is a prime in O K of good reduction for X and each C j . Let p be the rational prime contained in r . (1) Assume that the supports of Q , . . . , Q h are pairwise disjoint, and no point in the supportof any Q i for i ≥ has ramification degree under ρ i divisible by p . (2) Let N be the ramification index of p in K ( Q ) . Assume that p ≥ N + 2 . OINTS ON MODULAR CURVES USING GENERALISED SYMMETRIC CHABAUTY 7
Write λ = ℓ + . . . + ℓ h . Let V be the space of vanishing differentials on X with trace zero withrespect to each ρ j , and consider a basis e ω , . . . , e ω q for the image of V ∩ H ( X , Ω) under the mod r reduction map on differentials. Let p be a prime in K ( Q ) above r , and denote reductions of pointswith respect to p with a tilde. Write Q = n Q + . . . + n k Q k with Q , . . . , Q k ∈ X distinct pointsand each n i ≥ , and let t e Q i be a uniformiser at e Q i for each i . (3) Assume that the matrix e A := v ( e ω , t e Q , n ) v ( e ω , t e Q , n ) · · · v ( e ω , t e Q k , n k ) v ( e ω , t e Q , n ) v ( e ω , t e Q , n ) · · · v ( e ω , t e Q k , n k ) ... . . . . . . ... v ( e ω q , t e Q , n ) v ( e ω q , t e Q , n ) · · · v ( e ω q , t e Q k , n k ) has rank n − λ .Then every P ∈ X ( n ) ( K ) in the mod r residue disc of Q is in fact contained in Q + ρ ∗ C ( ℓ )1 ( K r ) + . . . + ρ ∗ h C ( ℓ h ) h ( K r ) . Remark . By Proposition 2.17 in the next section, n − λ is in fact the maximal possible rank ofthis matrix. Moreover, note that condition (3) is not satisfied when two distinct points Q i = Q j have the same reduction mod p . If they do, at least two columns in e A would agree, reducing itsrank to less than n − λ due to the nature of the linear equations between the columns obtainedfrom Proposition 2.17. Remark . In fact, from the proof we see that condition (2) can be replaced by p ≥ N ′ + 2,where N ′ is the maximum of the ramification indices of p in K ( Q i , Q j ) for i, j ∈ { , . . . , k } . Thisleads to a general lower bound p ≥ n ( n −
1) + 2. In particular, for n = 2, p ≥ n = 3, p ≥
11 suffices; and for n = 4, p ≥
17 suffices.
Remark . This theorem generalises [Sik09, Theorems 3.2 and 4.3]. More precisely, the case Q = Q corresponds to [Sik09, Theorem 3.2] and the case Q = Q corresponds to [Sik09, Theorem 4.3].In both cases, we significantly improve the lower bound on the prime p compared to [Sik09]. Partof the improvement is due to the introduction in the proof of elementary symmetric polynomialsreplacing power sums. This removes denominators in the power series expansion and hence in thematrix e A ; this is an idea also used in [DKSS]. Another improvement comes from a Galois theoryargument, exploiting the fact that P is K -rational. Remark . When p is small, it sometimes happens that X ( n ) ( Q ) surjects onto X ( n ) ( F p ), inwhich case this theorem may be used to determine X ( n ) ( Q ) directly (i.e. without sieving). Thishas the advantage of requiring no information on the Mordell–Weil group of X . When Q = Q and the vanishing differentials come from a rank zero quotient of J ( X ), it is best to use the formalimmersion criterion of Derickx, Kamienny, Stein and Stoll [Der16, Chapter 3 Proposition 3.7]instead, which gives the same statement but works for all primes p ≥ + , § X (22) (3) ( Q ), X (25) (3) ( Q ) with p = 3. Ourtheorem covers all relative and positive rank cases, and can often be used with p = 3. Remark . While we have stated the theorem for multiple maps ρ , . . . , ρ h , we only need the case h = 1 in our examples, in which case P ∈ Q + ρ ∗ C ( ℓ )1 ( K ) (the pull-back part is K -rational since P and Q are). Also note that condition (1) is quite limiting when h >
1. If, for example, there aretwo degree 2 maps ρ : X → C and ρ : X → C and Q ∈ X ( Q ) then ρ ∗ ρ ( Q )+ ρ ∗ ρ ( Q ) ∈ X (4) ( Q )does not satisfy (1).2.6. Trace maps and ramification.
We note that all statements in this section are essentiallywell-known results in algebraic number theory. We nonetheless give proofs because we could notfind the exact statements in the literature.Any map ρ : X → C of curves as in the statement of Theorem 2.6 may be ramified at certainpoints of degree at most d = deg( ρ ). Example 2.12.
When ρ is the quotient map ρ : X (67) → X +0 (67), there is a non-cuspidalrational point Q ∈ X (67)( Q ) that ramifies. Ramifying here means that w ( Q ) = Q . In order todeal with degree 3 effective divisors such as 3 Q = Q + ρ ∗ ρ ( Q ) ∈ X (67) (3) ( Q ), we study in moredetail how differentials transform under (ramified) maps of curves. JOSHA BOX, STEVAN GAJOVI´C AND PIP GOODMAN
First, we briefly recall some facts about maps between Dedekind domains.We consider an embedding A → B of Dedekind domains. Given a prime ideal p ⊂ A and anelement f ∈ A , denote by k ( p ) the residue field of A at p , and by f ( p ) the image of f in k ( p ).Assume that Frac( B ) / Frac( A ) is a finite extension. This field extension comes with a trace mapTr B/A : Frac( B ) → Frac( A ) . Write p B = Q i q e i i for the factorisation of p B into prime ideals. We similarly obtain local tracemaps Tr b B q i / b A p and Tr k ( q i ) /k ( p ) . Denote by loc r : R → b R r be the localisation map at the primeideal r in the ring R . Lemma 2.13.
We have loc p ◦ Tr B/A = X i Tr b B q i / b A p ◦ loc q i , and for each f ∈ B we have Tr B/A ( f )( p ) = X i e i Tr k ( q i ) /k ( p ) ( f ( q i )) . Proof.
Upon replacing A by A p , we may assume A is a DVR. Consider the completion of B withrespect to p and apply the Chinese Remainder Theorem: b B := lim ←− n B/ p n B = Y i b B q i . (7)Let b ∈ b B . We define the trace Tr b B/ b A ( b ) by taking the trace of the Frac( b A )-linear map givenby multiplication-by- b on b B ⊗ b A Frac( b A ). If b ∈ B , then the global trace Tr B/A ( b ) equals Tr b B/ b A ( b ),and corresponds to a block matrix on Q i b B q i by (7). This gives the first equality.For the second equation, we begin by reducing the first modulo p . It then suffices to show theequality e i Tr k ( q i ) /k ( p ) ( f ( q i )) = Tr b B q i / b A ( f )( p ) for each i . To compute Tr b B q i / b A ( f )( p ), we may firstdetermine the image of f in the k ( p )-vector space b B q i / p , and then take the trace down to k ( p ).We note that b B q i / p = b B q i / q e i i ≃ b B q i / q i × q i / q i × . . . × q e i − i / q e i i as k ( p )-vector spaces. The right-hand side is a k ( q i )-vector space, and to compute the trace to k ( p )we may first compute the trace to k ( q i ). Finally, if t ∈ b B q i is a uniformiser and f = P ∞ n =0 a n t n ∈ b B q i then the multiplication-by- f map on b B q i / p has, by the above, e i diagonal entries equal to a ,so the trace becomes e i a = e i f ( q i ). (cid:3) When M is an R -module and r is a prime ideal in R , denote by M r the localisation of M at r ,and by c M r its completion. Now suppose that K is a field such that the Dedekind domains A and B are K -algebras and A → B is an embedding of K -algebras.We obtain an embedding Ω A/K → Ω B/K of the spaces of K¨ahler differentials. Now given s ∈ A ,d s is a Frac( B )-basis for Ω B/K , and we obtain a trace mapTr
B/A : Ω
B/K → Ω A/K , f d s Tr B/A ( f )d s which is independent of the choice of s . Corollary 2.14.
The equation loc p ◦ Tr B/A = X i Tr b B q i / b A p ◦ loc q i also holds true on Ω B/K . We note that traces of n -th roots are easily computed. Lemma 2.15.
Suppose that F ( α ) /F is a field extension of degree n defined by the minimal poly-nomial α n = a for some a ∈ F . Then Tr F ( α ) /F ( α i ) = 0 unless n | i. Proof.
This follows directly from the shape of the minimal polynomial. (cid:3)
OINTS ON MODULAR CURVES USING GENERALISED SYMMETRIC CHABAUTY 9
We now consider a map ρ : X → C of curves over a field K and a K -rational point R ∈ C ( K ).As divisors, we write ρ ∗ R = P Q R e Q/R Q as a sum of points Q on X . Assume that we havechosen K such that Q ∈ X ( K ) for each Q mapping to R . Denote by v Q the valuation of thediscrete valuation ring b O X,Q .From now on, suppose that K is a finite extension of Q p for a prime p of good reduction for X and C , with ring of integers O K and prime ideal p . Then ρ extends to an O K -morphism X → C between the minimal proper regular models of X and C .Suppose no two distinct points mapping to R have equal reduction mod p and p ∤ e Q/R for allpoints Q R . Lemma 2.16.
Let t ∈ b O C,R be a well-behaved local coordinate, and consider Q ∈ X mapping to R . Then, after base changing to an unramified extension of K of degree at most e Q/R , there existsa well-behaved local coordinate s Q ∈ b O X,Q such that s e Q/R Q = t . Moreover, ρ ∗ : b O C,R → b O X,Q induces an embedding of fraction fields of degree e Q/R defined by the equation x e Q/R − t = 0 .Proof. Consider Q ∈ ρ − ( { R } ). As Q is a K -rational point, its residue field is k ( Q ) = K . Let π ∈ b O X,Q and t ∈ b O C,R be two well-behaved uniformisers, and write e = e Q/R . We can write ρ ∗ ( t ) = b π e + b π e +1 + . . . , with each b i ∈ O K . Define u = b + b π + b π + . . . ∈ b O × X,Q , so that ρ ∗ ( t ) = uπ e . Then the mod p reduction of u is e u = P ∞ n =0 e b n e π n since π is well-behaved. Since also t is well-behaved and e ρ still has ramification degree e at e Q/ e R by assumption, we must have e b = 0(c.f Section 2.1). Let k ′ /k be an extension of the residue field of K containing a root of X e − e b .Then k ′ corresponds to an unramified extension K ′ /K and by Hensel’s Lemma (applicable because b ∈ O × K and p ∤ e ), K ′ contains a root of X e − b . Then also b O X K ′ ,Q contains an e th root of u by Hensel’s lemma, and we simply define s Q = u /e π . The map between fraction fields now hasdegree e because ρ ∗ is an isomorphism on residue fields. (cid:3) Using the lemma, we extend K and define t and s Q (for each Q R ) to be well-behaveduniformisers satisfying s e Q/R Q = t . We consider the Dedekind domain O X,ρ ∗ R := { f ∈ K ( X ) | v Q ( f ) ≥ Q R } . and the DVR O C,R . The Dedekind domain has localisations O X,Q at all places Q mapping to R .Denote by Tr Q/R the trace map from Frac( b O X,Q ) to Frac( b O C,R ). Now suppose that ω is a globalmeromorphic differential on X/K . We can interpret ω as a K¨ahler differential in Ω O X,ρ ∗ R /K . Fromthe embedding ρ ∗ : O C,R → O
X,ρ ∗ R of Dedekind domains, we thus obtain a trace mapTr : Ω O X,ρ ∗ R /K → Ω O C,R /K , which for ω ∈ H ( X, Ω) equals the trace map defined in Section 2.1 (followed by localisation at R ). Proposition 2.17.
