Curves of genus 2 over Q whose Jacobians are absolutely simple abelian surfaces with torsion points of high order

New results
Explicit formulas
Other amusing examples


For which integers N can the Jacobian J of a curve C of genus 2 over Q contain a rational torsion point p of order N?

Without further assumptions on C or J, there is no hope at present for bounding N from above, so the best we can do is seek examples with large N.

The problem naturally splits into two parts, according as J is simple or split. Here an abelian variety is said to be “simple” if and only if it is not isogenous to a product of abelian varieties of lower dimension, and “(completely) split” if it is isogenous to a product of elliptic curves; in our case J is two-dimensional, so it must be one or the other. Note that the elliptic curves are allowed to be defined over some extension of Q; thus we use “simple” to mean “absolutely simple”: not split over any number field.

In the non-simple (split) case, the state of the art is represented by the paper “Large torsion subgroups of split Jacobians of curves of genus two or three” by Everett Howe, Franck Leprévost, and Bjorn Poonen (Forum Math.12 (2000) #3, 315-364). There one finds examples of curves or families of (C,J,p,N) with J split for various values of N, the largest of which is 63. The constructions hinge on the torsion structure of the elliptic curves whose product is isogenous with J. These are controlled by various “modular curves”, whose study is an active and highly developed topic in number theory.

In the simple case, the theory of modular curves is no longer available. Torsion points of high order are usually constructed by forcing C to have two or three non-Weierstrass points that support several divisors linearly equivalent to zero. This method is described in pages 84-87 of J.W.S.Cassels and E.V.Flynn's book Prolegomena to a middlebrow arithmetic of curves of genus 2 (LMS Lecture Notes 230), Cambridge University Press, 1996. This method was used by Leprévost to construct rational N-torsion points for N as high as 29 (in his paper “Jacobiennes décomposables de certaines courbes de genre 2: torsion et simplicité”, J. Théorie des Nombres de Bordeaux 7 (1995) #1, 283-306) and 30 (in an unpublished 4/1996 preprint, reported by Everett Howe). While Leprévost found only a single curve with a 29-torsion divisor, he obtained an infinite (one-parameter) family of curves with N=30.

Announcement of new results

By systematically pursuing Leprévost's method and natural variations of it, I found several new and larger values of N: a one-parameter family of curves whose Jacobian has a torsion point of order 32; two curves each of whose Jacobians has a torsion point of order 34; and one curve each for 39 and 40. The N=39 curve is also remarkable for having minimal automorphism group (only the identity and the hyperelliptic involution) and four pairs of non-Weierstrass points, each of which differs from any Weierstrass point by a torsion divisor. Without extra symmetry, parameter counts lead one to expect only three such pairs, because the moduli space of genus-2 curves has dimension 3; the N=39 curve is the only one known (as of 6/2002) with a fourth pair.

Explicit formulas

We exhibit the four curves with N=34, 34, 39, 40. We refrain from giving an explicit formula for the N=32 family, because the coefficients of the curves are very complicated rational functions of the family's parameter; we thus content ourselves with exhibiting a curve in that family with reasonably small coefficients. The simplicity of each of the curves' Jacobians was checked by Leprévost's criterion (Lemme 3.1.2 of his paper): a genus-2 curve C over Q has simple Jacobian if there is a prime l of good reduction such that the Galois group of the characteristic polynomial of Frobenius of the reduced curve is the 8-element dihedral group D4. [The Galois group is a priori contained in D4, because the roots of the polynomial are four complex numbers of the same absolute value; Leprévost shows that if J is isogenous to a product of elliptic curves then the Galois group is strictly smaller than D4.]


Let C be the curve
y2 = (9x2+2x+1) (32x3+81x2-6x+1).
This has a rational Weierstrass point at infinity, and a non-Weierstrass rational point (x,y)=(1,36). The difference between these points is a torsion divisor of order 34. The rational points (0,1), (1/2,27/4), and their hyperelliptic conjugates also arise in this 34-torsion group. The curve has good reduction mod 5, where the characteristic polynomial of Frobenius F4+F3+2F2+5F+25 has Galois group D4 according to the GP function polgalois.

