Torsion of genus 2 Jacobians over ℚ

groups›[2,28]›impossibility proof

Rational \(7\)-torsion and the \(2\)-primary torsion of geometrically split Jacobians

11 November 2026 · Geometrically split, simple over ℚ · Principal polarization required

Theorem and scope

Theorem. Let \(A/\mathbb Q\) be a principally polarized abelian surface that is simple over \(\mathbb Q\) and geometrically split. If \(A(\mathbb Q)\) has a point of order \(7\), then \[ \#A(\mathbb Q)[2^\infty]\leq 4. \]

Corollary. No genus-\(2\) Jacobian in this class has rational torsion containing \(C_2\times C_{28}\). In particular, the exact group \([2,28]\) is impossible in the census column geometrically split, simple over \(\mathbb Q\). More generally, every group with \(7\)-torsion and \(2\)-primary subgroup of order at least eight is excluded in this class, including \([56]\) and \([2,2,14]\).

Here geometrically split means that \(A_{\overline{\mathbb Q}}\) is isogenous to a product of elliptic curves. There is no restriction on the splitting degree or the integral endomorphism ring. The principal polarization is used in the cyclic cubic case. The theorem does not exclude \([2,28]\) for Jacobians split over \(\mathbb Q\), or for geometrically simple Jacobians. The combined geometrically split existence and infinitude question for \([2,28]\) therefore remains open. The bound also leaves \([28]\) and \([2,14]\) unexcluded.

The complete proof is available as PDF and LaTeX. The exact Magma certificate and its completed log accompany the proof.

1. Conventions and the existing reduction

For a number field \(k\), put \(G_k=\operatorname{Gal}(\overline{\mathbb Q}/k)\). Write \(T_\ell\) for the Tate module, \(V_\ell=T_\ell\otimes\mathbb Q_\ell\), and \(\chi_\ell\) for the cyclotomic character. We use residual semisimplification to compare different stable lattices in one rational Tate representation; rational torsion is not moved along arbitrary isogenies.

The published rational-\(7\)-torsion and \(2\)-rank proof, especially Proposition 2, the cyclic-field reduction, and its remarks on \([2,28]\), gives the following inputs.

In these last two cases put \[ D=\operatorname{End}^0(A_{\overline{\mathbb Q}})\simeq M_2(\mathbb Q), \qquad F=\operatorname{End}^0_{\mathbb Q}(A)=\mathbb Q(\sqrt{-3}), \] and let \(L\) be the smallest field over which all geometric endomorphisms are defined. The normalization and cubic untwisting below are those proved in the order-\(63\) proof, Sections 5–7 and 10.2, and the order-\(26/39\) proof, Section 8. We show that, in either remaining case, rational \(7\)-torsion is incompatible with rational \(2\)-torsion.

2. Cyclic cubic endomorphism field: polarization at 3

Suppose \(\operatorname{Gal}(L/\mathbb Q)=C_3\). Cubic untwisting expresses the rational Tate representation as \[ V_\ell(A)\simeq V_\ell(E_0)\otimes_{\mathbb Q_\ell}W_\ell, \] where \(E_0/\mathbb Q\) is non-CM and \(W\) is the two-dimensional rational representation given by a faithful character \[ \psi:G_{\mathbb Q}\longrightarrow\mu_3\subset F^\times, \qquad \ker\psi=G_L. \] This is the construction in Section 10.2 of the order-\(63\) proof before its specialization to an elliptic isogeny class. Replacing \(\psi\) by its inverse does not affect the argument.

2.1. Principal polarization forces a rational 3-isogeny on \(E_0\)

Suppose that \(E_0[3]\) is irreducible over \(\mathbb F_3\). Let \[ \mathcal O=\mathbb Z_3[\zeta_3],\qquad \pi=\zeta_3-1, \qquad T_0=T_3(E_0)\otimes_{\mathbb Z_3}\mathcal O. \] The lattice \(T_0\) is \(G_{\mathbb Q}\)-stable, with \(G_{\mathbb Q}\) acting on its second factor by \(\psi\). Since \(\pi^2\) is a unit times \(3\), the module \(T_0/3T_0\) is an extension of \(W_0=E_0[3]\) by itself, with natural submodule \(\pi T_0/3T_0\).

This extension is nonsplit. Modulo \(\pi^2\), write \[ \psi(g)=1+c(g)\pi, \] where \(c:G_{\mathbb Q}\to\mathbb F_3\) is a nonzero additive character. A \(G_{\mathbb Q}\)-linear splitting would give an \(\mathbb F_3\)-linear endomorphism \(S\) of \(W_0\) satisfying, up to an irrelevant overall sign, \[ \rho(g)^{-1}S\rho(g)-S=c(g)I. \] Taking traces gives \(2c(g)=0\) in \(\mathbb F_3\), a contradiction. As \(W_0\) is irreducible, a nonsplit length-two self-extension has a unique proper nonzero submodule.

