Castaño-Bernard, C.

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Castaño-Bernard, Carlos

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A note on the rational points of X₀⁺(N)

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arXiv:math/0508115v3  [math.NT]  22 Jul 2006A NOTE ON THE RATIONAL POINTS OF X+0 (N)CARLOS CASTAN˜O-BERNARDAbstract. Let C be the image of a canonical embedding φ of theAtkin-Lehner quotient X+0 (N) associated to the Fricke involutionwN . In this note we exhibit some relations among the rationalpoints of C. For each g = 3 (resp. the first g = 4) curve C wefound that there are one or more lines (resp. planes) in Pg−1 whoseintersection with C consists entirely of rational Heegner points orthe cusp point, where N is prime. We also discuss an explanationof the first non-hyperelliptic exceptional rational point.Contents1. Introduction 12. Preliminaries 23. Genus three: collinearity relations 44. Genus four: coplanarity relations 95. Concluding remarks 10References 101. IntroductionFix an integer N > 1 and let X0(N) be the moduli space of (or-dered) pairs (E,E ′) of generalised elliptic curves E and E ′ linked bya cyclic isogeny ϕ : E −→ E ′ of degree N . Consider the Atkin-Lehnerquotient curve X+0 (N) defined by the involution wN of X0(N) inducedby mapping an isogeny ϕ : E −→ E ′ to its dual ϕˆ : E ′ −→ E. The quo-tient curve X+0 (N) has been studied by Galbraith [3], Mazur [8], andMomose [9], among others. Galbraith [3] studied the rational pointsof a canonical image C ⊂ Pg+N−1 of X+0 (N), where g+N is the genusof X+0 (N). In particular he exhibits explicit formulæ for C for primeconductors N for which the curve has genus g ≤ 5. In each case helocates the cusp and rational CM points and, moreover, for N = 137 heexhibits a rational point which is neither a cusp point nor a CM point.In this note we exhibit an explicit set of hyperplanes {H1, . . .Hs} inPg+N−1 such that the intersection of each Hi with C (over the complexnumbers) consist entirely of rational points of C, for each prime levelN such that g+N = 3, i.e. N = 97, 109, 113, 127, 139, 149, 151, 179,and 239, and the first prime level N such that g+N = 4, i.e. N = 137.12 CARLOS CASTAN˜O-BERNARDFor the latter case we found a further plane defined by 3 different CMpoints that contains the exceptional point.The material is organised as follows. Section 2 introduces some basicresults that we used to compute an equation for C. The collinearityrelations are discussed in Section 3, while the coplanarity relations arediscussed in Section 4.Acknowledgments I would like to thank the referee of Experi-mental Mathematics for his/her insightful comments, which led to somesignificant improvements on a previous version of this paper. I wouldalso like to thank Barry Mazur for his interesting remarks. I would liketo heartily thank Jenny and Kenneth Cowan for their hospitality whilepreparing this paper.2. PreliminariesLet X be an algebraic curve defined over a field k and let Ω1X be thek-vector space of its holomorphic differentials. Also let {ω1, . . . ωg} bea basis of Ω1X . The integer g is called the genus of X. The canonicalmap φ of X in projective space Pg−1 is the morphismXφ// Pg−1P// (ω1(P ) : · · · : ωg(P ))It is well-known that