Torsion of genus 2 Jacobians over ℚ

1. Inputs on elliptic torsion and reduction

By Derickx–Najman, Theorem 1.1, no elliptic curve over a quartic number field has a point of order \(35\). The same holds over fields of degree \(1\) or \(2\), by extending the ground field to degree \(4\).

We also use Clark's Theorem 5: if an abelian surface over \(\Q\) has everywhere potentially good reduction, the order of its rational torsion subgroup belongs to

None of these integers is divisible by \(35\).

2. Combining torsion primes in an elliptic isogeny class

Proof. Choose a \(K\)-isogeny \(f:E_p\to E_q\), and let \(H\) be the \(q\)-primary subgroup of \(\ker f\). Factor \(f\) as

The degree of \(a\) is a power of \(q\), so \(a\) preserves the order of the given \(p\)-torsion point. The degree of \(b\) is prime to \(q\), so its dual maps the given \(q\)-torsion point injectively into \(E'(K)\). The sum of the resulting points has order \(pq\). ∎

In the rest of the proof, suppose that \(A(\Q)\) contains a point of order \(35\). Choose nonzero points \(P_p\in A(\Q)[p]\) for \(p\in\{5,7\}\).

3. Nonisogenous factors and the CM case

Suppose first 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 \(A\), and Galois permutes them. Since \(A\) is simple over \(\Q\), this action is nontrivial. Thus the two subvarieties \(E\) and \(E^\sigma\) are defined over a quadratic field \(K\), and its nontrivial automorphism \(\sigma\) interchanges them.

Put \(F=A_K/E\). The restriction \(E^\sigma\to F\) of the quotient map is a \(K\)-isogeny. Indeed, its kernel is finite, and its image is a nonzero elliptic subvariety of \(F\). Conjugating, we find that \(F^\sigma\) is \(K\)-isogenous to \(E\).

For each \(p\in\{5,7\}\), the image of \(P_p\) in \(F(K)\) either has order \(p\) or is zero. In the first case \(F^\sigma(K)\) contains a point of order \(p\). In the second case \(P_p\in E(K)\). Thus, for each \(p\in\{5,7\}\), the \(K\)-isogeny class of \(E\) contains a curve with a \(K\)-rational point of order \(p\).

By Lemma 1, this class contains an elliptic curve over \(K\) with a point of order \(35\). This contradicts the quartic torsion classification, after a quadratic extension of \(K\). This argument also covers nonisogenous CM factors.

Suppose next 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 \(A_{\overline{\Q}}\sim E^2\), with \(E\) non-CM.

4. The endomorphism field

In this case

Let \(L/\Q\) be the smallest field over which all geometric endomorphisms of \(A\) are defined, and put \(\Gamma=\Gal(L/\Q)\). The extension \(L/\Q\) is finite Galois. For a number field \(K\subset\overline{\Q}\), write \(G_K=\Gal(\overline{\Q}/K)\). By Skolem–Noether, the faithful action of \(\Gamma\) on \(D\) identifies it with a finite subgroup of

A finite subgroup of \(\PGL_2(\mathbb R)\) is cyclic or dihedral. Indeed, its action on the hyperbolic plane fixes a point, and the 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 \(\zeta\), and

Hence \(n\in\{3,4,6\}\). It follows that

where the subscript on a dihedral group denotes its order. In particular, \(D_4=C_2\times C_2\).

5. An irreducible projective factor and the cyclotomic field

Proof. Write \(\mathcal R=\End_{\Z_p}(\Lambda)\) for a lattice \(\Lambda\subset\Q_p^2\). Let \(T=T_p(A)\), and put

Since \(\mathcal R\) is finite free over \(\Z_p\), this is a full lattice in \(V_p(A)\). It is \(G_{\Q}\)-stable, since \(T\) and \(\mathcal R\) are stable. After choosing a basis of \(\Lambda\), the matrix units of \(\mathcal R=M_2(\Z_p)\) give

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

Put \(\overline V=T'/pT'\). By independence of the semisimplified residual representation from the choice of a stable lattice, \(\overline V\) and \(T/pT=A[p]\) have the same composition factors. Since \(A(\Q)[p]\neq0\), the module \(\overline V\) has a trivial composition factor.

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

along this projective action of \(G_{\Q}\), and denote the resulting group by \(\widetilde G\). There are two-dimensional \(\widetilde G\)-modules \(U\) and \(W\) with

Explicitly, an element \((g,B)\in\widetilde G\) acts on \(U\) through \(B\). Multiplying the action of \(g\) on \(\overline V\) by \(B^{-1}\) on the first factor gives an operator commuting with \(M_2(\F_p)\); it therefore acts on the second factor and defines the action on \(W\). All these actions factor through a common finite quotient of \(\widetilde G\).

The module \(U\) is irreducible by assumption. The normal subgroup

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

If \(W\) were reducible, its two composition factors would be characters. Tensoring a composition series with \(U\), the composition factors of \(U\otimes W\) would then be irreducible two-dimensional twists of \(U\). This contradicts the existence of a trivial composition factor. Thus \(W\) is irreducible.

The restriction \(W|_H\) is semisimple. To see this in modular characteristic, choose a simple \(H\)-submodule of \(W\). Its \(\widetilde G\)-translates are simple \(H\)-submodules, by normality of \(H\). Their sum is a nonzero \(\widetilde G\)-submodule of \(W\), hence all of \(W\).