It follows that every \(G_{\mathbb Q}\)-stable \(\mathbb Z_3\)-lattice in this rational representation is \(\pi^jT_0\) for some integer \(j\). Indeed, scale such a lattice \(T\) by a power of \(3\) so that \(T\subseteq T_0\) but \(T\nsubseteq3T_0\). If its image modulo \(3T_0\) is the whole module, Nakayama gives \(T=T_0\). Otherwise its image is the unique proper nonzero submodule, so \(T\subseteq\pi T_0\). Then \(\pi^{-1}T\subseteq T_0\) but \(\pi^{-1}T\nsubseteq\pi T_0\). Its reduction cannot be that proper submodule, and Nakayama gives \(\pi^{-1}T=T_0\).

The space of \(G_{\mathbb Q}\)-equivariant alternating forms on \(V_3(A)\) with cyclotomic similitude is one-dimensional over \(\mathbb Q_3\). To see this, restrict first to \(G_L\). By Faltings’s theorem and \(\operatorname{End}_L(E_0)=\mathbb Z\), equivariant bilinear forms on \(V_3(E_0)\) with cyclotomic multiplier form the line spanned by the elliptic Weil pairing \(e\). Thus the alternating forms on \(V_3(A)\) have the form \(e\otimes b\), with \(b\) symmetric on the multiplicity space. Invariance under \(\mu_3\) forces \(b\) to be a scalar multiple of \[ b_0(a,b)=\operatorname{Tr}_{\mathbb Q_3(\zeta_3)/\mathbb Q_3}(a\overline b). \]

For this trace pairing, the dual of \(\mathcal O\) is the inverse different \(\pi^{-1}\mathcal O\). Since \(T_3(E_0)\) is self-dual for \(e\), the dual of \(\pi^jT_0\) under \(q(e\otimes b_0)\), for \(q\in\mathbb Q_3^\times\), is \[ \pi^{-j-1-2v_3(q)}T_0. \] Self-duality would require \[ 2j=-1-2v_3(q), \] which is impossible by parity. A principal polarization supplies exactly such a perfect alternating form on \(T_3(A)\). Hence \(E_0[3]\) is reducible, and \(E_0\) has a rational cyclic \(3\)-isogeny. The argument in fact excludes every polarization of degree prime to \(3\) in the isogeny class under the irreducibility hypothesis.

2.2. Rational 7-torsion and \(X_0(21)\)

Since \(7\) splits in \(F\), the two characters of \(W\) over \(\mathbb F_7\) are \(\psi\) and \(\psi^{-1}\). A trivial constituent coming from \(A(\mathbb Q)[7]\ne0\) forces \(E_0[7]\) to have one of these character constituents. Thus \(E_0[7]\) is reducible and \(E_0\) has a rational cyclic \(7\)-isogeny. The coprime kernels of the \(3\)- and \(7\)-isogenies give a rational cyclic \(21\)-isogeny.

The exact certificate enumerates all rational points of \(X_0(21)\), using algebraic rank zero and its torsion group \(C_2\times C_4\). Its four noncuspidal \(j\)-invariants are \[ -\frac{189613868625}{128},\qquad \frac{3375}{2},\qquad -\frac{1159088625}{2097152},\qquad -\frac{140625}{8}. \] For each, the elliptic \(2\)-division polynomial is irreducible with nonsquare discriminant, so its splitting field has Galois group \(S_3\). These properties are unchanged by quadratic twists; none of the four values is \(0\) or \(1728\).

An \(S_3\) extension has no cyclic cubic subextension, so it is linearly disjoint from \(L\). Therefore \(E_0[2]\) remains irreducible on \(G_L\). The representation \(W\) becomes trivial there, so the residual semisimplification of \(A[2]\) restricted to \(G_L\) has two copies of this nontrivial irreducible constituent and has no trivial constituent. In particular \(A(\mathbb Q)[2]=0\). This completes the cubic case.

3. Cyclic sextic field: a level-21 point with square discriminant

Suppose \(\operatorname{Gal}(L/\mathbb Q)=C_6\), and let \(K\) be its quadratic subfield. The construction in Section 8 of the order-\(26/39\) proof gives the following facts without a torsion assumption.

