Torsion of genus 2 Jacobians over ℚ

1. Inputs on elliptic torsion and potentially good reduction

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

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

   Consequently an elliptic curve over a number field of degree \(1\), \(2\), or \(4\) has no point of order \(23\) or \(29\). For degrees \(1\) and \(2\), extend the field to one of degree \(4\).

2. Khawaja, Theorem 1.2, proves that

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

   In particular, an elliptic curve over a degree-\(8\) number field has no point of order \(29\).

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

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

   None of these integers is divisible by \(23\) or \(29\).

We also use the standard properties of Tate modules and geometric endomorphisms, and the independence of the semisimplified residual representation from the choice of a stable lattice.

2. Nonisogenous factors and the CM case

Fix \(p\in\{23,29\}\), and suppose that \(P\in A(\Q)\) has order \(p\). First suppose that \(A_{\overline{\Q}}\) is isogenous to \(E_1\times E_2\), where \(E_1\) and \(E_2\) 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 set \(F=A_K/E\). There is an exact sequence over \(K\)

The image of \(P\) in \(F(K)\) either has order \(p\) or is zero. In the latter case \(P\in E(K)\). Both possibilities contradict the degree-\(4\) elliptic torsion bound 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

3. The field of definition of the endomorphisms

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 \(D\) identifies \(\Gamma\) with a finite subgroup of

A finite subgroup of \(\PGL_2(\mathbb R)\) is cyclic or dihedral: its action on the hyperbolic plane has a fixed point, and the action on the tangent plane embeds it in \(O(2)\). Let an element of \(\PGL_2(\Q)\) have finite order \(n>2\), and let \(M\) be a representative. A non-semisimple \(M\) is a scalar times a nontrivial unipotent matrix and has infinite order in \(\PGL_2(\Q)\). Hence \(M\) is semisimple, and the ratio of its eigenvalues is a primitive \(n\)-th root of unity \(\zeta\). Thus

which implies \(n\in\{3,4,6\}\). It follows that \(\Gamma\) 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 \(L\). Indeed, it is the image of an integral multiple of an idempotent in \(D\), and that endomorphism is defined over \(L\). If \([L:\Q]\in\{1,2,4\}\), the elliptic exact sequence used in the previous section, now over \(L\), excludes \(P\). This includes the Klein four case. If \(p=29\), the degree-\(8\) elliptic torsion bound also excludes the case \([L:\Q]=8\).

In every remaining case, \(\Gamma\) has a normal cyclic subgroup \(\langle r\rangle\) of order \(n\), with

For a cyclic group take the whole group; for a dihedral group take the rotation subgroup.

4. A stable maximal order

Set \(k=\Q_p\) and \(\mathcal O=\Z_p\), and identify \(D\otimes_{\Q}k\) with \(M_2(k)\). Choose a matrix representative of \(r\), and divide it by its nonzero trace. The resulting matrix \(R\) has characteristic polynomial

Its discriminant is respectively \(-3,-1,\) or \(-1/3\). Each is a nonsquare unit modulo \(p\) in the cases retained above: \(23\equiv-1\pmod3\) and \(23\equiv-1\pmod4\), while \(29\equiv-1\pmod3\). Therefore \(k[R]/k\) is the unramified quadratic extension, and \(\mathcal O[R]\) is its maximal order.

An \(R\)-stable \(\mathcal O\)-lattice in \(k^2\) is a rank-one \(\mathcal O[R]\)-module, hence a fractional ideal of \(\mathcal O[R]\). Since the extension is unramified, all such lattices are homothetic under \(k^\times\). Moreover, \(\det R\) is a unit, so a homothety class fixed by the projective transformation \(r\) is represented by an \(R\)-stable lattice. Thus \(r\) fixes a unique homothety class of lattices.

The subgroup \(\langle r\rangle\) is normal in \(\Gamma\), and conjugation sends \(r\) to \(r\) or \(r^{-1}\). These elements have the same fixed lattice classes. It follows that \(\Gamma\) preserves the unique class above. Choose a representative \(\Lambda\). Then

is a \(\Gamma\)-stable maximal order in \(M_2(k)\). Every element of \(\Gamma\) has a projective representative in \(\GL_2(\mathcal O)\) acting on \(\Lambda\). Reduction gives a projective action

Its natural two-dimensional projective module is irreducible over \(\F_p\), since the reduction of \(R\) has irreducible characteristic polynomial.