the canonical map φ is an embedding, if the genusg exceeds 2 and X is not hyperelliptic. Now fix an integer N > 1 andrecall thatX0(N) = Γ0(N)\H∗,whereΓ0(N) ={µ =(α βγ δ)∈ SL2(Z) : γ ≡ 0 (mod N)}.The Atkin-Lehner quotient curve X+0 (N) may be defined as the curveX0(N) modulo de action of the Fricke involution wN , which is definedbyτ 7→ −1Nτ.Every holomorphic differential ω on X0(N) may be expressed uniquelyas ω = fdτ , where f(τ) =∑∞n=1 a(n)qn (where q = e2piτ ) is a modularform for Γ0(N) and of weight 2 , and conversely every modular form forΓ0(N) and of weight 2 gives rise to a holomorphic differential ω = fdτ .A NOTE ON THE RATIONAL POINTS OF X+0 (N) 3In particular the canonical maps for X+0 (N) are of the formX+0 (N)φ// Pg−1P// (f1(τ) : · · · : fg+N(τ)),where {f1, . . . fg+N} constitute a basis of the +1-eigenspace S+2 (Γ0(N))of the C-vector space of modular forms S2(Γ0(N)) with respect to theaction of the Fricke involution wN . It is a classical fact that the vec-tor space S2(Γ0(N)) has a basis consisting of modular forms with ra-tional integer coefficients only. The same is true for the eigenspaceS+2 (Γ0(N)). A set of equations with integral coefficients S for thecurve C may be computed by finding combinations of powers of theq-expansions which yield identically zero series. Using the methoddescribed in Galbraith [2, p. 19], with the help of William Stein’s com-puter package Hecke [12] and Pari [11] it is easy to compute theseequations S for small genera, such as g+N = 3 and g+N = 4. (See alsoElkies [1] for related methods and results on the computation of equa-tions for modular curves.) In these two cases Galbraith [2] has shownthat the curves X+0 (N) are non-hyperelliptic. So in fact C is a completeintersection. (See Examples IV.5.2.1 and IV.3.3.2 of Hartshorne [7].)Once we have a set of such equations S for C in practice it only takesonly a brute-force search to obtain as many rational points on C aspredicted by the theory of Complex Multiplication (see Gross [4]), thecusp rational point, or some other rational points that come as a fixedpoint of certain hyperelliptic involution. Sometimes there are more ra-tional points than these “obvious” ones; Galbraith [2] has shown thatthere exist rational points in C that are neither cusps nor CM rationalpoints for non-hyperelliptic X+0 (N) of genus at least 4.From now on let us assume that N is prime. Then using Proposition3.1 of Gross [5, p. 347] and the Riemann-Hurwitz formula we may seethatg+N =12(gN + 1−H(N)) ,whereH(N) ={12h(−4N), if N ≡ 1 (mod 4)12(h(−N) + h(−4N)), otherwise,with h(D) the class number of the imaginary quadratic order of dis-criminant D, and gN is genus of X0(N), which is given by gN = ⌊N+112⌋,unless N = 12q + 1 when g = q − 1. In particular, by using explicitupper bounds on the class number h(D) it may be found that the primeconductors N such that X+0 (N) has genus 3 are indeed N = 97, 109,113, 127, 139, 149, 151, 179, and 239. and all these curves X+0 (N) arenon-hyperelliptic. Similarly it may be found that there are exactly 54 CARLOS CASTAN˜O-BERNARDprime numbers N such that X+0 (N) has genus g = 4, namely N = 137,173, 199, 251, and 311.3. Genus three: collinearity relationsLet us assume N is one of the 9 prime conductors N such thatX+0 (N) has genus g+N = 3. So the canonical map φ associated toa basis of modular forms with integral Fourier coefficients {g1, g2, g3}is an embedding φ defined over Q of C into the projective plane P2such that its image C is of degree 4 = 2(g+N − 1). So we may obtain aprojective equation F (X, Y, Z) = 0 for the plane curve C by computinga (non-trivial) linear relation among the elements in the set{ga1gb2gc3 ∈ S8(Γ0(N)) : a, b, c ∈ Z≥0, a+ b+ c = 4}.Note that a line L in the projective plane P2 will intersect the curveC in 4 points, if we take into account intersection multiplicities. Soheuristically, the line defined by two rational points P1 and P2 of C(which is the tangent line of C at say P1, in case P1 = P2) is not ex-pected to intersect C in further rational points; the remaining pointsof intersection are expected to be in general two quadratic irrationals(one conjugate to the other). However, as we shall see it is possibleto exhibit for each of the levels under consideration a non-empty setof lines {L1, . . . Ls} such that the intersection of each Li with C (overthe complex numbers) consists entirely of rational points. We describethese lines with the help of some diagrams below. Each diagram con-tains an equation for C and a table of rational points. Since all theserational points are CM points, or the cusp point, we label these rationalpoints according to the discriminant D of the CM point. By the workof Galbraith [2] we know that in this case the relevant values of D areD = −3,−4,−7,−8,−11,−12,−16,−19,−27,−28,−43,−67,−163.To ease notation we regard the cusp i∞ labeled by D = 0. The dis-criminant D of each rational point of C exhibited was recognized inmost cases by computing a suitable approximation of the image in Cof the Heegner point of X+0 (N) of discriminant D so that it would beclear which of the rational points of C found previously corresponds toD.Remark 3.1. Sometimes simply too many Fourier coefficients were nec-essary to obtain a suitable approximation for the image in C of a ratio-nal Heegner point, e.g. for N = 97. In these cases there were exactlytwo rational points of C, say P and Q, which needed further work to belabeled. We were able to resolve these kind of ambiguities by using avariant of a result of Ogg [10] which may be applied to describe the reallocus X+0 (N)(R) in terms of certain indefinite binary quadratic forms.A NOTE ON THE RATIONAL POINTS OF X+0 (N) 5More precisely, first we were able to locate (exactly) in a parametri-sation pi of X+0 (N)(R) all the Heegner (i.e. CM) rational points, andthen by “walking along” a good polygonal approximation of the reallocus of C in the direction determined by pi, we paired the unmatcheddiscriminants, say D1 and D2, with the points P and Q.The diagrams are self explanatory; the intersection multiplicity ∂of Li with C at a specific point P ∈ Li ∩ C is indicated only whennecessary.D P0 (−4 : −4 : 3)−3 (−12 : −5 : 9)−4 (−9 : −2 : 5)−8 (−5 : 2 : 2)−11 (1 : 1 : 1)−12 (−6 : 1 : 1)−16 (1 : −6 : 1)−27 (0 : 1 : 0)−43 (−3 : −3 : 4)−163 (−13 : −27 : 8)−48Z4 − 91Z3Y − 15Z2Y 2 + 4ZY 3 + 3Z3X − 26Z2Y X + 25ZY 2X+3Y 3X + 54Z2X2 + 29ZYX2 + 18Y 2X2 + 29ZX3 + 11Y X3 + 4X4 = 0−27−8−12−4−163N = 97−11−3−43−160−4−7−16∂ = 2−430−28−12N = 109−27−3D P0 (−2 : −2 : 