  1. Both \(L\) and \(K\) are totally real.
  2. After an isogeny normalizing the action of \(\mathbb Z[\zeta_3]\) and a cubic twist, one obtains \(B/\mathbb Q\) whose geometric endomorphisms are all defined over \(K\).
  3. There is \(\theta\in\operatorname{End}_{\mathbb Q}(B)\) with \(\theta^2=-3\). For each elliptic \(E_0\subset B_K\), the restriction of \(\theta\) gives a cyclic \(K\)-isogeny of degree \(3\) to its distinct conjugate \(E_0^\tau\).
  4. On \(G_K\), the original representation differs from that of \(B\) by a faithful cubic multiplicity character \(\psi\), with kernel \(G_L\).

3.1. The level-7 structure and its conjugation

Rational \(7\)-torsion supplies a one-dimensional constituent in one of the two \(7\)-adic \(F\)-blocks. The lattice argument in Proposition 9.1 of the order-\(26/39\) proof applies with \(7\) in place of \(13\). Here are the steps needed for this application.

Reversing the direction if necessary, we obtain a cyclic isogeny \(\nu:E\to E^\tau\) of degree \(3\) with \(\nu(C_7)=\tau C_7\). Its kernel and \(C_7\) define \[ x=(E,\ker\nu+C_7)\in Y_0(21)(K),\qquad \tau(x)=w_3(x). \] The last equality is the moduli definition of \(w_3\): quotient by \(\ker\nu\) sends \(E[3]/\ker\nu\) to the kernel of the conjugate degree-three isogeny, and sends \(C_7\) to \(\tau C_7\). This is the same compatibility argument used at level \(39\). The point \(x\) cannot be rational, since a \(w_3\)-fixed point would give a degree-three endomorphism of a non-CM elliptic curve.

3.2. The elliptic 2-division field

Suppose \(A(\mathbb Q)[2]\ne0\). After extension of scalars to \(\mathbb F_4\), the multiplicity characters on \(G_K\) are \(\psi\) and \(\psi^{-1}\). A trivial residual constituent forces \(E[2]\otimes\mathbb F_4\) to have one of these nontrivial cubic characters as a constituent. Its determinant over \(\mathbb F_2\) is \(1\), so its two characters are \(\psi\) and \(\psi^{-1}\). The only subgroup of \(\operatorname{GL}_2(\mathbb F_2)=S_3\) with these constituents is \(C_3\); the natural \(S_3\) module remains irreducible over \(\mathbb F_4\). The intervening \(7\)-power isogeny leaves \(E[2]\) unchanged. Thus the \(2\)-division field of \(E\) is exactly \(L/K\), and its discriminant is a square in \(K\).

For \(j(E)\notin\{0,1728\}\) the identity \[ j(E)-1728=\frac{c_6(E)^2}{\Delta(E)} \] shows that \(j(E)-1728\) is a square in \(K\). Hence \(x\) lifts to a \(K\)-point of the double cover of \(X_0(21)\) obtained by adjoining \(\sqrt{j-1728}\). The next section determines the relevant points of this genus-three cover.

4. The discriminant cover and its rank-zero quotients

Use the standard model \[ X_0(21):y^2+xy=x^3-4x-1, \qquad t=\frac{y-x-3}{x+1}. \] The certificate checks that the \(j\)-map has degree \(32\), that its four cusp pole orders are \(21,7,3,1\), and that the designated \(w_3\) satisfies \(\Phi_3(j,j\circ w_3)=0\), whereas \(w_7\) and \(w_{21}\) do not. It also checks the point counts against the Hecke traces at primes through \(43\) away from \(3\) and \(7\).

In the function field one has \[ x^2-(t^2+3t+3)x-t^2-6t-10=0, \qquad j-1728=H\cdot(\text{a rational function})^2, \qquad H=t(x+2). \] Write the cover as \(z^2=H\). The involution \(w_3\) fixes \(t\). Put \(q=t/(x+2)\). Then \[ \frac{w_3(H)}{H}=q^2,\qquad q\,w_3(q)=1. \] The two involutions lifting \(w_3\) have the elliptic quotients \[ \begin{aligned} C_+:&\quad U^2=t(t^2+5t+7),& U&=z(1+q),\\ C_-:&\quad U^2=t(t^2+t+7),& U&=z(1-q). \end{aligned} \] The identities defining these quotients are checked exactly. Both elliptic curves have rank zero by algebraic higher \(2\)-isogeny descent. Their full rational torsion groups and the corresponding finite \(t\)-values are

Curve Rational torsion Finite \(t\)-values
\(C_+\) \(C_2\) \(0\)
\(C_-\) \(C_6\) \(0,1,7\)

Each also has the point at infinity. The fibers \(t=0,\infty\) on \(X_0(21)\) are exactly its four cusps. At the remaining values the two \(x\)-coordinates are \[ \begin{array}{c|c} t&x\\\hline 1&(7\pm3\sqrt{13})/2\\ 7&(73\pm21\sqrt{13})/2. \end{array} \]