Write loc Q ( ω ) = P ∞ i =0 a i ( Q ) s iQ d s Q for the expansion of ω in Ω b O X,Q /K . Then loc R (Tr( ω )) = P Q R P j ≥ a je Q/R − t j − d t .In particular, if Tr( ω ) = 0 , then for each j ≥ we have X Q R a j · e Q/R − ( Q ) = 0 . Remark . Note that this equality determines a linear relation between the columns of thematrix e A defined in the statement of Theorem 2.6. For example, when ρ has degree 2 and Q ∈ X ramifies, the equation says a ( Q ) = 0, and e A has a vanishing column, c.f. Example 4.3. Proof.
We also obtain local trace maps Tr
Q/R : Ω O X,Q /K → Ω O C,R /K . By Corollary 2.14, we haveloc R (Tr( ω )) = X Q R Tr Q/R (loc Q ( ω )) = X Q R Tr Q/R ∞ X i =1 a i − ( Q ) s i − Q d s Q ! . Next, we recall that by Lemma 2.15 we have Tr
Q/R ( s iQ ) = 0, unless i is a multiple of e Q/R ,in which case Tr
Q/R ( s je Q/R Q ) = e Q/R t j . Now i Tr Q/R ( s i − Q d s Q ) = dTr Q/R ( s iQ ) , from which wededuce that loc R (Tr( ω )) = P Q R P ∞ j =1 a je Q/R − t j − d t . Finally, we recall from Section 2.1 thatΩ b O C,R /K = K [[ t ]]d t , so the equality of power series yields a coefficient-wise equality. (cid:3) Finally, we show that traces of uniformisers can be computed as expected.
Lemma 2.19.
Consider Q such that ρ ( Q ) = R , and P ∈ D ( R ) with D := ρ ∗ P ∩ D ( Q ) = P ei =1 P i (with possible repetition). We base change X and C from K to K ( D ) , the extension of K containingall coordinates of points in the support of D . Then for a well-behaved uniformiser s at Q we have Tr Q/R ( s )( P ) = e X i =1 s ( P i ) . Consequently, if f ( X ) = 1 − b X + b X − . . . + ( − e b e X e is the reverse minimal polynomial of s over b O C,R , then b i ( P ) is equal to the i th symmetric polynomial in s ( P ) , . . . , s ( P e ) .Proof. Let O X,D be the ring of functions f ∈ K ( D )( X ) such that v P i ( f ) ≥ i ∈ { , . . . , e } .We obtain a map ρ ∗ : O C,P → O
X,D , and denote by b O X,D the completion of O X,D with respectto ρ ∗ m P , where m P ⊂ O C,P is the maximal ideal. Well-behaved means that s ∈ b O X , e Q . Now werecall from Section 2.1 that b O X , e Q is contained in b O X,Q ′ for any Q ′ ∈ D ( Q ). We find that O X , e Q is contained in both b O X,Q and b O X,D , and all three have the same field of fractions. Similarly, b O C , e R ⊂ b O C,P , b O C,R .Now we apply the second part of Lemma 2.13 to b O C,P → b O X,D , using the fact that k ( P i ) = k ( P )for each i (due to extending the base field). This yields Tr b O X,D / b O C,P ( s )( P ) = P ei =1 s ( P i ). Finally,the equality of fractions fields in each of the rows of the below diagram makes it commute: b O X,Q b O X , e Q b O X,D b O C,R b O C , e R b O C,P Tr Q/R
Tr Tr , from which we conclude the first statement. For the second statement, we apply the previous to s, s , . . . , s e , and note that the power sums generate the algebra of symmetric polynomials. (cid:3) Lemmas to control the prime bound.
The following lemmas are needed to control thelower bound on the prime p in Theorems 2.6 and 2.24. Lemma 2.20.
Let N , T and ℓ > T be positive integers, and p > N + T a prime number. Thenwe have (8) ℓ − T ≥ N ord p ( ℓ ) + 1 . Proof.
Write ℓ = p a b , where a = ord p ( ℓ ). When a = 0, this is true because ℓ > T . Otherwise, wenote that ℓ ≥ p a , and p a − T ≥ ( N + T + 1) a − T ≥ a ( N + T ) − T ≥ aN, as desired. (cid:3) Next, we consider the inclusions of polynomial rings Q [ s , . . . , s n ] = Q [ e , . . . , e n ] ⊂ Q [ z , . . . , z n ],where s k = P nℓ =1 z kℓ for each k ≥ k > n ), and e k is the k thelementary symmetric polynomial in z , . . . , z n . We adopt the conventions e k = 0 for k > n and e = 1. Let m ⊂ Q [ e , . . . , e n ] be the ideal generated by e , . . . , e n . Then in the ring of formalpower series Q [ z , . . . , z n ] J X K , Newton’s identities can be given the following compact form: ∞ X k =0 ( − k e k X k = n Y i =1 (1 − z i X ) = exp − ∞ X k =1 s k k X k ! . (9) Lemma 2.21. (1)
For each k ≥ , the difference s k /k − ( − k − e k is in m , and equals asum of terms b · e i · · · e i ℓ (for ℓ ≥ ) with coefficient b ∈ Q of denominator dividing ℓ . (2) If k > n we have s k k − ( − k n X i = k − n e i e k − i ! ∈ m , and it equals again a sum of terms b · e i · · · e i ℓ (for ℓ ≥ ) with coefficient b of denominatordividing ℓ . In particular, if k > n then s k /k ∈ m . (3) Suppose that S is a ring with an ideal I , and we have a , . . . , a n ∈ I satisfying for each i ≤ n that a i ≡ a n + i mod I w , where w ∈ Z ≥ . Then a · · · a n ≡ a n +1 · · · a n mod I w + n − . OINTS ON MODULAR CURVES USING GENERALISED SYMMETRIC CHABAUTY 11
Proof.
For part (3), it suffices to note the equality a · · · a n − a n +1 · · · a n = n X i =1 ( a i − a i + n ) a i +1 a i +2 · · · a i + n − . Parts (1) and (2) follow directly from (9) after applying Log to both sides of the equation andcomparing coefficients for X k . (cid:3) Proof of Theorem 2.6.
For simplicity of exposition (to avoid using triple indices), we assumethat h = 1. Then there is one map ρ : X → C , and we assume moreover that Q = ρ ∗ ( m · R ) for R ∈ C ( K ) and m ∈ Z > . For the general case, one can simply sum all arguments that follow overthe points in the curves C j , because Q , . . . , Q h are distinct. We write ρ ∗ R = P ki =1 m i Q i , where m i ≥ i , and Q = k X i =1 n i Q i , where n i ≥ m · m i for each i . We choose the ordering so that m i = 0 for all i > k ′ , where k ′ ≤ k .Note that Q = P ki =1 ( n i − mm i ) Q i and Q = P k ′ i =1 mm i Q i .Let L = K ( Q ) be the field obtained by adjoining the coordinates of Q , . . . , Q k and consider allpoints, maps and curves as base-changed to L . Let t R be a well-behaved uniformiser at R . UsingLemma 2.16, at each of the distinct points Q i , consider a well-behaved uniformiser t Q i ∈ b O X,Q i satisfying t m i Q i = t R if i ≤ k ′ . For this we need to extend L by an unramified extension, whichwe still denote by L . Consider F := L ( P ) and let q be a prime in F above p . We write P = P ki =1 ( P i, + . . . + P i,n i ), where P i,j ≡ Q i mod q for each j ∈ { , . . . , n i } .We write z i,j := t Q i ( P i,j ) and s i,ℓ = P n i j =1 z ℓi,j , and let e i,ℓ be the ℓ th elementary symmetricpolynomial in z i, . . . , z i,n i . Note that each z i,j ∈ q because t Q i is well-behaved. Fix one vanish-ing differential ω of trace zero for all maps and write loc Q i ( ω ) = P j ≥ a j ( Q i , ω ) t jQ i d t Q i . Aftermultiplying ω with a scalar, we may assume ω ∈ H ( X , Ω) and each a j ( Q i , ω ) ∈ O L p .For i > k ′ , we know that e i, = . . . = e i,n i = 0 implies that z i,j = 0 for all j ∈ { , . . . , n i } , andhence P i,j = Q i for all j ∈ { , . . . , n i } by injectivity of t Q i on residue discs. For i ≤ k ′ , we wouldlike to be able to read off from the e i,ℓ whether P is (partially) a pullback. Lemma 2.22.
Consider R, P , Q , ρ, z i,j , e i,ℓ etc. as defined so far, and define P ′ = P k ′ i =1 ( P i, + . . . + P i,n i ) (sum only up to k ′ ).Then P ′ = P k ′ i =1 ( n i − mm i ) Q i + ρ ∗ S for some S ∈ C ( m ) ( K r ) if (and only if ) there exists apolynomial f ∈ K r [ X ] of degree m with constant coefficient 1 such that for all i ∈ { , . . . , k ′ } wehave an equality of polynomials − e i, X + . . . + ( − n i e i,n i X n i = f ( X m i ) . Proof.