3)−3 (11 : −3 : 1)−4 (0 : 1 : 0)−7 (−3 : 4 : 1)−12 (−3 : −3 : 1)−16 (−6, 1, 2)−27 (1 : −6 : 2)−28 (5 : −2 : 3)−43 (−5 : 2 : 4)24Z4 + 39Z3X + 25Z2X2 + 2ZX3 − 2X4 + 77Z3Y + 122Z2Y X+41ZYX2 − 3Y X3 + 51Z2Y 2 + 79ZY 2X + 23Y 2X2 + 13ZY 3 + 9Y 3X = 06 CARLOS CASTAN˜O-BERNARDD P0 (1 : 0 : 0)−4 (1 : 2 : −1)−7 (−1 : 1 : 0)−8 (1 : 1 : −1)−11 (0 : 1 : 0)−16 (−1 : 0 : 1)−28 (1 : 1 : −2)−163 (3 : 3 : −1)Y 2 + Y 3 +X + 3Y X + 5Y 2X + 2Y 3X + 4X2+8Y X2 + 7Y 2X2 + 6X3 + 7Y X3 + 3X4 = 00−163 −11−16−28−8−4N = 113−7D P0 (−1 : −1 : 1)−3 (−1 : 5 : 0)−7 (−1 : 2 : 0)−12 (1 : 1 : 0)−27 (−2 : 1 : 0)−28 (−3 : 0 : 2)−43 (0 : 0 : 1)−67 (−1 : −4 : 1)10X4 + 17X3Y + 45X3Z − 12X2Y 2 + 9X2Y Z + 81Z2X2 − 13XY 3−99XY 2Z − 81XYZ2 + 54XZ3 − 2Y 4 − 36ZY 3 − 162Y 2Z2 − 135Y Z3 = 0−67−3−12N = 127−43−28−270−7A NOTE ON THE RATIONAL POINTS OF X+0 (N) 7D P−3 (−5 : 2 : 1)−8 (1 : 1 : −1)−12 (−1 : 0 : 1)−19 (−2 : 0 : 1)−27 (2 : 1 : −1)−43 (0 : −1 : 1)X2Y 2 +XY 3 − Y 4 +X3Z + 3X2Y Z + 3XY 2Z − 2Y 3Z+5X2Z2 + 8XY Z2 − 2Y 2Z2 + 8XZ3 + 3Y Z3 + 4Z4 = 0−43−8−27−12N = 1390−3−19−40−7N = 149−16−19−28−67D P0 (1 : 0 : 0)−4 (0 : −1 : 1)−7 (−1 : 0 : 1)−16 (2 : 1 : −1)−19 (1 : 1 : −1)−28 (1 : 2 : −1)−67 (−3 : 1 : 1)X2Y 2 +XY 3 + Y 4 +X3Z + 2X2Y Z + 6XY 2Z + 4Y 3Z+4X2Z2 + 7XYZ2 + 7Y 2Z2 + 4XZ3 + 5Y Z3 + Z4 = 08 CARLOS CASTAN˜O-BERNARD−3N = 151−7 ∂ = 1∂ = 3−12 −163−67−27−280D P0 (1 : 0 : 0)−3 (0 : −4 : 1)−7 (−1 : 1 : 0)−12 (−2 : 0 : 1)−27 (0 : −1 : 1)−28 (1 : 1 : −2)−67 (1 : −2 : 1)−163 (1 : 0 : 1)XY 3 + Y 4 −X3Z −X2Y Z + 5XY 2Z + 8Y 3Z−3X2Z2 + 6XY Z2 + 20Y 2Z2 + 17Y Z3 + 4Z4 = 0−11−28−7−8∂ = 1∂ = 3−163N = 1790D P0 (1 : 0 : 0)−7 (−1 : 0 : 1)−8 (0 : 1 : 0)−11 (1 : 1 : −1)−28 (1 : 2 : −1)−163 (3 : 3 : −1)XY 3 +X3Z + 2X2Y Z + 2XY 2Z + 2Y 3Z + 4X2Z2+7XYZ2 + 5Y 2Z2 + 6XZ3 + 7Y Z3 + 3Z4 = 0A NOTE ON THE RATIONAL POINTS OF X+0 (N) 9−19−28−7−430N = 239D P0 (1 : 0 : 0)−7 (−1 : 0 : 1)−19 (−2 : 0 : 1)−28 (−1 : 2 : 1)−43 (−2 : 1 : 1)X2Y 2 −XY 3 − Y 4 +X3Z − 2X2Y Z + 4XY 2Z + 2Y 3Z+5X2Z2 − 5XY Z2 + 8XZ3 − Y Z3 + 4Z4 = 04. Genus four: coplanarity relationsNow we will study the rational points of the first genus 4 Atkin-Lehner quotient curve X+0 (N). Again by the work of Galbraith [2] weknow that C may be given as the intersection in P3 ofXY +WY + 2Y 2 + 2WZ +XZ + 6Y Z + 3Z2 = 0,andX3 +WX2 + 6X2Z − 2XY 2 − 5XY Z +XZW + 13XZ2 + 2Y 3+3WY 2 +W 2Y + 3WY Z − 6Y Z2 + ZW 2 − 4Z2W + 14Z3 = 0.(See also Example IV.5.2.2 of Hartshorne [7].) By a brute-force searchwe may find in C the rational pointsD P0 [1 : 0 : 0 : 0]−4 [2 : −4 : −3 : 2]−7 [2 : −1 : −2 : 1]−8 [−1 : 1 : 0 : 0]−11 [1 : 1 : −1 : 0]−16 [2 : 0 : −1 : 0]−19 [1 : −2 : −1 : 1]−28 [0 : 1 : 2 : −1]We may also find in C the non-CM rational point [19 : 2 : −16 : 4].The fact that the degree of C is 6 implies that the planes Π in P3will intersect C (over a fixed algebraic closure of Q) at 6 points, ifwe take into account multiplicities. So if Π is a plane defined by 3rational points of C, then the other intersection points of Π with C arein general expected to be defined over a cubic extension of Q. Hovever,10 CARLOS CASTAN˜O-BERNARDit turns out that there are 3 different planes Π1, Π2, and Π3 in P3 suchthat each of these planes intersects C at exactly 6 rational points (withmultiplicities):• Π1: z = 0:2(0) + 2(−8) + (−11) + (−16)• Π2: x+ 2y + 