Géométrie algébrique / algebraic geometry

GENUS-TWO CURVES WITH 22 TORSION POINTS

BJORN POONEN

Abstract. We prove the existence of infinitely many genus 2 curves over C whose hyperelliptic torsion packet contains at least 22 points.

Courbes de genre deux ayant 22 points de torsion

Resumé. On démontre qu’il y a une infinité de courbes de genre 2 sur C pour lesquelles le paquet de torsion hyperelliptique a au moins 22 points.

Version française abrégée

Soient k un corps de caractéristique zéro et \overline{k} une clôture algébrique de k. Soit X une courbe (lisse, projective, géométriquement intègre) de genre g ≥ 2 sur k. On choisit un plongement d’Albanese de X dans sa jacobienne J, associé à un point P ∈ X(k). On désigne par X ∩ J_{tors} l’ensemble des points de X(\overline{k}) dont l’image dans J(\overline{k}) est d’ordre fini. La conjecture de Manin-Mumford, prouvée par Raynaud [8] en 1983, affirme que X ∩ J_{tors} est fini.

Mazur [6, p. 234] a posé la question suivante: peut on borner le cardinal de l’intersection de X avec un sous-groupe de J(\overline{k}) de rang fini r en fonction seulement de g et r? La réponse à cette question n’est pas connue même pour r = 0. Dans ce cas, le problème revient à borner #(X ∩ J_{tors}) en fonction de g.

Supposons que g = 2 et que P soit un point de Weierstrass. Dans ce cas, l’ensemble T := X ∩ J_{tors} est appelé le “paquet de torsion hyperelliptique.” La littérature contient quelques courbes de genre 2 pour lesquelles on sait que #T est grand. Par exemple, Coleman [4, p. 155] a remarqué que #T ≥ 22 pour la courbe modulaire X_{1}(13). Dans [7], on donnera un algorithme qui montre que #T = 22. Boxall et Grant [2] ont montré que #T = 22 aussi pour y^{2} = x^{5} + x. On a #T = 22 encore pour X_{1}(18), comme le montre l’algorithme de [7]. (Mais X_{1}(16), l’autre X_{1}(N) de genre 2, n’a que #T = 14.) On ne sait pas s’il existe une courbe de genre 2 avec #T > 22. On démontre le résultat suivant:

Théorème. Il y a une infinité de courbes non-isomorphes de genre 2 sur C pour lesquelles #T ≥ 22.

Esquissons la preuve. On commence avec la famille X_{t} : y^{2} = (x^{3} − 1)(t^{3} − x^{3}) sur C, où t > 1 est un paramètre réel. Chaque courbe de cette famille possède un groupe d’automorphismes G d’ordre 12. Comme remarqué dans [4], T contient quatre points fixés par un automorphisme d’ordre 3, et les six points de Weierstrass. Avec un argument réelanalytique, on trouve qu’il y a une infinité dénombrable de valeurs t > 1 tel que X_{t} acquière un point Q de torsion supplémentaire qui n’est fixé par aucun automorphisme non-trivial. Donc, la G-orbite de Q fournit 12 nouveaux points de T, augmentant le total à (au moins) 4 + 6 + 12 = 22.

Date: January 27, 2000.

This research was supported by NSF grant DMS-9801104, a Sloan Fellowship, and a Packard Fellowship.

I thank John Boxall for several helpful comments.

English Version

Let k be a field of characteristic zero, and let \overline{k} denote an algebraic closure. Let X be a (smooth, projective, geometrically integral) curve of genus g ≥ 2 over k, and suppose X is embedded in its Jacobian J using an Albanese map associated to some P ∈ X(k). We write X ∩ J_{tors} for the set of points in X(\overline{k}) that map to torsion elements of J(\overline{k}). The Manin-Mumford Conjecture, proved by Raynaud [8] in 1983, asserts that X ∩ J_{tors} is finite.

Mazur [6, p. 234] asked whether the size of the intersection of X with a subgroup of J(\overline{k}) of finite rank r can be bounded in terms of g and r alone. For r = 0, this would imply a bound on #(X ∩J_{tors}) depending only on g. Some partial results in this direction are known. Coleman suggested at the end of [5] that his methods should give a bound on #(X ∩J_{tors}) for curves of fixed genus with good reduction over unramified extensions of a fixed p-adic field assuming that the Albanese map is also defined over such an extension, and a very explicit bound of this type was given in [3]. A uniform bound exists also if one fixes X but varies the basepoint P for the Albanese map; in fact, #(X ∩ J_{tors}) ≤ 2 for all but finitely many choices of P [1].

We now restrict to the case g = 2, and consider the “hyperelliptic torsion packet” T, which is defined as X ∩ J_{tors} when P is chosen as a Weierstrass point. Since the difference of two Weierstrass points represents a 2-torsion point of J, T is independent of the choice of P. It follows that T is preserved by any automorphism of X. Despite the results mentioned above, a uniform bound for #T in this situation is not known. In fact, it seems that there is not a single family of genus 2 curves over P_{Q}^{1} with non-constant moduli for which it is known that #T can be bounded uniformly for all nonsingular fibers above rational points.

There are a few curves known to have large T. Coleman remarked on page 155 of [4] that the genus-two modular curve X_{1}(13) has #T ≥ 22, and the algorithm to be described in [7] shows that in fact #T = 22. This record has not been broken, although Boxall and Grant [2] proved that (the smooth projective model of) the curve y^{2} = x^{5} + x also has #T = 22. The algorithm of [7] shows that X_{1}(18) also has #T = 22: it turns out that T consists of the 16 cusps and the 6 Weierstrass points. The other remaining X_{1}(N) of genus 2, namely X_{1}(16), has #T = 14 only: [7] shows that T consists of the 14 cusps, which this time include the 6 Weierstrass points. See [2] for some other examples where T is known.