Denote by R the set of inverses of roots of f . Choose i ∈ { , . . . , k ′ } . By assumption, wefind that n i Y j =1 (1 − z i,j X ) = f ( X m i ) . In particular, after reordering we find that(a) z i,mm i +1 = 0 , . . . , z i,n i = 0, and moreover(b) { z m i i,j | j ∈ { , . . . , mm i }} = R .Because uniformisers are injective on residue classes, (a) implies that Q i = P i,mm i +1 = . . . = P i,n i .Recall that z m i i,j = t R ( ρ ( P i,j )) for j ≤ mm i . From (b) and because t R is injective on D ( R ), we findthat { ρ ( P ij ) | i ∈ { , . . . , k ′ } , j ∈ { , . . . , mm i }} has size m . Denote these points by S , . . . , S m .They satisfy S ℓ ≡ R mod q for each ℓ , and { t R ( S ℓ ) | ℓ ∈ { , . . . , m }} = R .Write ρ ∗ ( S ℓ ) = P ki =1 P m i j =1 P ′ ℓij , where again P ′ ℓij ≡ Q i modulo a fixed prime above q in F ( ρ ∗ S ℓ )for each i . Write z ′ ℓij = t Q i ( P ′ ℓij ), and define e ′ ℓij to be the j th elementary symmetric polynomialin z ′ ℓ,i, , . . . , z ′ ℓ,i,m i . Recall that t m i Q i = ρ ∗ t R . From Lemma 2.19, we see that e ′ ℓ,i,j = 0 for each j ∈ { , . . . , m i − } and e ′ ℓ,i,m i = ( − m i +1 t R ( S ℓ ). We conclude from Lemma 2.21 (1) that m Y ℓ =1 m i Y j =1 (1 − z ′ ℓ,i,j X ) = m Y ℓ =1 (1 − t R ( S ℓ ) X m i ) = f ( X m i ) for each i , from which it follows by injectivity of t Q i that P i, + . . . + P i,mm i = m X ℓ =1 ( P ′ ℓ,i, + . . . + P ′ ℓ,i,m i ) , and thus P = P ki =1 ( n − mm i ) Q i + ρ ∗ ( S + . . . + S m ). (cid:3) We continue the proof of Theorem 2.6. As [
P − Q ] ∈ J ( K ) and ω is a vanishing differential, wefind that R PQ ω = 0 . This is a sum of tiny integrals equal by Proposition 2.5 (i) to(10) k X i =1 ∞ X ℓ =1 a ℓ − ( Q i , ω ) s i,ℓ ℓ = 0 . For i ∈ { , . . . , k ′ } , write p i,ℓ := m k ′ P n k ′ j =1 Tr Q i /R ( t ℓQ i )( ρ ( P k ′ ,j )). Then p i,ℓ = 0 when m i ∤ ℓ and p i,jmi jm i = s k ′ ,jmk ′ jm k ′ for j ≥
1. Now we consider i = k ′ , and subtract0 = 1 m k ′ Z P k ′ , + ... + P k ′ ,nk ′ n k ′ Q k ′ ρ ∗ Tr( ω ) = k ′ X i =1 ∞ X ℓ =1 a ℓ − ( Q i , ω ) ℓ p i,ℓ from (10). Here we used Proposition 2.17 to evaluate the integral.We thus obtain(11) k ′ X i =1 ∞ X ℓ =1 a ℓ − ( Q i , ω ) (cid:18) s i,ℓ − p i,ℓ ℓ (cid:19) + k X i = k ′ +1 X ℓ ≥ a ℓ − ( Q i , ω ) s i,ℓ ℓ = 0 . For i ≤ k ′ , define r i,ℓ for ℓ ∈ Z ≥ by r i,ℓ = 0 when m i ∤ ℓ and r i,jm i = ( − jm i − jm k ′ e k ′ ,jm k ′ . For i > k ′ , define r i,ℓ = 0 for all ℓ . We now use Lemma 2.21 (1) to rewrite traces/power sums in termsof elementary symmetric polynomials: p i,jm i /jm i = s k ′ ,jm k ′ /jm k ′ = ( − jm i − r i,jm i + h . o . t and s i,ℓ /ℓ = ( − ℓ − e i,ℓ + h . o . t. This yields0 = k X i =1 n i X ℓ =1 ( − ℓ − a ℓ − ( Q i , ω )( e i,ℓ − r i,ℓ ) + h . o . t ., (12)where the higher order terms are of the form αs ( e i,i · · · e i,i s − r i,i · · · r i,i s ) with s ≥ α ∈ O L p . (13)Now define f ( X ) = 1 + ( − m k ′ e k ′ ,m k ′ X + . . . + ( − mm k ′ e k ′ ,mm k ′ X m . Then for each i ≤ k ′ ,we find that f ( X m i ) = 1 − r i, X + . . . + ( − n i r i,n i X n i . In view of Lemma 2.22, it is our aim toshow that for each i ∈ { , . . . , k ′ } , we have f ( X m i ) = 1 − e i, X + . . . + ( − n i e i,n i X n i , and for i > k ′ we have e i,ℓ = 0 for all ℓ ∈ { , . . . , n i } . In other words, we aim to show e i,ℓ = r i,ℓ for all i ∈ { , . . . , k } and ℓ ≤ n i .We first show that e i,ℓ , r i,ℓ ∈ L p . Note that F/L is Galois. Suppose that σ ∈ Gal( F q /L p ). Foreach P i,j ∈ P (that is, in the support of P ) reducing mod q to g P i,j = f Q i , also P σi,j must reduce to f Q i = f Q σi . Also P σi,j ∈ P as P is K -rational, so P σi,j = P i,j ′ for some j ′ ∈ { , . . . , n i } . As moreover t σQ i = t Q i , we find that e σi,ℓ = e i,ℓ . By definition, then also r σi,ℓ = r i,ℓ , and all e i,ℓ and r i,ℓ are in L p . We define ν := min i ≥ ℓ ≤ n i v p ( e i,ℓ − r i,ℓ ) , where v p is p -adic valuation. We argue by contradiction and assume that ν < ∞ . Recall that ν ≥ P and Q are in the same residue disc.Recall that N is the ramification index of p /p . We apply Lemma 2.21 (3) with I = p , whichyields e i,i · · · e i,i s ≡ r i,i · · · r i,i s mod p ν + s − . By Lemma 2.20 with T = 1, and because p ≥ N + 2,we find that s − ≥ N ord p ( s ) + 1. We conclude that each of the higher order terms (13) vanishesmod p ν +1 .We obtain equation (12) for each ω ∈ { ω , . . . , ω q } . Let A be the matrix made up of the v ( ω j , t Q i , n i ), so that its reduction mod p is the matrix e A from the statement of the theorem.Denote by B the matrix obtained from A by removing for each j ∈ { , . . . , m } the column withentries (( − jm k ′ a jm k ′ − ( Q k ′ , ω ) , . . . , ( − jm k ′ a jm k ′ − ( Q k ′ , ω q )) . OINTS ON MODULAR CURVES USING GENERALISED SYMMETRIC CHABAUTY 13
By Proposition 2.17, the columns of A satisfy m linear equations, each of which has coefficient 1for exactly one of the removed columns. Since e A has rank n − m , we conclude that the reduction e B of B has full rank n − m . Denote by v the vector of length n − m with as entries the list e , − r , , . . . , e ,n − r ,n , . . . , e k,n k − r k,n k , from which we remove the elements e k ′ ,jm k ′ − r k ′ ,jm k ′ for j ∈ { , . . . , m } (each of which is zero by definition of r i,ℓ ). Then v = 0 mod p ν by definition of ν , and from (12) and the analysis of the higher order terms, we conclude that B · v = 0 mod p ν +1 . Since e B has full rank, we find that v = 0 mod p ν +1 , contradicting the definition of ν , unless indeed ν = ∞ and v = 0. We find that e i,ℓ = r i,ℓ for all values of i and ℓ , as desired.For i > k ′ , this means z i,j = 0 for all j ∈ { , . . . , n i } , so that P i,j = Q i for each such i and j .We then apply Lemma 2.22 with P ′ = P k ′ i =1 ( P i, + . . . + P i,n i ) and polynomial f . We concludethat P is of the desired form.2.9. Chabauty using the higher order terms.
In some situations, there is an obstruction thatprevents the matrix in Theorem 2.6 to have sufficiently high rank at any prime, but we do expectthe outcome of the theorem to hold true.
Example 2.23.
We consider the degree 2 map ρ : X (73) → X +0 (73). Let c , c ∞ ∈ X (73)( Q ) bethe two cusps, and consider Q = 3 c . Note that w interchanges these cusps, i.e. ρ ( c ) = ρ ( c ∞ ).Let t be the pullback under ρ of a well-behaved uniformiser at ρ ( c ). Then t is a well-behaveduniformiser at c and at c ∞ . The curve X (73) has genus 5, and its quotient X +0 (73) has genus 2.We find a 3-dimensional space of vanishing differentials with trace zero on X +0 (73) (c.f. [Box21]).Denote by ω , ω , ω a basis. By Proposition 2.17, if we write loc c ( ω i ) = P j ≥ a i,j t j d t thenloc c ∞ ( ω i ) = P j ≥ ( − a i,j ) t j d t . Consider the matrices A = ( a i,j ) ≤ i ≤ ≤ j ≤ and A ∞ = −A . Then A := ( A | A ∞ ) satisfies rk( A ) = rk( A ). Also, A is the matrix whose mod p reductioncorresponds to the matrix in condition (3) of Theorem 2.6 for the point Q = 3 c + 3 c ∞ . If itsreduction e A modulo any large prime p had rank 3, the entire mod p residue class of 3 c + 3 c ∞ would be contained in ρ ∗ (cid:0) ( X +0 (73)) (3) ( Q ) (cid:1) This is not the case, however. We compute that the Riemann–Roch space L (3 c + 3 c ∞ ) is3-dimensional, which is unusually large. In fact, we find a degree 6 function f ∈ L (3 c + 3 c ∞ )such that w ∗ f = f , i.e. the corresponding map f : X → P does not factor via ρ . Now note that3 c + 3 c ∞ = f ∗ (1 : 0). So for each prime p of good reduction, the points R ∈ P ( Q ) such that R ≡ (1 : 0) mod p satisfy also that f ∗ R ≡ c + 3 c ∞ . But, as f does not factor via X +0 (73), f ∗ R isin general not the pullback of a degree 3 divisor on X +0 (73). By Theorem 2.24, e A therefore cannothave rank 3, and neither can e A . Indeed, we find that ω satisfies a , = a , = a , = 0, so thatthe entire bottom row of A is zero. Therefore, condition (3) in Theorem 2.6 is never satisfied for Q = 3 c .A similar situation occurs on X (57).In such cases, it can help to look further into the expansion of the 1-forms. We begin with somenotation. Consider a point Q on a curve X and a regular 1-form ω ∈ H ( X, Ω). We expand ω around Q in terms of a uniformiser t Q at Q , giving loc Q ( ω ) = P j ≥ a j t jQ , and define v ( ω, t Q , ℓ, k ) := (cid:0) ( − ℓ − a ℓ , ( − ℓ a ℓ +1 , . . . , ( − k − a k − (cid:1) . With this notation, we have v ( ω, t Q , k ) = v ( ω, t Q , , k ). For integers j and i such that j +1 ≤ i ≤ j and any x , . . . , x j in some field, we define ψ i ( x , . . . , x j ) = j X ℓ = i − j x ℓ x i − ℓ . Theorem 2.24.