3z + 1 = 0(−7) + (−8) + 2(−11) + (−16) + (−19)• Π3: x+ y + 3z = 0(0) + 2(−7) + (−11) + (−19) + (−28)There are some other remarkable properties about the set of rationalpoints of C. The set {−7,−11,−19} is contained in one line, say, L1,and also the set {−8,−11,−16} is contained in one line, say, L2. Thesetwo lines meet at D = −11 and, moreover, both are contained in planeΠ2. Finally, the 3 planes Π1, Π2, Π3 meet at D = −11..Another property worth mentioning is that the plane Πe defined bythe points D = 0,−4,−112x+ 2y + 7z = 0also contains the exceptional point, which is quite unexpected sincethis may not be heuristically explained by the small size of the coeffi-cients of the Jacobi form involved. (See Gross, Kohnen and Zagier [6].)However, the remaining two points of intersection are not rational;these are two conjugate points defined over the real quadratic field ofdiscriminant ∆ = 8.5. Concluding remarksIt seems to be an open question whether these relations may have aninteresting explanation, or else if these relations are only a consequenceof an “accident” due to the small size of the genus g+N of X+0 (N),perhaps related to the fact that the field Kf generated by the Fouriercoefficients of the newform f of level N and “−1” sign in the functionalequation of its Λ-function, is an abelian extension of Q for each primeNwith g+N = 3. It seems worthwhile to extend the above list of examplesto higher levels, hoping that a more extensive experimental evidencemay help to grasp the nature of this phenomenon. This may shed somemore light into the nature of X+0 (N)(Q) for prime levels N , which is“extremely interesting”, as expressed in Mazur [8].References1. Noam D. Elkies, Elliptic and modular curves over finite fields and related com-putational issues, Computational perspectives on number theory (Chicago, IL,1995), AMS/IP Stud. Adv. Math., vol. 7, Amer. Math. Soc., Providence, RI,1998, pp. 21–76.A NOTE ON THE RATIONAL POINTS OF X+0 (N) 112. S.D. Galbraith, Equations for modular curves, Oxford Ph.D. thesis (1996),v+91.3. , Rational points on X+0 (p), Experiment. Math. 8 (1999), no. 4, 311–318.4. B.H. Gross, Heegner points on X0(N), Modular forms (Durham, 1983), Hor-wood, Chichester, 1984, pp. 87–105.5. , Heegner points and the modular curve of prime level, J. Math. Soc.Japan 39 (1987), no. 2, 345–362.6. B.H. Gross, W. Kohnen, and D.B. Zagier, Heegner points and derivatives ofL-series. II, Math. Ann. 278 (1987), no. 1–4, 497–562.7. R. Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977, GraduateTexts in Mathematics, No. 52.8. B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes E´tudes Sci.Publ. Math. (1977), no. 47, 33–186 (1978).9. F. Momose, Rational points on the modular curves X+0 (N), J. Math. Soc. Japan39 (1987), no. 2, 269–286.10. A. P. Ogg, Real points on Shimura curves, Arithmetic and geometry, Vol. I,Progr. Math., vol. 35, Birkha¨user Boston, Boston, MA, 1983, pp. 277–307.11. The PARI Group, Bordeaux, PARI/GP, Version 2.1.5,http://www.parigp-home.de/.12. W.A. Stein, HECKE: Modular Forms Calculator, Version 0.4,http://modular.fas.harvard.edu/Tables/hecke-cpp.html.E-mail address : ccastanobernard@gmail.com

A note on the rational points of X₀⁺(N)

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