In this note we prove the following:

Theorem. There are infinitely many non-isomorphic genus-two curves over C with a hyperelliptic torsion packet T of size at least 22.

Proof. For real t > 1, let X_{t} be the curve y^{2} = (x^{3} − 1)(t^{3} − x^{3}). The automorphisms (x, y) ↦ (x, −y), (x, y) ↦ (e^{2πi/3}x, y), and (x, y) ↦ (tx^{−1}, t^{3/2}x^{−3}y) generate a group G of order 12. In particular, as in [4], the fixed points of a cubic automorphism give rise to four points of T distinct from the six Weierstrass points.

For t > 1, X_{t}(R) is a single oval projecting to [1, t] on the x-axis. Let ω_{1} = dx/y and ω_{2} = x dx/y on X_{t}. For 1 < u < t, let γ_{u} denote the shortest clockwise path from (1, 0) to the point (u, v) ∈ X_{t}(R) with v > 0, and define

f(t, u) = (f_{1}(t, u), f_{2}(t, u)) := (\frac{∫_{γ_{u}} ω_{1}}{∫_{X_{t}(R)} ω_{1}} , \frac{∫_{γ_{u}} ω_{2}}{∫_{X_{t}(R)} ω_{2}} ) ∈ R^{2},

where the denominator integrals are taken clockwise. As u → 1^{+} for fixed t > 1, we have f(t, u) → (0, 0), and f_{1}(t, u)/f_{2}(t, u) → ∫_{X_{t}(R)} ω_{2}/ ∫_{X_{t}(R)} ω_{1} by L’Hôpital’s Rule. For t > 1, this ratio of periods is strictly between 1 and t, so it is a non-constant function of t. Since one can approach (0, 0) along curves of variable slope in the image im(f) of the real-analytic map f, dim(im(f)) = 2, and im(f) contains a nonempty open subset of R^{2}. On the other hand, for each t > 1 consider the finite subset S_{t} ⊂ X_{t}(R) of points in the upper half of X_{t}(R) that are fixed by some non-trivial g ∈ G. Let S = ⋃_{t>1} f(t, S_{t}). Then S is a finite union of real-analytic curves, so im(f) \ S contains a nonempty open subset U ⊂ R^{2}. By the Abel-Jacobi theory, the points in U ∩ Q^{2} come from torsion points Q with x-coordinate u on the curve X_{t}, and infinitely many values of t must occur since U ∩ Q^{2} has a 2-dimensional closure. For these t, the orbit of Q under G gives 12 new points in T, bringing the total to 4 + 6 + 12 = 22 (at least). Our family is non-isotrivial, so we obtain infinitely many non-isomorphic curves.

Remarks.

1. Standard specialization arguments show that one can replace C with \overline{Q} in the theorem.

2. The proof shows also that for any sufficiently large N, one can find a curve as above with #T ≥ 22, and such that the Albanese map associated to some Weierstrass point maps at least one of the points in T to a point of exact order N on J.

3. The curves X_{1}(13) and X_{1}(18) have automorphisms of order 3 (diamond operators). The curve y^{2} = x^{5} + x of [2] also has an automorphism of order 3, namely

(x, y) ↦ (ζ^{−1} (\frac{x + ζ}{x − ζ} ) , \frac{−(i\sqrt{8})y}{(x − ζ)^{3}} )

where ζ = e^{2πi/8}. It follows that each of these curves is isomorphic over C to a curve of the form y^{2} = (x^{3} − 1)(t^{3} − x^{3}) for some t ∈ C.

4. If a Weierstrass point is chosen as basepoint, then of the 22 points in T for X_{1}(13), one has order 1, five have order 2, four have order 6, and twelve have order 38. The numbers of points of each order are exactly the same for X_{1}(18), except that 38 should be replaced by 42. (This is related to the facts that J_{1}(13)(Q)≅Z/19Z and J_{1}(18)(Q)≅Z/21Z.) The six points of order dividing 2 are the Weierstrass points, and the four of order 6 are the fixed points of an automorphism of order 3. Choosing the cusp ∞ as basepoint instead would produce different orders, although of course the same prime factors would occur. For y^{2} = x^{5} + x, all the non-Weierstrass points in T have order 6, as shown in [2].

References

[1] Baker, M., Poonen, B., Torsion packets on curves, preprint, 1999.

[2] Boxall, J., Grant, D., Examples of torsion points on genus two curves, to appear in Trans. Amer. Math. Soc.

[3] Buium, A., Geometry of p-jets, Duke Math. J. 82 (1996), no. 2, 349–367.

[4] Coleman, R., Torsion points on curves and p-adic abelian integrals, Ann. of Math. (2) 121 (1985), no. 1, 111–168.

[5] Coleman, R., Ramified torsion points on curves, Duke Math. J. 54 (1987), no. 2, 615–640.

[6] Mazur, B., Arithmetic on curves, Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 207–259.

[7] Poonen, B., Computing torsion points on curves, in preparation.

[8] Raynaud, M., Courbes sur une variété abélienne et points de torsion, Invent. Math. 71 (1983), no. 1, 207–233.

Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

  E-mail address: poonen@math.berkeley.edu