Torsion of genus 2 Jacobians over ℚ

1. Inputs on torsion

For a positive integer \(d\), let \(S(d)\) be the set of primes that occur as the order of a torsion point on an elliptic curve over a number field of degree \(d\). We use the following results.

1. Derickx–Kamienny–Stein–Stoll, Theorem 1.2, prove that

   \[ S(4)=\{2,3,5,7,11,13,17\},\qquad S(6)=\{2,3,5,7,11,13,17,19,37\}. \]

   Hence an elliptic curve over a field of degree \(1,2,3,4,\) or \(6\) has no point of order \(31\). For degrees \(1\) and \(2\), extend to a field of degree \(4\); for degree \(3\), extend to one of degree \(6\).

2. Khawaja, Theorem 1.2, proves that

   \[ S(8)=\{2,3,5,7,11,13,17,19,23\}. \]

   Thus an elliptic curve over a degree-\(8\) number field has no point of order \(31\).

3. Clark, Theorem 5, proves that, if an abelian surface over \(\Q\) has everywhere potentially good reduction, its rational torsion order belongs to

   \[ \{1,\ldots,16,18,19,20,22,24,25,28,30,36,48,60,72\}. \]

   None of these integers is divisible by \(31\).

We also use the standard independence of the semisimplified residual representation from the choice of a stable lattice. It applies to isomorphic \(p\)-adic representations after transporting a lattice through the isomorphism.

2. The elementary cases and the endomorphism field

Suppose that \(P\in A(\Q)\) has order \(31\). First suppose that \(A_{\overline{\Q}}\) is isogenous to \(E_1\times E_2\), where the elliptic curves \(E_1,E_2\) are not geometrically isogenous. There are exactly two geometric elliptic subvarieties of \(A\). Galois permutes them, so both are defined over a field \(K\) with \([K:\Q]\leq2\). Choose one of them, \(E\), and put \(F=A_K/E\). The exact sequence

shows that \(P\) either belongs to \(E(K)\) or maps to a point of order \(31\) in \(F(K)\). Both possibilities contradict the elliptic torsion bounds above.

Next suppose that \(A_{\overline{\Q}}\) is isogenous to \(E^2\), where \(E\) has complex multiplication. CM elliptic curves have everywhere potentially good reduction. This property is preserved by products and isogenies, and descends from a finite extension of the ground field. Hence \(A\) has everywhere potentially good reduction, and Clark's theorem gives a contradiction.

It remains to consider the case where \(A_{\overline{\Q}}\) is isogenous to \(E^2\) for a non-CM elliptic curve \(E\). Then

Let \(L/\Q\) be the smallest field over which all geometric endomorphisms of \(A\) are defined. It is a finite Galois extension. Put \(\Gamma=\Gal(L/\Q)\). By Skolem–Noether, its faithful action on \(D\) embeds \(\Gamma\) into \(\PGL_2(\Q)\).

A finite subgroup of \(\PGL_2(\mathbb R)\) is cyclic or dihedral: it fixes a point of the hyperbolic plane, and its action on the tangent plane embeds it in \(O(2)\). If an element of \(\PGL_2(\Q)\) has order \(n>2\), the ratio of the eigenvalues of a representative \(M\) is a primitive \(n\)-th root of unity \(\zeta\). Thus

which implies \(n\in\{3,4,6\}\). Hence \(\Gamma\) is cyclic of order \(1,2,3,4,\) or \(6\), or dihedral of order \(4,6,8,\) or \(12\). Here the dihedral group of order \(4\) is the Klein four group. In particular,

Every geometric elliptic subvariety of \(A\) is defined over \(L\): it is the image of an integral multiple of an idempotent in \(D\). Applying the elliptic exact sequence over \(L\) and the torsion bounds from Section 1 excludes every listed degree except \(12\). We are therefore left with

where the subscript denotes the group order. Its rotation subgroup \(C=\langle r\rangle\) has order \(6\). Put \(K=L^C\), so \([K:\Q]=2\).

3. A stable split quadratic order

Set \(p=31\), \(k=\Q_p\), and \(\mathcal O=\Z_p\). Identify \(D\otimes_{\Q}k\) with \(M_2(k)\), and normalize a matrix representative \(R\) of \(r\) to have trace \(1\). Since \(r\) has projective order \(6\), the characteristic polynomial of \(R\) is

Its discriminant \(-1/3\) is a nonzero square modulo \(31\): it is \(10\) modulo \(31\), and \(14^2\equiv10\pmod{31}\). Thus its eigenvalues \(\alpha,\beta\) lie in \(\mathcal O^\times\), and \(\alpha-\beta\) is a unit. It follows that

Write \(e_1,e_2\) for its two orthogonal idempotents.