Restriction to \(H\) preserves the existence of a trivial composition factor. Since \(\overline V|_H\) is two copies of the semisimple module \(W|_H\), it follows that \(W^H\neq0\). Normality of \(H\) makes \(W^H\) a \(\widetilde G\)-submodule. Irreducibility gives \(W^H=W\). Hence \(\overline V|_{G_L}\) is the trivial four-dimensional representation.

Choose a polarization of \(A\) over \(\Q\), which exists for every abelian variety over \(\Q\). It gives an isomorphism of rational \(p\)-adic representations

Transport the lattice \(\operatorname{Hom}_{\Z_p}(T',\Z_p)\otimes_{\Z_p}\Z_p(1)\) into \(V_p(A)\) by this isomorphism. Its reduction and \(\overline V\) have the same semisimplification as \(G_L\)-modules. The latter consists of four copies of the trivial character, while the former consists of four copies of the mod-\(p\) cyclotomic character \(\chi_p|_{G_L}\). Therefore \(\chi_p|_{G_L}=1\), which is equivalent to \(\Q(\zeta_p)\subseteq L\). ∎

The polarization is used on the rational Tate module. Thus the argument applies even if its degree is divisible by \(p\). The comparison of lattices uses only their residual semisimplifications.

6. The primes \(5\) and \(7\)

Suppose first that \(\Gamma\) has a normal rotation subgroup \(\langle r\rangle\) of order \(n=3\) or \(6\). This occurs for

Normalize a matrix representative of \(r\) to have trace \(1\). The resulting matrix \(R\) has characteristic polynomial

The discriminants \(-3\) and \(-1/3\) are nonsquare units modulo \(5\). Hence \(\Q_5[R]\) is the unramified quadratic extension of \(\Q_5\), with ring of integers \(\Z_5[R]\).

An \(R\)-stable \(\Z_5\)-lattice in \(\Q_5^2\) is a rank-one \(\Z_5[R]\)-module, hence a fractional ideal of this ring. Since the extension is unramified, all these lattices are homothetic under \(\Q_5^\times\). Moreover, \(\det R\) is a unit, so a homothety class fixed by \(r\) has an \(R\)-stable representative. Thus \(r\) fixes a unique homothety class of lattices.

Conjugation by \(\Gamma\) sends \(r\) to \(r\) or \(r^{-1}\). These elements have the same invariant lattice classes. It follows that \(\Gamma\) preserves the unique class above. For a representative \(\Lambda\),

is a \(\Gamma\)-stable maximal order. Every projective element of \(\Gamma\) has a representative in \(\GL(\Lambda)\), so it acts projectively on \(\Lambda/5\Lambda\). This residual projective action is irreducible, since the reduction of \(R\) has irreducible characteristic polynomial.

Since \(A(\Q)[5]\neq0\), Lemma 2 gives \(\Q(\zeta_5)\subseteq L\). Hence \(\Gamma\) has a cyclic quotient of order \(4\). The groups \(C_3\) and \(C_6\) have orders not divisible by \(4\), while the abelianizations of \(D_6\) and \(D_{12}\) are \(C_2\) and \(C_2\times C_2\), respectively. None has a cyclic quotient of order \(4\), a contradiction.

The remaining groups with a rotation of order greater than \(2\) are \(C_4\) and \(D_8\). For a rotation of order \(4\), the trace-\(1\) representative has characteristic polynomial

Its discriminant is \(-1\), a nonsquare unit modulo \(7\). The same lattice argument, now over \(\Q_7\), produces a stable maximal order with irreducible residual projective action. Since \(A(\Q)[7]\neq0\), Lemma 2 gives \(\Q(\zeta_7)\subseteq L\). Its degree is \(6\), which divides neither \(4\) nor \(8\), again a contradiction.

We are left with

In particular, \([L:\Q]\in\{1,2,4\}\).

7. Completion of the proof

Every geometric elliptic subvariety of \(A\) is defined over \(L\), since it is the image of an integral multiple of an idempotent in \(D\). Choose one, \(E\), and put \(F=A_L/E\). The elliptic curves \(E\) and \(F\) are \(L\)-isogenous. Indeed, all endomorphisms in \(D\) are defined over \(L\), and \(D\simeq M_2(\Q)\) acts on the two-dimensional space \(\operatorname{Hom}^0(E,A_{\overline{\Q}})\) through its standard representation, which has no invariant line. Hence some \(\varphi\in\End(A_L)\) satisfies \(\varphi(E)\not\subseteq E\), and the composite \(E\hookrightarrow A_L\xrightarrow{\varphi}A_L\to F\) is a nonzero \(L\)-homomorphism of elliptic curves, that is, an \(L\)-isogeny.

For each \(p\in\{5,7\}\), the exact sequence

shows that either \(E(L)\) or \(F(L)\) contains a point of order \(p\). Thus their common \(L\)-isogeny class contains curves with rational points of orders \(5\) and \(7\). By Lemma 1, it contains an elliptic curve over \(L\) with a point of order \(35\). Since \([L:\Q]\in\{1,2,4\}\), this contradicts the quartic torsion classification. This proves the theorem. ∎

8. Consequences for the census

The groups [35] and [70] are impossible in the class “geometrically split, simple over \(\Q\)”. The theorem places no restriction on the geometric splitting degree or on the integral geometric endomorphism ring. It applies to abelian surfaces without assuming a principal polarization.

References

1. M. Derickx and F. Najman, Classification of torsion of elliptic curves over quartic fields, arXiv:2412.16016v2 (2025), Theorem 1.1.

2. 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.