N=34, again

Let C be the curve
y2 = (10-x) (3x+2) (72x4+96x3+45x2-38x+5).
This has a rational non-Weierstrass point (x,y)=(0,10) that differs from the Weierstrass point (10,0) by a 17-torsion divisor; since the difference between (10,0) and the Weierstrass point (-2/3,0) must be 2-torsion, we conclude that (0,10) - (-2/3,0) is a torsion divisor of order 34. Replacing (0,10) by the rational point (1,90) yields a multiple of this divisor. This curve has good reduction mod 17, where the characteristic polynomial of Frobenius is F4-4F3+20F2-68F+289, again with dihedral Galois group.


Let C be the curve
y2 = x6+4x4+10x3+4x2-4x+1.
[This curve has good reduction away from 2, 3, 31; at 2 one does better with
y2 + (x3+1) y = x4 + 2 x3 + x2 - x;
-- as often happens, we get somewhat smaller coefficients by going from strict to extended Weierstrass form -- but there is still a singular point at (x,y)=(1,1). Thanks to Dino Lorenzini for drawing my attention to this singularity (March 2010); an earlier version of this webpage (June 2003) incorrectly asserted that the curve has good reduction at 2.] Like all our other curves, this one has minimal automorphism group. Unlike the others, the N=39 curve has four pairs of nonrational Weierstrass points, at x=-1, 0, 1, and infinity. The pairwise difference among these points generate a cyclic subgroup of J of order 39. Leprévost's criterion is satisfied for l=5, with characteristic polynomial F4+2F3+F2+10F+25.

Bjorn Poonen has a program that computes, for most genus-2 curves C, all points that differ from their hyperelliptic conjugates by torsion divisors; his paper that develops the requisite theory and describes the program is available online in pdf or compressed dvi. He confirmed that for the N=39 curve these four pairs of points, together with the six Weierstrass points, are the only points of C, even over the complex numbers, that differ from their hyperelliptic conjugates by torsion divisors.


Let C be the curve
y2 = (3x+4) (x4+5x3+8x2+(19/4)x+1),
with two rational Weierstrass points (at x=-4/3 and infinity) and a non-Weierstrass rational point at (-2,1). The difference between this point and either of the Weierstrass points is a torsion divisor of order 40, the highest known (as of 6/2001) for a simple genus-2 Jacobian. The rational points (0,2) and (-1,1/2) yield multiples of this divisor. Simplicity is again proved by Leprévost's criterion, this time with l=17 and characteristic polynomial F4+F3+12F2+17F+289.

N=32: an example

One of infinitely many curves with N=32 is
y2 = (15x-1) (1056x4+156183x3+26297x2+649x-121).
In addition to the Weierstrass points at (1/15,0) and infinity, this curve has the rational non-Weierstrass points (0,11) and (-1,1440). The difference between either of these and either of the Weierstrass points is a torsion divisor of order 32. The simplicity criterion holds for l=7, with dihedral quartic F4+F3+6F2+7F+49.

Other amusing examples

N=31, almost...

Let C be the curve
y2 = 5x6-4x5+20x4-2x3+24x2+20x+5.
This curve has 3 pairs of points (x,7x2) where x is one of the roots of the cubic x3+x2-2x-1, that is, x=2cos(m*Pi/7) for m=1,2,3. Any two of these six points differ by a 31-torsion divisor on J. Thus J has a rational 31-torsion subgroup, but without a rational generator; the Galois group of Q(x)/Q multiplies this group by {1,5,25} mod 31. As with N=39, these six points are known to be the only non-Weierstrass points that differ from their hyperelliptic conjugates by torsion divisors; this was the first case that required Matt Baker's refinement of Bjorn Poonen's method for computing the “Weierstrass torsion packet”.