Conjugation by \(\Gamma\) sends \(r\) to \(r\) or \(r^{-1}\). The trace-normalized representative of \(r^{-1}\) is \(1-R\), so \(\Gamma\) preserves \(\mathcal B\) and permutes \(\{e_1,e_2\}\). The rotation subgroup fixes each idempotent, and every element outside it exchanges them.

Let \(T=T_p(A)\), and set

Since \(\mathcal B\) is finite free over \(\mathcal O\), this is a full lattice in \(V_p(A)\). It is stable under \(G_{\Q}=\Gal(\overline{\Q}/\Q)\), since both \(T\) and \(\mathcal B\) are stable. We obtain

Each \(T_i\) is free of rank \(2\) over \(\mathcal O\): the four-dimensional \(M_2(k)\)-module \(V_p(A)\) is the sum of two copies of its standard module. The group \(G_K\) preserves each \(T_i\), while every element of \(G_{\Q}\setminus G_K\) exchanges the two summands.

4. The two summands differ by a character

Put \(V_i=T_i\otimes_{\mathcal O}k\), and choose an off-diagonal matrix unit \(X\in e_1M_2(k)e_2\). The action of \(G_K\) on \(D\otimes k\) is through \(C\). Consequently

where \(\eta:G_K\to k^\times\) is the eigenvalue-ratio character, with value \(\alpha/\beta\) on a lift of \(r\). It factors through \(\Gal(L/K)=C\) and has exact order \(6\). Its values are units, and its reduction \(\overline\eta\) also has exact order \(6\), since reduction is injective on roots of unity of order prime to \(31\).

The matrix unit \(X:V_2\to V_1\) is an isomorphism, and the equality \(gXg^{-1}=\eta(g)X\) gives

as \(G_K\)-representations. Put \(M_i=T_i/pT_i\). Applying lattice independence to this isomorphism gives

This step does not require \(X\) to carry \(T_2\) onto \(T_1\) integrally.

5. The determinant contradiction

The point \(P\) generates a trivial submodule of \(A[p]=T/pT\). By lattice independence, \(T'/pT'\) therefore has a trivial \(G_{\Q}\)-composition factor. Restriction to \(G_K\) is exact, so at least one of \(M_1,M_2\) has a trivial \(G_K\)-composition factor. An element of \(G_{\Q}\setminus G_K\) conjugates these two \(G_K\)-modules. Since a trivial module remains trivial under conjugation, both have a trivial composition factor.

The relation \(M_1^{\mathrm{ss}}\simeq\overline\eta\otimes M_2^{\mathrm{ss}}\) shows that \(\overline\eta^{-1}\) is also a composition factor of \(M_2\). Since \(M_2\) has dimension \(2\) and \(\overline\eta\) is nontrivial, we obtain

Hence

Every abelian variety over \(\Q\) has a polarization over \(\Q\). The induced nondegenerate alternating pairing on \(V_p(A)\) shows that the determinant of its \(p\)-adic representation is the square of the \(p\)-adic cyclotomic character. A change of stable lattice does not change this determinant. Writing \(\chi_p:G_{\Q}\to\F_p^\times\) for the mod-\(p\) cyclotomic character, we therefore have

The character \(\chi_{31}^2\) has order \(15\), and its kernel fixes \(\Q(\zeta_{31}+\zeta_{31}^{-1})\), a field of degree \(15\) over \(\Q\). The displayed identity would put this field inside the quadratic field \(K\), a contradiction. This proves the theorem. ∎

6. Consequence for the census

The group [31] is impossible in both the class “split over \(\Q\)” and the class “geometrically split, simple over \(\Q\)”. The argument applies to every geometric splitting degree, including degrees divisible by \(31\). It uses a stable split quadratic order at \(31\), without requiring a stable maximal order or a maximal integral endomorphism ring.

References

1. M. Derickx, S. Kamienny, W. Stein and M. Stoll, Torsion points on elliptic curves over number fields of small degree, Algebra & Number Theory 17 (2023), 267–308, Theorem 1.2.

2. M. Khawaja, Torsion primes for elliptic curves over degree 8 number fields, Research in Number Theory 10 (2024), article 48, Theorem 1.2.

3. P. L. Clark, Bounds for torsion on abelian varieties with integral moduli, arXiv:math/0407264v2 (2004), Theorem 5. The same list is Theorem 1.4 of P. L. Clark and X. Xarles, Local bounds for torsion points on abelian varieties, Canad. J. Math. 60 (2008), 532–555, for abelian surfaces with everywhere anisotropic reduction, which includes potentially good reduction.