Since \(\tau(x)=w_3(x)\), conjugation of any lift to the cover agrees with one of the two lifts of \(w_3\). Its image on the corresponding elliptic quotient is rational. The complete rational point sets above therefore force \[ K=\mathbb Q(\sqrt{13}),\qquad t\in\{1,7\}. \] The corresponding \(j\)-invariants are \[ \begin{aligned} t=1:&\quad6565897584689283072\pm1821052339310297088\sqrt{13},\\ t=7:&\quad75497472\pm20938752\sqrt{13}. \end{aligned} \] They are neither \(0\) nor \(1728\). Thus every elliptic curve over \(K\) with one of these values is a quadratic twist of the model \(E_j=\operatorname{EllipticCurveFromjInvariant}(j)\).

5. Frobenius at 29

The prime \(29\) splits in \(\mathbb Q(\sqrt{13})\). For both displayed \(j\)-invariants and both primes above \(29\), the certificate checks that \(E_j\) has good reduction, its \(2\)-division polynomial splits completely, and its Frobenius trace has absolute value \(6\).

The actual \(E\) is a quadratic twist of \(E_j\), with unchanged \(2\)-division field. Therefore \(\psi\) is unramified and trivial on Frobenius at these primes: it is the faithful cubic \(2\)-division character. If the quadratic twist were ramified, inertia would act on the original mod-\(7\) representation as \(-I\), since \(E_j\) and \(\psi\) are both unramified. This contradicts the trivial constituent supplied by rational \(7\)-torsion. Hence the quadratic twist is unramified and its trace is \(a\in\{6,-6\}\).

A rational Frobenius at \(29\) belongs to \(G_K\), where the cubic multiplicity twist is now trivial. The Frobenius polynomial of \(A\) is consequently \[ P_{29}(T)=(T^2-aT+29)^2. \] The same representation is unramified, so \(A\) has good reduction at \(29\). Prime-to-\(29\) rational torsion injects into its reduction, and its order must divide \[ P_{29}(1)=(30-a)^2\in\{576,1296\}. \] Neither integer is divisible by \(7\). This contradiction completes the sextic case.

6. Conclusion

In both the cubic and sextic cases, rational \(7\)-torsion precludes rational \(2\)-torsion. The entire rational \(2\)-primary group is then trivial, because every nontrivial finite \(2\)-group has an element of order two. In all other geometric cases the existing argument bounds its order by four. This proves the theorem.

The exclusion of \([2,28]\) concerns only the Q-simple geometrically split column. Its geometrically simple example is retained, its split-over-\(\mathbb Q\) case is not settled here, and the combined geometrically split infinitude grade remains open. The same general bound excludes groups outside the census, such as cyclic \([56]\), in the Q-simple geometrically split class.

7. Verification

The standalone certificate qsimple_228_checks.m checks every new numerical input with an assertion: the modular model and \(j\)-map, cusps and Hecke traces, identification of \(w_3\), the genus of the discriminant cover, the exact quotient identities, algebraic rank-zero upper bounds, complete elliptic torsion sets, the four rational \(j\)-classes and their \(S_3\) \(2\)-division groups, and both embeddings above \(29\) for each remaining quadratic fiber. Its log ends with VERIFIED_228_MODULAR_CERTIFICATE_DONE.

The proof has been independently reviewed, and the final certificate was independently rerun. The finite-field calculations at \(29\) were also recomputed separately. The completeness statements use algebraic descent and complete torsion enumeration, with no analytic-rank or BSD assumption. No bounded search is used as a nonexistence proof.

References

  1. Rational 7-torsion and the 2-rank of geometrically split abelian surfaces, census proof of 10 November 2026: Proposition 2, the cyclic-field reduction, and the remarks on \([2,28]\).
  2. No rational 63-torsion on a geometrically split abelian surface simple over Q, census proof of 21 October 2026: Sections 5–7 and 10.2, including the general cubic untwisting construction.
  3. No rational points of order 26 or 39 on geometrically split abelian surfaces, census proof of 21 October 2026: Section 8 and the lattice and level-structure steps of Proposition 9.1. The present proof repeats the transfer at \(7\) and supplies the new modular-cover calculation.
  4. G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349–366. Published paper. We use the isogeny and Tate-module endomorphism theorems; in Section 2.1 the required consequence is \(\operatorname{End}_{G_L}(V_3(E_0))=\mathbb Q_3\) for non-CM \(E_0\).
  5. J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. 88 (1968), 492–517. Primary paper: the good-reduction criterion and prime-to-residue-characteristic torsion injection.
  6. Magma’s small modular curve and j-map tools. The model and involution conventions are checked in the accompanying script. Its TwoPowerIsogenyDescentRankBound computations are algebraic upper bounds.