Poonen also observed that J, while simple, appears to have real multiplication by Q(sqrt(2)), which would make J conjecturally modular; he reports using Qing Liu's program to determine the arithmetic conductor of J: it is 2452=7452, so J should be isogenous to a simple factor of the Jacobian J0(245) of the modular curve X0(245)=X0(725), and not to a factor of J0(M) for any M<245. Again as with N=39, good reduction at 2 can be confirmed by “uncompleting the square” mod 4 to obtain an equivalent formula for the curve, here

y2 + (x3-1) y = x6 - x5 + 5 x4 + 6 x2 + 5 x + 1.
The traces of Frobenius mod p for primes p<100 (other than the primes 5, 7 of bad reduction) coincide with those of the two-dimensional factor of J0(245) labeled 245H in William Stein's Modular Forms database [go to the "Modular Forms Explorer", click the start button, enter 245,2 (for level 245, weight 2) and click "Change space to", then hit the “[Click]” link for 245H1 to get the coefficients indicating that J is isogenous to that factor].

Rank 1, six point pairs

Much the same dimension count that suggested at most three pairs of non-Weierstrass torsion points also indicates that we can force at most five non-Weierstrass pairs of points of C on a rank-1 subgroup of J -- again assuming that C has minimal automorphism group. But again I found one example with a lucky extra point-pair. Namely, the curve
y2 = x6 - 2 x5 - 4 x4 + 2 x3 + (37/4) x2 - (15/2) x + (9/4)
[or in extended Weierstrass form y2 + (x3-1) y = (x2+1)2 (x-1) (x-3)] has the following six points: (x,y)=(inf,inf3), (0,3/2), (1,1), (-1,-4), (3,6), and (1/2,7/8). The elements of J represented by the differences between these points and their hyperelliptic conjugates are all in the subgroup generated by q := (inf,inf3) - (inf,-inf3), namely q, 5q, 13q, 29q, 61q, 83q. That there are no further points in the intersection of this rank-1 group with C can be deduced from the Coleman-Chabauty bound over Q7.

There are over a hundred examples over Q with five point pairs (both with and without a rational Weierstrass point), and many nonconstant families with four point pairs parametrized by P1. The above six-point curve was found by specializing the (q, 5q, 13q, 29q) family

C(b): y2 = [x3 - x2 + b x + (b+1)]2 - 4 b (b+1) x
over the b-line to b=-5/2. It is still (as of June 2002) the only case known of six point pairs whose differences generate a subgroup of J isomorphic with Z; but in September 2001 Michael Stoll found specializations of two other four-point families that yield curves of genus 2 six point pairs whose differences generate a rank-1 subgroup of J isomorphic with Z+(Z/2Z). The curve
y2 = [x3 + tx2 - x + t-1)]2 - 4(t2-t) x2

has four points (x,y)=(inf,inf3), (0,1-t), (-1,1), and (1,-1) over C(t) giving q,2q,8q,13q, and the specialization t=5 has two further points, with x=-2 and x=-130/77, that yield divisors that differ from 4q and 38q by a 2-torsion point. A more complicated family of curves with a rational Weierstrass point as well as 6q,8q,9q,13q specializes to

y2 = (x+2) (x2-3x+6) (4x3+8x2-3x+3)
with rational points at x=-2 (Weierstrass), -3,-1,0,1,4,inf -- the two extra point pairs differ by 10q+T and 50q+T (2T=0). In both cases the lists of points proportional to q modulo torsion are complete, again by Coleman-Chabauty over Q7.

The points on C(b) giving q,5q,13q,29q are (x,y)=(inf,inf3), (0,-(b+1)), (1,1), and (-1,2b+1). The choice of coefficients 1,5,13,29 may seem bizarre, but in fact {-29,-13,-5,-1,1,5,13,29} is the lexicogrphically first symmetric set of 2*4 odd integers satisfying the necessary condition that all nonzero sums of elements of the set be distinct. Likewise {-13,-8,-2,-1,1,2,8,13} is one of the first such sets if we drop the parity condition.

Michael Stoll also computed the canonical heights of q on C(b), and of its specialization to b=-5/2, finding 1/357 and .000797... As far as we know, .000797... is the smallest positive canonical height yet exhibited for a point on the Jacobian of a curve of genus 2 over Q. Over P1, the record is 1/546, belonging to the q,2q,8q,13q curve over Q(t) exhibited above.

In the more familiar case of genus 1 (elliptic curves), there are questions of similar flavor concerning points of low positive height; one can also consider integral points on the minimal (Néron) model of an elliptic curve as an analogue of rational points on a curve of genus 2. Some remarkable examples of nontorsion points with low height and/or many integral multiples are listed here.