Consider a number field K , a curve X/K and Q = P ki =1 n i Q i ∈ X ( n ) ( K ) , where n = P ki =1 n i and Q , . . . , Q k ∈ X are distinct points. Let r be a prime of K , of good reductionfor X , containing the rational prime p ∈ r . Denote by X / O K r a minimal proper regular model of X/K . Let N be the ramification index of p in K ( Q ) . (1) Suppose that p ≥ N + 3 . (2) Suppose that the space V of vanishing differentials has dimension at least n , and considerlinearly independent ω , . . . , ω n ∈ V on X . For each Q i , choose a uniformiser t Q i , andsuppose that the n × n -matrix A := v ( ω , t Q , n ) · · · v ( ω , t Q k , n k ) ... . . . ... v ( ω n , t Q , n ) · · · v ( ω n , t Q k , n k ) has rank r < n .We thus choose uniformisers t e Q i at e Q i , and a basis e ω , . . . , e ω n for the image of V ∩ H ( X , Ω) under reduction mod r , such that the vectors v ( e ω j , t e Q i , n i ) for all ≤ i ≤ k and r < j ≤ n are zerovectors. (3) Assume that the r × n matrix e A defined below has rank r . (4) Let F be the residue field of a prime in K ( Q ) above r . Suppose moreover that there are nonon-zero vectors ( e x , . . . , e x k ) ∈ F n solving the system of equations e L · e w = , where e x i = ( e x i, , . . . , e x i,n i ) , e L = e A e A ! , e A := v ( e ω , t e Q , n ) · · · v ( e ω , t e Q k , n k ) ... . . . ... v ( e ω r , t e Q , n ) · · · v ( e ω r , t e Q k , n k ) , e A = − v ( e ω r +1 , t e Q , n , n ) · · · − v ( e ω r +1 , t e Q k , n k , n k ) ... . . . ... − v ( e ω n , t e Q , n , n ) · · · − v ( e ω n , t e Q k , n k , n k ) , and e w = ( e x , . . . , e x k , ψ n +1 ( e x ) , . . . , ψ n ( e x ) , ψ n +1 ( e x ) , . . . , ψ n ( e x ) , . . . , ψ n k +1 ( e x k ) , . . . , ψ n k ( e x k )) T . Then
Q ∈ X ( n ) ( K ) is alone in its residue class.Remark . We emphasize that rk( A ) < n means that modulo any prime p of K ( Q ) the reductionof A mod p has rank smaller than n , and Theorem 2.6 therefore cannot be applied. Proof.
Let p be a prime above r in L := K ( Q ) with uniformiser π ∈ p . Let P = P ki =1 P n i j =1 P i,j ∈ X ( n ) ( K ) belong to the residue disc of Q , where the sum is arranged such that points P i,j reduceto the same point as Q i modulo a prime q of F := L ( P ) above p . For each i , choose t Q i to bea well-behaved uniformiser at Q i , and for each i and j , define z i,j = t Q i ( P i,j ) ∈ O F q . Define s i,ℓ = P n i j =1 z ℓi,j , and let e i,ℓ be the ℓ th elementary symmetric polynomial in z i, . . . , z i,n i .In the statement of the theorem, condition (3) implies that Span { e ω r +1 , . . . , e ω n } is exactlythe space of differentials e ω on e X satisfying v ( e ω, t e Q i , n i ) = 0 for all i . Note that condition (4)is independent of the choice of bases for Span { e ω , . . . , e ω r } and Span { e ω r +1 , . . . , e ω n } . Now let ω , . . . , ω n ∈ V ∩ H ( X , Ω) be linearly independent such that v ( ω j , t Q i , n i ) = 0 for all i and foreach j ∈ { r + 1 , . . . , n } . Having chosen well-behaved uniformisers and integral differential forms weconclude that condition (4) holds for the reductions e ω , . . . , e ω n of ω , . . . , ω n modulo r . Moreover,the matrices L , A and A are the mod p reductions of corresponding matrices over O L p , definedin terms of ω , . . . , ω n .As in the proof of Theorem 2.6, we find that each e i,j ∈ O L p by Galois theory. Define ν := min i ∈{ ,...,k } j ∈{ ,...,n i } v p ( e i,j )and assume that ν < ∞ . We aim to find a contradiction, giving ν = ∞ . Recall that ν ≥ P and Q are in the same residue disc. Let π ∈ p be a uniformiser, and define x i,ℓ by e i,ℓ = π ν x i,ℓ .Their mod p reductions e x i,ℓ will correspond to a non-zero solution of e L · e w = , which we shallshow in two parts.For ω ∈ { ω , . . . , ω n } , we have at each Q i the expansion loc Q i ( ω ) = P ℓ ≥ a ℓ ( Q i , ω ) t ℓQ i d t Q i . Asbefore, we obtain from R PQ ω = 0 the equation(14) 0 = k X i =1 ∞ X ℓ =1 a ℓ − ( Q i , ω ) s i,ℓ ℓ = 0 . OINTS ON MODULAR CURVES USING GENERALISED SYMMETRIC CHABAUTY 15
We again use Lemma 2.21 (1) to rewrite this in terms of e i,ℓ s, and apply Lemma 2.21 (3) andLemma 2.20 (with T = 1) to the higher order terms to obtain(15) k X i =1 n i X ℓ =1 ( − ℓ − a ℓ − ( Q i , ω ) e i,ℓ ≡ p ν +1 ) . Dividing by π ν and ranging over ω ∈ { ω , . . . , ω r } , we obtain e A · ( e x , . . . , e x k ) T = , the “upperhalf” of e L · e w = .Consider now any of the remaining vanishing differential forms ω ∈ { ω r +1 , . . . , ω n } . Recall thatall a ℓ − ( Q i , ω ) = 0 for 1 ≤ i ≤ k , 1 ≤ ℓ ≤ n i . Hence, using Lemma 2.21 (2), equation (14) in thiscase becomes(16) k X i =1 2 n i X ℓ = n i +1 a ℓ ( Q i , ω ) · ( − ℓ n i X m = ℓ − n i e i,m e i,ℓ − m + h . o . t = 0 , where each of the higher order terms is of the form bs · e i,i · · · e i,i s with s ≥ b ∈ O L p . Now s ≥ T = 2) that v p ( e i,i · · · e i,i s ) ≥ ν + ( s − ≥ ν + 1 + N ord p ( s ).We conclude that all higher order terms vanish mod p ν +1 . Multiplication by 2 (note that 2 / ∈ p )and division by π ν of (16) now yields e A · ( ψ n +1 ( e x ) , . . . , ψ n ( e x ) , ψ n +1 ( e x ) , . . . , ψ n k ( e x k )) T = 0 , as desired. Note that ( x , . . . , x k ) = mod p by maximality of ν . This solution of e L · e w = contradicts our assumption, so we must have ν = ∞ and e i,ℓ = 0 for all i and ℓ , so that also z i,j = 0for all i and j and P = Q by injectivity of well-behaved uniformisers on residue classes. (cid:3) The Mordell–Weil sieve
A formal description.
As mentioned in Section 2.3, Theorem 2.6 can in some cases be usedto compute the set of rational points on symmetric powers of curves when combined with theMordell–Weil sieve. In this section we describe this sieve, which is similar to the sieves in [Sik09],[Box21] and [Box].We consider a curve X with maps ρ i : X → C i of degree d i , for i ∈ { , . . . , s } , and an integer e .Next, we suppose given the following:(i) A finite list of points L ′ ⊂ X ( e ) ( Q ).(ii) A Q -rational degree e divisor D on X .(ii) Explicit independent generators D , . . . , D r for a subgroup G ⊂ J ( X )( Q ) and I ∈ Z ≥ such that I · J ( X )( Q ) ⊂ G . Here r is the rank of J ( X )( Q ).(iii) A list p , . . . , p n of primes of good reduction for X .Extend L ′ to a (possibly infinite) set L by adding for each point of the form P + ρ ∗ ( R ) + . . . + ρ ∗ s ( R s ) ∈ L ′ , with R i ∈ C ( m i ) i ( Q ) ( m i ≥
0) for each i , the entire set P + ρ ∗ C ( m ) ( Q ) + . . . + ρ ∗ s C ( m s ) ( Q ) to L . The purpose of the sieve is to show that X ( e ) ( Q ) = L .To this end, we first consider the map ι : X ( e ) ( Q ) −→ G, D I · [ D − D ] . Next, define A to be the abstract finitely generated abelian group isomorphic to G , with basis e , . . . , e r ∈ A , and set φ : A → J ( X )( Q ) , e i D i . For each p ∈ { p , . . . , p n } , we obtain a commutative diagram L ′ X ( e ) ( Q ) J ( X )( Q ) A e X ( e ) ( F p ) J ( e X )( F p ) ι red p φφ p ι p , where ι p : D I [ D − e D ], e D is the reduction of D , and φ p = red p ◦ φ . Definition 3.1.
Define M p ⊂ J ( e X )( F p ) to be the subset of elements of red p ( G ) ∩ ι p ( e X ( e ) ( F p ))that are either(1) not in the image of L ′ , or (2) the image of Q = P + ρ ∗ ( R ) + . . . + ρ ∗ s ( R s ) ∈ L ′ , with R i ∈ C ( m i ) i ( Q ) ( m i ≥
0) for each i , such that Q does not satisfy the conditions of Theorems 2.6 and, when applicable, 2.24.By definition, any hypothetical point Q ∈ X ( e ) ( Q ) \ L satisfies red p ◦ ι ( Q ) ∈ M p . We concludethe following. Proposition 3.2 (Mordell–Weil sieve) . If n \ i =1 φ − p i ( M p i ) = ∅ then X ( e ) ( Q ) = L .Remark . Naturally, this will only work in practice if the finite set L ′ contains all isolated points,as well as sufficiently many points (partially) composed of pullbacks.3.2. Implementing the sieve efficiently.