This construction uses the full endomorphism algebra \(D\). It therefore requires no maximality assumption on \(\End(A_{\overline{\Q}})\otimes\Z_p\).

5. A change of Tate lattice

Let \(V=V_p(A)\) and \(T=T_p(A)\). Define

Since \(\mathcal R\) is finite free over \(\mathcal O\), this is a full \(\mathcal O\)-lattice in \(V\). It is stable under \(G_{\Q}=\Gal(\overline{\Q}/\Q)\), since both \(T\) and \(\mathcal R\) are stable. After choosing a basis of \(\Lambda\), we have \(\mathcal R=M_2(\mathcal O)\), and the matrix units give

where \(W_0=e_{11}T'\) is free of rank \(2\) over \(\mathcal O\). The off-diagonal matrix units identify \(e_{11}T'\) and \(e_{22}T'\), which gives the displayed decomposition.

Put \(\overline V=T'/pT'\). The reductions of two stable lattices in the same \(p\)-adic representation have the same semisimplification, by Brauer–Nesbitt. Since \(T/pT=A[p]\) has a trivial submodule generated by \(P\), the module \(\overline V\) has a trivial composition factor. Only this composition factor is used; a fixed vector need not survive a change of lattice.

The action of \(G_{\Q}\) on \(\overline V\) normalizes the algebra \(M_2(\F_p)\) acting on \(U=\Lambda/p\Lambda\). Its induced projective action on \(U\) factors through \(\Gamma\). Pull back

along this projective action of \(G_{\Q}\), and denote the resulting group by \(\widetilde G\). Then \(U\) is an actual two-dimensional \(\widetilde G\)-module. The centralizer of \(M_2(\F_p)\) gives a two-dimensional \(\widetilde G\)-module \(W\) such that

Explicitly, for \((g,B)\in\widetilde G\), the action on \(U\) is given by \(B\), and \(B^{-1}\) times the action of \(g\) on \(\overline V\) commutes with \(M_2(\F_p)\), so acts on the second factor. All these representations factor through a common finite quotient of \(\widetilde G\).

The module \(U\) is irreducible by the preceding section. The subgroup

is normal in \(\widetilde G\) and acts trivially on \(U\). The trivial composition factor of \(\overline V\) remains trivial on pullback to \(\widetilde G\).

6. The determinant contradiction

We first show that \(W\) is irreducible. Otherwise its two composition factors are one-dimensional. Tensoring a composition series with \(U\), the composition factors of \(U\otimes W\) would then be one-dimensional twists of the irreducible two-dimensional module \(U\). There could be no trivial composition factor, a contradiction.

Since \(H\) is normal, the restriction \(W|_H\) is semisimple. For completeness, this form of Clifford's theorem holds in modular characteristic: choose a simple \(H\)-submodule of \(W\). Its \(\widetilde G\)-translates are simple \(H\)-submodules, and their sum is a nonzero \(\widetilde G\)-submodule, hence all of \(W\). Thus \(W|_H\) is a sum of simple \(H\)-modules.

Restriction to \(H\) preserves the existence of a trivial composition factor. But \(\overline V|_H\) is two copies of the semisimple module \(W|_H\), so \(W\) has a nonzero \(H\)-fixed vector. Since \(H\) is normal, \(W^H\) is a \(\widetilde G\)-submodule of \(W\). Irreducibility gives \(W^H=W\). Hence \(H\) acts trivially on \(\overline V\).

Write \(\chi_p:G_{\Q}\to\F_p^\times\) for the mod-\(p\) cyclotomic character. Every abelian variety over \(\Q\) has a polarization over \(\Q\). The resulting 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. Reducing modulo \(p\), the trivial action of \(H\) therefore gives

The character \(\chi_p^2\) has order \((p-1)/2\), and its fixed field is the maximal real cyclotomic field \(\Q(\zeta_p+\zeta_p^{-1})\). Consequently

For \(p=23\), this field has degree \(11\), and for \(p=29\) it has degree \(14\). Neither degree divides \([L:\Q]\), whose only prime factors are \(2\) and \(3\). This contradiction completes the proof of the theorem. ∎

7. Consequences for the census

The groups [23] and [29] are therefore impossible in the class “geometrically split, simple over \(\Q\)”, as well as in the class “split over \(\Q\)”. The argument applies to every geometric splitting degree, including degrees divisible by \(23\) or \(29\).

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.