Torsion of genus 2 Jacobians over ℚ

1. The family

For \(s\in\Q\), put

Let \(C_s\) be the smooth projective model of

We prove exactness for every rational parameter. In particular, the theorem requires no exceptional thin set or congruence restriction.

2. Parameters and elliptic gluing

Put

A direct calculation gives

The polynomials \(3s^2-5\), \(3s^2-10s+5\), and \(3s^2-6s+5\) have no rational zeros. Hence \(a,b\ne0\) and \(t<-1/4\) for every rational \(s\). The discriminant of \(D\) is \(-14175\), so \(D(t)>0\) for real \(t\).

Let \(m=t^2-1/16\), and consider

After completing the square in \(y\), their monic cubic models have the following ordered roots:

The roots are distinct for \(t<-1/4\). Let \(\alpha:E_t[2]\longrightarrow F[2]\) send the point at \(e_i\) to the point at \(f_i\), and let \(\Gamma\) be its graph.

We apply the gluing construction of Howe–Leprévost–Poonen, Propositions 3 and 4. In their notation,

where subscripts are taken modulo \(3\). Thus the correspondence of roots is not induced by an affine change of the elliptic \(x\)-coordinate, and \(\alpha\) is not the restriction of an isomorphism of elliptic curves over \(\overline{\Q}\). Consequently

with its induced principal polarization is a genus-2 Jacobian. Write

for the gluing isogeny and its dual, so that \(\psi\phi=[2]\) and \(\phi\psi=[2]\).

The quotient is precisely the Jacobian of the displayed family. For completeness, let \(h(X)\) be the sextic in Howe–Leprévost–Poonen, Proposition 4, using the root order above. Its constants \(A,B\) are

If \(g_t(x)\) denotes the right-hand side of the family equation, then

All factors in this change of variables are nonzero for our parameters. The sextic discriminant is

Thus every \(C_s\) is smooth of genus \(2\), and its Jacobian splits over \(\Q\).

3. The torsion Kummer images

We use multiplicative notation for \(\Q^\times/\Q^{\times2}\). On an elliptic curve with monic cubic model \(Y^2=\prod_{i=1}^3(x-r_i)\), the \(2\)-Kummer map is

For a point away from the nonzero \(2\)-torsion, it is represented by \((x-r_1,x-r_2,x-r_3)\). At the \(2\)-torsion point with \(x=r_i\), the \(i\)-th component is \(\prod_{j\ne i}(r_i-r_j)\), and the other components are \(r_i-r_j\). The identity has image \((1,1,1)\). We use the ordered roots above for the coordinate order on the two curves, so that their Kummer coordinates agree under \(\alpha\).

Let \(P=(0,0)\) on the displayed generalized Weierstrass model of \(E_t\), and let \(Q=(0,0)\) on that of \(F\). Let \(T\in E_t[2]\) have \(x=e_2\), and let \(U\in F[2]\) have \(x=f_3\). The addition law gives

In particular, \(P,T\) generate a subgroup of type \([2,4]\), and \(Q,U\) generate a subgroup of type \([2,8]\).

The Kummer images, as triples of square classes, are

By the conic identities, \(m=a^2b^2/15\). Therefore

Proof. Mazur's theorem gives the assertion for \(F\). For \(E_t\), the only possibilities are \([2,4]\) and \([2,8]\). In the latter case its four points of order \(4\) are \(\pm P,\pm(P+T)\), and some such point is twice a rational point of order \(8\). However, the first Kummer coordinates of \(P\) and \(P+T\) have square classes \(-15\) and \(15\), respectively. Neither is a square in \(\Q\). Their negatives have the same Kummer images, so none of these points is divisible by \(2\) in \(E_t(\Q)\). This excludes \([2,8]\). □

Put \(T_E=E_t(\Q)_{\tors}\) and \(T_F=F(\Q)_{\tors}\). The first coordinates of their Kummer images are, respectively,

The two sets have exactly the common square classes \(1\) and \(-15\). By the matching identity, the complete triples with first coordinate \(-15\) also agree. It follows that

4. Exact torsion of the Jacobian

Define

We claim that there is an exact sequence

Indeed, choose halves \(\widetilde A,\widetilde B\) of \(A,B\). The point \(\phi(\widetilde A,\widetilde B)\) is rational if and only if every Galois difference of the pair \((\widetilde A,\widetilde B)\) lies in \(\Gamma\). Since \(E_t[2]\) and \(F[2]\) are rational, this is exactly the equality of Kummer classes in the definition of \(M\). It is then a lift of \((A,B)\) under \(\psi\). The lift is torsion: either this follows from its construction using halves of torsion points, or from \(2R=\phi\psi(R)\). Conversely, every rational lift gives the equality of Kummer classes. This proves the exact sequence.

The group scheme

is constant of order \(4\). The kernel of the Kummer map on the torsion subgroup of an elliptic curve is twice that torsion subgroup, since a half of a torsion point is torsion. Hence the lemma and the Kummer intersection give

By the exact sequence,

It remains to determine the group structure. Put \(V=Q+U\), which has order \(8\), and

The graph \(\Gamma\) is generated by \((2P,U+4V)\) and \((T,4V)\). In the quotient of \(T_E\times T_F\), these relations eliminate the generators corresponding to \(T\) and \(U\). The remaining relations are \(4X=0\) and \(8Y=0\). Thus

The Kummer match gives \(R\in J(\Q)_{\tors}\) with \(\psi(R)=(P,V)\). It satisfies

The point \(X+Y\) has order \(8\), so \(R\) has order \(16\). Moreover, \(R\notin H\), since \(2H\) has exponent \(4\). Therefore \(\langle H,R\rangle\) has order \(64\), and by the order calculation it is the whole rational torsion subgroup. It is generated by \(X,R\), since \(Y=2R-X\). The resulting surjection

is an isomorphism because both groups have order \(64\).

5. Variation in moduli

The varying elliptic factor has

This rational function is nonconstant, and so is \(t(s)\). Thus \(j(E_{t(s)})\) assumes infinitely many values for \(s\in\Q\). The gluing construction supplies a degree-\(2\) map \(C_s\longrightarrow E_t\) over \(\Q\). A fixed smooth genus-2 curve over \(\overline{\Q}\) has finitely many automorphisms, and hence finitely many involutions and degree-\(2\) elliptic quotients up to isomorphism. Consequently the infinitely many values of \(j(E_{t(s)})\) force infinitely many geometric isomorphism classes of \(C_s\). This proves the theorem. □

6. Reproducibility and the census example

The accompanying standalone Magma certificate verifies the elliptic point orders, all Kummer identities, the HLP change of variables, the discriminant, the conic parametrization, and the elliptic \(j\)-formula. All assertions passed in Magma V2.30-1; the verification log is available. The proof above uses only exact identities, elliptic torsion classification, and the stated gluing theorem.

At \(s=1\), we have \(t=-1\), and after dividing \(y\) by \(225\) the curve becomes

The change of variables \(z=3/x\), \(w=3y/x^3\) gives

Writing \(w=2v+z^3+z\) gives the census example 4.16.a,

Magma independently computes exact torsion \([4,16]\) for this specialization and for \(s=2\), where \(t=-19/49\). These specialization checks are consistency checks; they are not needed for the uniform torsion argument.

References

• E. W. Howe, F. Leprévost, and B. Poonen, Large torsion subgroups of split Jacobians of curves of genus two or three, Forum Math. 12 (2000), 315–364.
• B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. IHES 47 (1977), 33–186.