The groups J ( e X )( F p ) can be computed using theclass group algorithm of Hess [Hes02]. When e and p are large, it is important to implement thesieve in an efficient way. Rather than first computing M p i for each i and intersecting afterwards,we compute intersections recursively. It also turns out that computing e X ( e ) ( F p ) and ι p for largevalues of e and p is suboptimal, so we compute Riemann–Roch spaces instead.For a divisor D on a curve Y over a field K , define the Riemann–Roch space L ( D ) := { f ∈ K ( Y ) × | div( f ) + D ≥ } ∪ { } . Also define H i − := ker( φ p ) ∩ . . . ∩ ker( φ p i − ). Suppose that we have computed the finite set W i − of H i − -coset representatives for ∩ i − j =1 φ − p j ( M p j ), and we want to compute W i . First, we determinethe (larger) set W ′ i of H i -coset representatives for the same intersection. For each w ∈ W ′ i , wethen compute the F p -vector spaces L ( z + e D ) for each z such that I · z = φ p i ( w ) . Often these spaces will simply be 0-dimensional and w can be removed from W ′ i . For each non-zero f ∈ L ( z + D ), we obtain div( f ) + z + e D ∈ e X ( e ) ( F p ), and we verify whether it is in the image of L ′ and satisfies the conditions of Theorem 2.6 or Theorem 2.24. This way, we determine W i ⊂ W ′ i without the need to compute e X ( e ) ( F p ).We mention another improvement. When p and p are distinct primes such that J ( X )( F p )and J ( X )( F p ) are coprime, we have φ p ( W ) = φ p ( A ) by the Chinese Remainder theorem. Onethus tends to choose primes p such that the numbers J ( e X )( F p ) have prime factors in common.Often these factors are powers of small primes, whereas J ( e X )( F p ) tends to also be divisible bya large prime factor, let us call it r . It is unlikely that r will also divide J ( e X )( F q ) for any otherprime q we consider, hence the “mod r ” information is of little use. It does, however, slow the sievedown considerably, because information is stored in H i -cosets and the factor r increases the indexof H i . We remedy this by composing ι p and φ p with the multiplication-by- r map m r , thereby“forgetting” the mod r information.4. Applying Chabauty to modular curves
For each X ( N ) with N ∈ { , , , , } , let ρ : X ( N ) → X +0 ( N ) be the quotient map by w N . Denote by X ∗ ( N ) the quotient of X ( N ) by the full Atkin-Lehner group. To determine thecubic points on these curves, we use the Mordell–Weil sieve described in Section 3 in combinationwith Theorems 2.6 and 2.24 to determine the isolated points in X ( N ) (3) ( Q ), i.e. those not ofthe form P + ρ ∗ Q with P ∈ X ( N )( Q ) and Q ∈ X +0 ( N )( Q ). For X (57), we determine theentire set X ( N ) (3) ( Q ). On X (65), we moreover determine the points in X (65) (4) ( Q ) that arenot of the form ρ ∗ D for D ∈ X +0 (65) (2) ( Q ) or of the form P + ρ ∗ Q with P ∈ X (65) (2) ( Q ) and Q ∈ X +0 (65)( Q ).In this section, we determine the information necessary to apply Theorems 2.6 and 2.24 andrun the sieve. OINTS ON MODULAR CURVES USING GENERALISED SYMMETRIC CHABAUTY 17
Models, points and differentials.
All the modular curves we consider were studied in[Box21], so we use the models, subgroups of Mordell–Weil groups and vanishing differentials ascomputed there.We recall that these (canonical) models for each X ( N ) were computed in [Box21] using eitherthe Small Modular Curves package in Magma or the code written by ¨Ozman and Siksek [OS19]. Bothalgorithms simply compute equations satisfied by q -expansions of weight 2 cusp forms, followingGalbraith [Gal02].To find cubic and quartic points a modular curve X ( N ) ⊂ P n (with coordinates x , . . . , x n ),we have used code written by ¨Ozman and Siksek [OS19] to decompose intersections H ∩ X ( N ) ofhyperplanes H ⊂ P n given by H : a x + . . . + a n x n = 0 , with each a i ∈ Z satisfying | a i | ≤ C for a (small) bound C . Each irreducible component of X ( N ) ∩ H of degree d yields a point of degree d on X ( N ).Decomposing such intersections can be time-consuming on genus 5 curves, and we did notalways find all points this way. In such cases, we ran the sieve described in Section 3 with thistoo small set of known points L ′ . Consider the notation introduced in Section 3. After a numberof primes p , . . . , p n , we found a non-empty set W n of representatives for the possible H n -cosetswhere unknown rational points can map into. As the index of H n in A is large, it is a priori unlikelythat one of these cosets w + H n , for w ∈ W n , contains a “small vector”, say with all coefficientsfor free generators of A smaller than 100 in absolute value. So, heuristically, small vectors shouldonly occur with good reason, for example as images of rational points.Using the LLL-algorithm [LLL82], we searched for small vectors in these cosets w + H n . For eachsmall vector v ∈ w + H n found, we compute L ( D + φ ( v )) and find that it is indeed 1-dimensional,generated by, say, f . Then in each case, indeed div( f ) + D + φ ( v ) was a new isolated rationalpoint on the symmetric power.Finally, in order to apply Theorems 2.6 and 2.24 to X ( N ), we need for each prime p (of goodreduction for X ( N )) a basis for the image e V of V ∩ H ( X N , Ω) under the reduction map, where X N / Z p is a proper minimal model of X ( N ) and V is the space of vanishing differentials, withtrace zero to X +0 ( N ) when N = 57 and X ∗ (57) otherwise. In [Box21, Section 3.4], it was shown foreach N ∈ { , , , , } that V = Ker(1 + w ∗ N ) and e V = Ker(1 + e w ∗ N ), where w ∗ N and e w ∗ N arethe pull-back morphisms on Ω X ( N ) / Q resp. Ω e X ( N ) / F p . Similarly, for N = 57, it was shown that V = Ker(1 + w ∗ ) ∩ Ker(1 + w ∗ ) and e V = Ker(1 + e w ∗ ) ∩ Ker(1 + e w ∗ ). This allows us to computethose vanishing differentials and verify the conditions of Theorems 2.6 and 2.24 in practice.4.2. Computing Mordell-Weil groups.
For each N ∈ { , , , , , } , the first namedauthor [Box21] computes a finite index subgroup of J ( N )( Q ). Apart from N = 57, the index ofthese subgroups is shown to divide 2. Thanks to our Mordell-Weil sieve §
3, this almost completelysuffices for our purposes. The only cases where they do not are N = 57 ,
65. For N = 57 weproduce a subgroup with index dividing 2; this appears to be needed purely for computationalreasons. Whereas when N = 65 the issue is somewhat deeper. In this case we verify the subgroupgiven in [Box21] is the entire Mordell-Weil group.The method used in [Box21] to compute subgroups of J ( N )( Q ) with bounded index can bebroken down into two steps. First, one finds a subgroup of bounded index in the free part of J ( N )( Q ). Then one computes the torsion subgroup of J ( N )( Q ). This is done by combiningtheorems of Manin–Drinfeld and Mazur with bounds given by J ( N )( F p ) for p ∤ N , see [Box21,Lemma 3.2]. We mention that bounds may also be obtained using Hecke operators as in [DEvH + ,Section 4.3], although this is not needed here.Let X/ Q be a (projective, non-singular) curve and Γ ≤ Aut Q ( X ) a finite subgroup. The quotientcurve C = X/ Γ is also defined over Q and the natural map ρ : X → C has degree X by J ( X ) and that of C by J ( C ). Choosing compatible base points for the maps ι X : X → J ( X ), ι C : C → J ( C ), we obtain a commutative diagram: X ι X / / ρ (cid:15) (cid:15) J ( X ) ρ ∗ (cid:15) (cid:15) C ι C / / J ( C ) . If the ranks of J ( C )( Q ) and J ( X )( Q ) are equal then ρ ∗ J ( C )( Q ) gives a subgroup of index dividing J ( X )( Q ) [Box21, Prop. 3.1].The problem of computing subgroups of bounded index in J ( X )( Q ) is thus reduced to computingthe free part of J ( C )( Q ). When C is an elliptic curve this may be done using Cremona’s algorithm[Cre97]. Else, if C has genus 2, this may be achieved through an algorithm of Stoll [Sto02].We shall always take X = X ( N ) and C = X +0 ( N ) with N ∈ { , , , , , } . For allthese values of N , the curve X +0 ( N ) has genus one or two. Example 4.1.
Let us give a few more details in the case X = X (57). Here we choose C = X +0 (57).This curve has a model defined by y = x − x + 3 x + 3 x − x + 1. Stoll’s algorithm [Sto17]may be used to show the free part of J ( C )( Q ) is generated by Q = (1 : 1 : 0) − (1 : − Q back to J (57) gives ρ ∗ ( Q ) = (1 : 1 : 0 : 1 : 0) + (3 : 9 / − / / − P + P , where P = (cid:0) − ( √− −
1) + 1 : − ( √− −
1) + 1 : ( √− −
1) : − ( √− − (cid:1) .4.2.1. The Mordell-Weil group of J (65) . Let ρ : X (65) → X +0 (65) be the quotient map. Thelatter is an elliptic curve and so we may determine its Mordell-Weil group. The group G = ρ ∗ ( J +0 (65)( Q )) J (65)( Q ) tors is isomorphic to Z × Z / Z × Z / Z as shown in [Box21, Section 4.5]. Denote by D a generatorof the free part of G . We use the Abel–Jacobi map ι : X (65)( Q ) (4) → J ( Q ) given by P [ P − ρ ∗ (0 : 1 : 0)].A priori G ⊂ J (65)( Q ) has at most index two. Whilst the sieve described in § p there seems to be a point on X (65)( F p ) (4) which under theAbel–Jacobi map ι p (compatible with ι and reduction modulo p ) doubles to the reduction of ρ ∗ ([( − − (0 : 1 : 0)]) (there also appear to be similar problems for four points comingfrom the quotient X +0 (65)). It seems likely that for each prime p there is an element in the set ofpreimages of ρ ∗ ([( − − (0 : 1 : 0)]) in J (65)( Q ) under muliplication by two whose reductionbelongs to ι p ( X (65)( F p ) (4) ). We have not been able to compute the full set of preimages to verifythis, but this does suggest the following method to verify G = J (65)( Q ).First find a divisor D defined over an extension K/ Q such that 2 D = D . Computing the2-torsion of J (65)( K ) then allows us to determine all points in J (65)( K ) which double to D .Indeed, if 2 D ′ = D , then 2( D − D ′ ) = 0. Given this set, it then suffices to decide whether anyelement belongs to J (65)( Q ).Owing in part to J +0 (65) being an elliptic curve, it was straightforward for us to find such a D .There are algorithms and explicit formulas which compute Mumford representations for preimagesunder multiplication by 2 on odd degree hyperelliptic jacobians [Sto17] [Zar19]. In principle, thisallows one to adapt our method to other curves. Theorem 4.2.
The Mordell-Weil group J (65)( Q ) of X (65) is the group generated by the twosubgroups ρ ∗ ( J +0 (65)( Q )) and J (65)( Q ) tors . Proof.
Let us abbreviate J (65) to J . Set G = h ρ ∗ ( J +0 (65)( Q )) , J ( Q ) tors i . In [Box21], it is shownthat G ∼ = Z × Z / Z × Z / Z . Explicit generators are given (they are the same as in § D, T , T respectively. So that, in particular, D = ρ ∗ ((1 : 0 : 1) − (0 : 1 : 0))generates the free part of G (here we use the models as given in § D + h T , T i is the double of a pointin J ( Q ). Equivalently, that none of D , D + T , D + T and D + T + T are doubles. The lastthree may easily be checked by reducing modulo 7. We now focus our attention on D .There are four points which double to (1 : 0 : 1) on X +0 (65). Two of these are defined over Q ( √ Q ( √ d be any of these points. The pullback D := ρ ∗ ( d − (0 : 1 : 0))then satisfies 2 D = D .We claim J ( K ) tors = J ( Q ) tors for K = Q ( √ , Q ( √ J ( F p ) asan abstract group for primes p = 5 ,
13 split in K . Since for these primes we have embeddings ofabstract groups J ( K ) tors ֒ → J ( F p ) which allow us to show the lower bound given by J ( Q ) tors ∼ = Z / Z × Z / Z is tight. The primes 11 and 19 split in Q ( √
5) and the corresponding groups give J ( F ) = Z / Z × Z · · · · Z and J ( F ) = Z · · Z × Z · · · · , OINTS ON MODULAR CURVES USING GENERALISED SYMMETRIC CHABAUTY 19 from which it is quickly deduced that J ( Q ( √ tors = Z / Z × Z / Z . For K = Q ( √
13) one mayuse 17 and 23.We thus have the equality of sets S := { x ∈ J ( K ) | x = D } = { D + T | T ∈ J ( Q )[2] } . Hence we are left to show D is not defined over Q . Denote by d c the Galois conjugate of d . Thenboth d + d c and 2 d are in X +0 (65)( Q ), so also Q := d − d c ∈ X +0 (65)( Q ). Now Q = Q c = − Q ,so Q is the unique non-trivial point in X +0 (65)( Q )[2]. We find that D − D c = ρ ∗ Q , and it sufficesto show that ρ ∗ Q = 0 ∈ J ( Q ). We can check this modulo 7.Hence none of the elements in S is in J ( Q ), and so we must have that J ( Q ) = G, as desired. (cid:3) Two examples.
We now give explicit examples of Theorems 2.6 and 2.24 in use.
Example 4.3.
Let us consider the non-cuspidal rational point Q ramified under the map ρ : X (67) → X +0 (67). This gives rise to the point 3 Q in X (67)( Q ) (3) . We will apply Theorem 2.6with r = 19 to show any other element in its residue disc also belongs to the set Q + ρ ∗ X +0 (67)( Q ).Writing 3 Q = Q + ρ ∗ ρ ( Q ), we see it is enough to show the matrix e A has rank two. Computing e A with respect to a uniformiser s at Q such that s = ρ ∗ t , where t is a uniformiser at ρ ( Q ), we find e A =
15 0 02 0 26 0 11 which clearly has rank 2. Note the second column is identically zero, in agreement with Proposition2.17. Example 4.4.
As explained in Example 2.23, Theorem 2.6 will always fail for the points 3 c , c ∞ ∈ X (73)( Q ) (3) , where c , c ∞ ∈ X (73)( Q ) are the cusps. To fix this we need expand further intothe coefficients of the differentials. We verify the conditions of Theorem 2.24 at r = 19. Given achoice of uniformiser and differentials, the corresponding matrices for 3 c ∞ are A = −
13 470 − which clearly has rank 2, and the F -matrices e A = (cid:18) (cid:19) and e A = (cid:0) (cid:1) . One then checks e L · ( x, y, z, y + 2 xz, yz, z ) T = has no non-zero solutions for x, y, z ∈ F .Note Theorem 2.24 does not require one to form e A from the reductions of the uniformiser anddifferentials used for computing A . For this reason, e A above does not coincide with the reductionof the first two rows of A . 5. Results Q -curves. An elliptic curve E defined over a number field K is said to be a Q -curve if itis Q isogenous to all of its Gal( Q / Q ) conjugates. For example, all elliptic curves with rational j -invariant are Q -curves, as are CM elliptic curves (for elliptic curves with CM by the maximalorder this may be deduced from Propositions 1.2 and 2.1 in § II of [Sil94]).For N ∈ { , , , , , } there are only finitely many cubic points on X ( N ). These pointsare listed § Q -curves exactly when the elliptic curves haveCM.The non-CM cubic points on X ( N ) for N ∈ { , , , , } do not give rise to Q -curves:this follows Theorem 1 . P ∈ X ( N )( K ) one may alwaysfind an elliptic curve defined over K with a K -rational N -isogeny. For N = 65, the same may bededuced from combining [CN, Theorem 2.7] and [CN, Corollary 3.4].Cremona and Najman describe an algorithm to determine whether an elliptic curve is a Q -curveor not in § X (65) whichgive rise to Q -curves are exactly the CM points. This algorithm has been implemented in Sage [Cre]. However, as part of this implementation the conductor is computed, and this appears tobe too costly for our elliptic curves. Instead, we partially implemented the algorithm in
Magma and avoid computing the conductor. In particular, for each of the non-CM elliptic curves
E/K under consideration there was a rational prime p and primes p , q of K above p such that either E had bad potentially multiplicative reduction at p but not q or E had good reduction at all primesabove p and the endomorphism algebras of the reductions at p and q were not isomorphic. Thesecontradict certain isogeny properties satisfied by Q -curves, see Propositions 5.1 and 5.2 of [CN].The curve X (65) has infinitely many quartic points: the quotient X +0 (65) is an elliptic curve.That is, we have a degree two map to an elliptic curve, and elliptic curves have infinitely manydegree two maps to P . This yields infinitely many degree four maps to P and in particularinfinitely many quartic points on X (65).In fact, by the Existence Theorem in Brill–Noether Theory [ACGH85, Chapter V, pg206], genus5 curves have infinitely many degree four maps to P . The fact X (65) admits a degree two mapto an elliptic curve is our saving grace in this instance. Indeed, each such map turns out to factorthrough this elliptic curve X +0 (65), allowing us to perform our relative Chabauty methods withrespect to the quotient. On the flip side, no such quotient exists for X (57), leaving us with anabundance of quartic points from infinitely many sources.Accordingly, every quartic point on X (65) either arises from a quadratic point on X +0 (65), oris a point listed in § Q -curves are exactlythose with CM, just as the case was for the cubic points. The points arising from the quotient maybe Q -curves (for example if they come from the full Atkin-Lehner quotient X ∗ (65)), but are notnecessarily. However, each of these points will give rise to a K -curve for some quadratic field K .Indeed, w determines the isogeny between such a quartic point and one of its Galois conjugates.5.2. Cubic Points.
Let N ∈ { , , , , , } . In this section we list all cubic points on X ( N ), or rather a representative from each Gal( Q / Q )-conjugacy class. As explained in § G ⊆ J ( N )( Q ) of index at most two are computed using J +0 ( N )( Q ). We provideexplicit generators for G and J +0 ( N )( Q ) below.For N = 57, we apply the Mordell-Weil sieve in conjunction with Theorems 2.6, 2.24 relative to X +0 ( N ) in order to provably show all points on X (3)0 ( N )( Q ) either arise as the sum of a rationalpoint and the pullback of a rational point from X +0 ( N ), or belong to a finite set. This finite setconsists of (Gal( Q / Q )-stable sums of) the cubic points and sums of rational and isolated degreetwo points. Rational and isolated degree two points are listed in [Box21, § N = 57 there is no need to use ‘relative Chabauty’ as there are only finitely manyquadratic points on X (57) and these can be determined provably using Chabauty. In fact Siksek’ssymmetric Chabauty Theorem [Sik09, Theorem 3.2] almost suffices to determine the cubic points.But, as mentioned in Example 2.23, there are two points on X (3)0 (57) which require Theorem 2.24.The only other curve where we are also required to apply Theorem 2.24 is X (73).Using the same models as in [Box21, § X ( N ), along with the corresponding j -invariant and indicate whether theclass of elliptic curves have CM. The only points which give rise to Q -curves are the CM points,see § Q -curves.We recall that the Magma [BCP97] code to verify all computations can be found at https://github.com/joshabox/cubicpoints/ . .2.1. X (53) . Model for X (53):9 x x − x x − x − x x + 78 x x − x x + 30 x x x − x x − x + 136 x x − x x + 49 x = 0 ,x x − x x − x + 5 x x − x + x x − x = 0 . Genus X (53): 4.Cusps: (1 : 0 : 0 : 0), (1 : 1 : 1 : 1). X +0 (53): elliptic curve y + xy + y = x − x of conductor 53.Group structure of J +0 (53)( Q ): Z · [ Q − O ], where Q := (0 : − G ⊂ J (53)( Q ): G = Z · D ⊕ Z / Z · D tor , where D tor = [(1 : 1 : 1 : 1) − (1 : 0 : 0 : 0)] and D = [ P + P − (1 : 1 : 1 : 1) − (1 : 0 : 0 : 0)] = ρ ∗ [ Q − O ]for P := (cid:18) ,
16 ( −√−
11 + 5) : 1 : 1 (cid:19) ∈ X (53)( Q ( √− ρ ( P ) = Q .Primes used in sieve: 31, 17.Points with α satisfying α − α + 5 α + 52 = 0.Coordinates j -invariant CM(12 α − α + 254 : − α + 22 α + 32 : 7 α − α + 252 : 178) / − α +1545152415129700179443672757063109789 α − NO( α − α + 7 : 3 α − / − α + 63003373 α − α satisfying α + 11 α − j -invariant CM(143 α + 63 α + 1740 : 53 α + 49 α + 1004 : 25 α + 33 α + 612 : 524) / α +299279229875227284830201161759246904 α +4934717748383394121661044694765075727) NO(107 α + 72 α + 1345 : 42 α + 16 α + 718 : 20 α − α + 420 : 328) 1 / α + 1110894791062576754 α +18317182399622734527) NO oints with α satisfying α − α + 53 α + 147 = 0.Coordinates j -invariant CM(3(47 α − α +3271) : 31 α − α +7331 : 3(13 α − α +1767) : 5286) / α − α +352389719111433433199614166785341899) NO(39 α − α + 2550 : 11 α − α + 927 : 4 α − α + 430 : 292) / α − α +361470319752701) NOPoints with α satisfying α − α − j -invariant CM( − α − α −
21 : 3 α + 7 α + 9 : − α − α + 1 : 4) − α − α −
74 NO( − α + 81 α + 120 : − α + 15 α + 220 : 18 α + 24 α + 174 : 267) − α − α − NOPoints with α satisfying α − α − j -invariant CM(11 α + 23 α + 20 : 3 α + 5 α + 8 : α + α + 4 : 6) 117843669992 α + 258763889291 α + 214668790692 NO(131 α + 98 α + 80 : 38 α + 80 α + 410 : α + 91 α + 396 : 563) α +65291506111938160265579873086771 α +54165390933943542489231382553348 NO5.2.2. X (57) . Model for X (57): x x − x + 2 x x + 2 x x − x − x x + 3 x x − x − x x − x = 0 ,x x − x x − x x + 4 x x − x x − x + 5 x x − x = 0 ,x x − x + x x − x x − x = 0 . Genus of X (57): 5.Cusps: (1 : 0 : 0 : 0 : 0) , (1 : 1 : 0 : 1 : 0) , (3 : 3 : 1 : 2 : 1) , (3 : 9 / − / / X +0 (57): hyperelliptic curve y = x − x + 3 x + 3 x − x + 1Group structure of J +0 (57)( Q ): Z · [(1 : 1 : 0) − (1 : − ⊕ Z / Z · [(0 : − − (1 : − G ⊂ J (57)( Q ): G = Z · D ⊕ Z / Z · D tor , ⊕ Z / Z · D tor , , where D tor , = [(1 : 1 : 0 : 1 : 0) − (1 : 0 : 0 : 0 : 0)], D tor , = [(3 : 3 : 1 : 2 : 1) − (1 : : 0 : 0 : 0)] and D = [(1 : 1 : 0 : 1 : 0) + (3 : 9 / − / / − P + P ] where P := (cid:18) −
13 ( √− −
1) + 1 : −
43 ( √− −
1) + 1 : 13 ( √− −
1) : −
23 ( √− −
1) + 1 : 1 (cid:19) . satisfying ρ ( P ) = (1 : − α satisfying α − α + α + 2 = 0.Coordinates j -invariant CM( α − α + 7 : 2( − α + α + 1) : α − α + 1 : − α + α + 3 : 2) − α + 16880722 α − α − α + 8 : 2 α − α + 11 : − α − α − α + 11 : 3) − α − α + 291907686304 NO( − α + 4 : α − α + 1 : α − α + 3 : 1) 114 α − α −
218 NO( α − α + 7 : 4( α + 1) : α − α − α − α + 3 : 4) − α + 33591312773593 α − α satisfying α − α − j -invariant CM(3 α + 9 : − α + 9 α + 12 : − α + 4) : 4) 1 / − α − α − α + 2 : − α + α + 4 : α − α : − α + α + 4 : 2) 1 / − α − α − α − α + 6 : 4( α − α + 3) : − α + 2 α − α − α + 10 : 4) 1 / − α + 14416014175 α − α + 3 : − α + 9 : 2 : − α + 7 : 2) 1 / − α − α − α − α −
243 : 4 α − α : − α + 5 α − α − α −
153 : 18) / − α +124697130934598013699081937354752 α +256826826563492878403632537894912) NO oints with α satisfying α − α −
27 = 0.Coordinates j -invariant CM(55 α − α + 983 : 24(2 α + 76) : 9( − α + 9 α + 35) :27( α + 3 α + 65) : 648) 1 / α − α − − α + 20 α + 90 : α − α : α − α −
15 : − α + 5 α + 24 : 3) 1 / − α − α − α + 45 α + 604 : − α − α + 661 : 9(3 α − α −
22) : − α − α + 577 : 279) / − α +3777138616478919143030784 α +20593798011121996168462336) NO5.2.3. X (61) . Model for X (61): x x + x x x − x x − x − x x + 5 x x − x x + 4 x x x − x + 14 x x − x x + 4 x = 0 ,x x − x − x x − x + 2 x x − x = 0Genus of X (61): 4.Cusps: (1 : 0 : 0 : 0), (1 : 0 : 1 : 1). X +0 (61): elliptic curve y + xy = x + 6 x + 11 x + 6 of conductor 61.Group structure of J +0 (61)( Q ): Z · [ Q − O ], where Q = ( − G ⊂ J (61)( Q ): G = Z · D ⊕ Z / Z · D tor , where D tor = [(1 : 0 : 1 : 1) − (1 : 0 : 0 : 0)] and D = [ P + P − (1 : 1 : 1 : 1) − (1 : 0 : 1 : 1)] = ρ ∗ [ Q − O ]for P := (cid:18) √− −
1) : 1 : 1 (cid:19) ∈ X (61)( Q ( √− ρ ( P ) = Q .Primes used in sieve: 31, 19, 53, 23.Points with α satisfying α − α −
20 = 0.Coordinates j -invariant CM(4 α + 16 α −
16 : − α + 12 α −
12 : 5 α + 2 α + 34 : 72) − / α +5909569160562647305266609005744561 α +13495086575687455058014736453622810) NO( α + 2 α + 8 : − α − / α − α − .2.4. X (65) . Model for X (65): x x − x + x x − x − x x + 3 x x − x + 2 x x − x = 0 ,x x − x x − x − x x + 4 x x − x + 2 x x − x = 0 ,x x − x x − x − x x + 5 x x + 3 x x − x = 0 . Genus of X (65): 5.Cusps: (1 : 0 : 0 : 0 : 0) , (1 : 1 : 1 : 1 : 1) , (1 / / / / , (1 / / / / X +0 (65): elliptic curve y + xy = x − x of conductor 65.Group structure of J +0 (65)( Q ): Z · [ Q − O ] ⊕ Z / Z · [(0 : 0 : 1) − O ], where Q = (1 : 0 : 1).Group structure of J (65)( Q ): Z · D ⊕ Z / Z · ( − D tor , + 2 D tor , ) ⊕ Z / Z · (17 D tor , + 13 D tor , ), where D tor , = [(1 : 1 : 1 : 1 : 1) − (1 : 0 : 0 : 0 : 0)], D tor , = [(1 / / / / − (1 : 0 : 0 : 0 : 0)] and D = [ P + P − (1 : 0 : 0 : 0 :) − (1 : 1 : 1 : 1 : 1)] = ρ ∗ ([ Q − O ]) for P = (cid:18) i ) : 1 : 1 (cid:19) ∈ X (65)( Q ( i ))satisfying ρ ( P ) = Q .Primes used in sieve: 17, 23.Points with α satisfying α − α + α − j -invariant CM( − α + 15 α + 23 : 9 α − α + 17 : 10 α − α + 38 : 7 α + 8 α + 18 : 43) − α − α − NO(28 α + 10 α + 42 : − α − α −
12 : 3 α + α + 6 : − α : 2) 16448 α + 49984 α − α − α + 3) : 16( α − α + 1) : 3 α + 7 α + 5 : 2( − α + 3 α + 9) : 32) 1629248 α − α + 946624 NO(4(3 α − α + 14) : 2(2 α − α + 40) : − α − α + 61 :2( − α − α + 35) : 92) 136553952252949568 α + 102819038189152064 α − .2.5. X (67) . Model for X (67): x x − x + 2 x x + 2 x x − x − x x + 3 x x − x − x x − x = 0 ,x x − x x − x x + 4 x x − x x − x + 5 x x − x = 0 ,x x − x + x x − x x − x = 0 . Genus of X (67): 5.Cusps: (1 : 0 : 0 : 0 : 0) , (1 / / / X +0 (67): genus 2 hyperelliptic curve y = x − x + x + 2 x + 2 x + 4 x + 1.Group Structure of J +0 (67)( Q ): Z · [ Q − (1 : − ⊕ Z [ Q − (1 : − Q = (1 : 1 : 0) and Q = (0 : 1 : 1).Group Structure of G ⊂ J (67)( Q ): G = Z · D ⊕ Z D ⊕ Z / Z · D tor , where D tor := [(1 / / / − (1 : 0 : 0 : 0 : 0)], D := [ P + P − (1 : 0 : 0 : 0 :0) − (1 / / / D := [ P + P − (1 : 0 : 0 : 0 : 0) − (1 / / / P , P as below satisfying ρ ( P ) = Q and ρ ( P ) = Q .Primes used in sieve: 73 , , , , , , , P = (cid:18)
118 ( −√− √− −√− −√− (cid:19) P = (cid:18) √−
11 + 11) : 122 ( −√−
11 + 11) : 122 ( √−
11 + 11) : 1 (cid:19)
Points with α satisfying α − α + α − j -invariant CM(7 α − α + 13 : − α − α + 11 : α + 2 α + 7 : − α − α + 11 : 18) − α − α − α − α + 79 : − α − α + 184 : 19 α − α + 79 : − α − α + 92 : 203) − α − α − α satisfying α − α − α + 5 = 0.Coordinates j -invariant CM(11(3 α − α + 29) : 7 α − α + 371 : 13( α − α + 9) :2( − α − α + 90) : 286) 1821644613377781192000 α − α +4744756895498802918000 − .2.6. X (73) . Model for X (73): x x − x + 2 x x − x x − x + 3 x x + 3 x − x = 0 ,x x − / x x − x x + 1 / x − / x x + x x − x + 9 / x x − / x = 0 ,x x − x x + x x − x x − x + 4 x x = 0 . Genus of X (73): 5.Cusps: (1 : 0 : 0 : 0 : 0) , (1 : 1 : 1 : 0 : 0). X +0 (73): genus 2 hyperelliptic curve y = x + 2 x + x + 6 x + 2 x − x + 1.Group Structure of J +0 (73)( Q ): Z · [ Q − (1 : 1 : 0)] ⊕ Z · [ Q − (1 : 1 : 0)], where Q := (0 : − Q := (0 : 1 : 1).Group Structure of G ⊂ J (73)( Q ): G = Z · D ⊕ Z · D ⊕ Z / Z · D tor , where D tor = [(1 : 0 : 0 : 0 : 0) − (1 : 1 : 1 : 0 : 0)], D = [ P + P − (1 : 0 : 0 : 0 : 0) − (1 :1 : 1 : 0 : 0)], D = [ P + P − (1 : 0 : 0 : 0 : 0) − (1 : 1 : 1 : 0 : 0)] for P , P as below and satisfying ρ ( P ) = Q and ρ ( P ) = Q .Primes used in sieve: 19 , , , , , P = (cid:18) / √− −
10) : 114 ( √− −
17) : 114 ( √− −
17) : 114 ( √−
19 + 11) : 1 (cid:19) P = (cid:18)
16 ( −√− −
8) : − −√− −
8) : 16 ( −√− (cid:19)
Points with α satisfying α − α + 7 α + 8 = 0.Coordinates j -invariant CM( − α + 3 α − − α − α − − α + 945810849040 α − α − α + 39 : α − α + 4 : 4 α − α + 29 : − α + 2 α − α − α − NO .3. Quartic Points.
Recall that X (57) and X (65) both have genus 5 and the Mordell–Weil groups of their jacobians over Q have rank one. This puts themwithin the bounds to apply Theorem 2.6. However, there is a problem. These curves have infinitely many degree four maps to P ! For X (65), we are in luck: allof the maps defined over Q factor through the quotient X +0 (65). On the other hand, X (57) is not bielliptic, making us unable to determine its isolated quarticpoints.Using the same model as in § X (65) which do not come fromthe quotient X +0 (65). As before, the points which give rise to Q -curves are exactly the CM points.Points with α satisfying α + 3 α + 1 = 0.Coordinates j -invariant CM( − α + α − α + 1 : α + α + 1 : − α : α + 1 : 1) 1728 − α + 2 α + 4 α + 12 : 4 α − α + 9 α + 11 : − α + 3 α − α + 19 :3 α + 2 α + 8 α + 19 : 18) 19691491018752 α + 51552986141376 − α satisfying α − α + 2 α + α + 1 = 0.Coordinates j -invariant CM (8 α + 18 α − α + 220 : 93 α − α + 128 α + 509 : − α + 102 α − α + 363 : − α + 128 α − α + 413 : 482) − α − − α − α + 6 α : α + α + 6 : − α + 2 α − α + 2 : − α + α − α + 1 : 4) 0 − α satisfying α − α − α + 4 α + 10 = 0.Coordinates j -invariant CM (3(9 α + 3 α − α + 65) : 5(4 α + 11 α − α + 74) : 23 α − α − α + 730 : − α + 46 α − α + 610 : 870) / − α + 42997532 α − α − (7 α − α − α + 65 : − α + 25 α + α −
29 : α − α + 14 α + 19 : − α + 18 α − α + 7 : 51) / α + 1124273732 α + 11474554 α − − α − α + 8 α + 12) : 3(3 α − α −
12) : 3( − α + 4 α + 6) : − α − α + 10 α + 20 : 6) 1 / − α + 25312371833066 α + 8043694506022 α − − α + 5 α + α + 5 : 3 α − α − α − α + 8 : − α + 4 α − α + 4 : 9) 1 / α − α − α + 782896654) NO oints with α satisfying α − α + 36 = 0.Coordinates j -invariant CM (8 α − α + 32 α + 160 : α + 3 α + 4 α + 84 : 6( α − α + 54) :6( α − α + 4 α + 60) : 144) 1 / α + 905537632545 α + 1655605513116 α +766437509340) NO( − α + 12 α − α −
300 : 12(2 α + 11 α + 17 α + 60) :5 α − α + 143 α + 954 : 4(16 α + 21 α − α + 78) : 1608) / α − α − α +9869446094702964951347100) NOPoints with α satisfying α − α + 64 = 0.Coordinates j -invariant CM( − α + 56 α − α −
184 : 2(3 α − α + 61 α + 24) :4( α − α + 72) : 8( − α + 9 α −
8) : 256) 1 / − α − α + 274492275 α + 6008246680) NO16( − α − α + 72) : 16( − α − α + 72) : − α + 8 α + 207 α + 1400 :2( − α − α + 11 α + 856) : 2048) / α +451625115647050834040533476755 α +191531841188243058702903432000 α − NOPoints with α satisfying α + 10 α − α + 6 = 0.Coordinates j -invariant CM(4(3 α + 11 α − α + 134) : 4(6 α + 22 α + 77 α + 77) : − α − α − α + 971 : 2(25 α + 28 α + 98 α + 289) : 764) / − α − α − α +985686437922287224950855418754958911083274126) NO( − α − α − α + 2589 : 3( − α − α − α + 1479) :9(4 α + 8 α + 47 α + 294) : 4617 / α + α + 22 α + 114) : 4617) / − α − α − α +197644133686780526062866403118) NO ( − α + 3 α + α + 102 : 2( − α − α − α + 78) : 2( − α − α − α + 36) :2( − α − α + 42) : 144) 1 / α − α +3322177705362944338498999061316877 α − NO(2( − α − α − α + 46) : 6(2 α + 3 α + 20 α + 4) :9( α + 10 α −
13) : 54 : 0) / − α − α +1305535708381 α − NO oints with α satisfying α − α + 33 α + 124 α + 324 = 0.Coordinates j -invariant CM ( α +79 α − α +1420:3( − α +29 α − α +1388):9( α +7 α +24 α +844):27( − α +5 α − α +188):7776) 1 / − α − α − α − NO (21384 / α − α +233 α +972):29 α − α +1837 α +11934:6( − α +11 α − α +1098):9( − α +22 α − α +1614):21384) 1 / α − α +555106151674649210688259872 α +153918822709072746459229012) NO (1053 α − α +36528 α +219596:10(127 α − α +2948 α +41036):4(33 α − α +868 α +81236):8(114 α − α − α +49558):518080) / − α +12586954158315417996000859906005034036565 α − α +344518838187266283365991746158951784244372) NO (8( − α +85 α − α − α − α +1609 α +2486):45 α − α +920 α +15980:2( − α − α +410 α − / − α +5228953638910671575533618290081115693190725 α +83338824414498902469112145034057200015322912 α +325680039927910810332592971628693692224201812) NOPoints with α satisfying α − α + 2 α + 2 α − j -invariant CM ( − α +1189 α − α +1526:150 α − α +744 α − − α +177 α − α +302:16 α − α +86 α +54:166) 1 / α − α +470785458314526794481074 α − NO(11(3 α − α + 6 α + 18) : − α + 52 α − α −
58 :27 α − α + 30 α + 162 : − α − α + 18 α + 62 : 88) / α − α − α +29993540624153550) NO (241 α +876 α − α +10510:2(317 α − α +2 α +6390):4( − α +604 α − α +2654):8( − α +116 α − α +1494):16288) 1 / α − α +46495731438962280331948301537 α − NO( α + 3 α : α − α − α + 4 : 2 α + 4 : 2 α : 4) / − α +95010271213628 α − α − NOPoints with α satisfying α − α − α + 3 α + 8 = 0.Coordinates j -invariant CM (27 α − α − α + 193 : 10( − α + 2 α + 6 α −
3) : 4( − α + 12 α + 4 α −
7) :8( α − α + 2 α + 9) : 80) / − α − α +119780562840206 α +177065002925201) NO(2(33 α + 21 α − α + 101) : 2( − α − α + 14 α + 301) :27 α + 73 α − α + 278 : 2(20 α − α − α + 331) : 614) / α − α − α +2112213001641271144506379601) NO( − α + 348 α + 3048 α + 2713 : 164 α − α − α −
946 : − α + 57 α + 569 α + 696 : 29 α − α − α + 90 : 289) / − α +25536131264162 α +56923104477966 α − NO( − α + 518 α + 178 α + 383 : − α + 590 α + 294 α −
241 : − α + 70 α + 562 α + 741 : 5 α − α + 166 α + 1091 : 1244) / α − α − α − NO oints with α satisfying α − α − α − α + 4 = 0.Coordinates j -invariant CM( α + 5 α − α + 10 : α − α + 9 α + 10 : − α + 3 α + 7 α + 6 : α − α + α + 10 : 16) / α +4085001869259831852583379 α +5405460540196338811614783 α − NO (2(63 α +197 α +285 α +548):67 α +103 α +367 α +2010:4(13 α +30 α − α +390):4(15 α − α − α +450):2684) 1 / α +101927451713689293321395 α +134875046777126873505951 α − NO( − α + 3 α + 47 α + 182 : 2(3 α − α − α −
18) :8( − α + α + 5 α + 16) : 32 : 64) 1 / α − α − α − (8( − α +75 α +407 α +1630):32(12 α − α − α +51): − α +289 α +2957 α +13446:2(13 α − α − α +3310):8864) 1 / α − α − α − NOPoints with α satisfying α − α + α + 15 α + 8 = 0.Coordinates j -invariant CM(4( α − α − α +6) : 8( α + α +1) : α − α +5 α +40 : 2( α − α +8) : 24) / − α +833547216099653533626822633 α − α − NO ( − α +891 α − α +496:202 α − α +1197 α +3602: − α +686 α − α +695: − α − α +84 α +2248:3554) 1 / α − α +8609137673795812100808665275 α +6567184709951911955583590440) NO(2( − α + 6 α − α + 4) : 3( α − α + 5 α + 18) :3( − α + 3 α − α + 5) : 36 : 54) / α − α +1745183842219 α +1331261559976) NO (2(7 α − α + 13 α + 112) : 4( − α + 3 α − α −
20) : 7 α − α + 19 α + 136 : − α − α + 16 : 48) / α − α − α − NOPoints with α satisfying α + 22 α − α − j -invariant CM ( − α +56049 α − α +4990147:10(31 α − α +9017 α +473485):4( − α +9049 α − α +1840387):8( − α − α − α +788591):8416960) / α − α +27551326444124285184035446453362223 α − NO (2(1540 α +24771 α +279122 α +2323657):5413 α +747 α +217423 α +5494649:6063 α +18979 α +101933 α +3263633:3317 α +23025 α +73939 α +3159503:4790452) / α +545528649047435090745590753332103525334768 α +5219097010417755802150885707010029194059739 α +26435647875049957400594991340415017745868228) NO (2 / − α +2133 α − α +131049):99 α − α +7209 α − − α +765 α − α +49527:3( α − α +463 α +7483):38628) / α − α +13628990993078146017200509961 α − NO (2255 α +12903 α +122561 α +667341:727 α +3783 α +38017 α +283941:205 α +1173 α +13483 α +128559:31 α +303 α +5305 α +93933:103008) / α +156777348815983490103 α +1499898372357530447969 α +7597242259694917251093) NO oints with α satisfying α − α − α − α −
376 = 0.Coordinates j -invariant CM (2( − α +819 α +338 α +3140):2( α − α +4030 α +7708):41 α − α − α +11516: − α +273 α +2756 α +11132:19032) 1 / α +87629530896959309713118771172 α +627432607595384179205760536627 α +363246817106763572205210032174) NO (1 / − α +2535 α +45058 α +1115008):1 / − α +130 α +3445 α +104558):1 / − α +1755 α +24388 α +953788):1 / − α +13 α +884 α +116148):1) 1 / − α +753052741393911437371620 α +11532547733418129305425109 α +275363866633306279112305730) NO (1 / α +7553 α +26728 α +269940):1 / α +110 α +211 α +7982):1 / α − α +780 α +246548):1 / α +273 α − α +89880):1) 1 / α +38465549476528877862288479220 α +275415602064898773665089788899 α +159449540270119811148234532430) NO (4( − α +29081 α +576784 α +14718564):4( − α +10907 α +310180 α +7975676): − α +48581 α +609596 α +17635140:2( − α +4823 α +91364 α +6258412):10977824) 1 / − α +12283824828743039994402351421616561892 α +188119355258177801813217995229096721397 α +4491745817999145473916690222308277724034) NO OINTS ON MODULAR CURVES USING GENERALISED SYMMETRIC CHABAUTY 33
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