Torsion of genus 2 Jacobians over ℚ

1. Conventions and inputs

All number fields are subfields of a fixed algebraic closure \(\bar\Q\). For a number field \(k\) we write \(G_k=\Gal(\bar\Q/k)\). For a prime \(\ell\), \(\chi_\ell\) is the \(\ell\)-adic cyclotomic character, \(T_\ell\) is the Tate module and \(V_\ell=T_\ell\otimes\Q_\ell\). For an abelian variety \(X\), \(\End^0=\End\otimes\Q\) and \(\Hom^0=\Hom\otimes\Q\). For a variety \(Y\) over \(\bar\Q\) and \(g\in G_{\Q}\), \(Y^g\) is the conjugate variety, \(g:Y(\bar\Q)\to Y^g(\bar\Q)\) the induced bijection, and a morphism \(f:Y\to Z\) has the conjugate \(g(f):Y^g\to Z^g\), characterized by \(g(f)(g(P))=g(f(P))\). For \(K\) a quadratic field we write \(\sigma\) for its nontrivial automorphism.

We use the following.

• (I1) Kenku–Momose [KM, Theorem (0.1)]: let \(N\) be a product of powers of \(2,3,5,7,11,13\), and let \(m\mid N\). If \(X_1(m,N)\) has positive genus and \((m,N)\) is not one of \((1,11)\), \((1,13)\), \((1,14)\), \((1,15)\), \((1,16)\), \((1,18)\), \((2,10)\), \((2,12)\), then \(Y_1(m,N)(k)=\emptyset\) for every quadratic field \(k\). The curves \(X_1(28)=X_1(1,28)\) and \(X_1(2,14)\) have genus \(10\) and \(4\). Hence for every quadratic field \(k\) and every elliptic curve \(E\) over \(k\), the group \(E(k)\) has no subgroup isomorphic to \(\Z/28\) or to \(\Z/2\times\Z/14\).

• (I2) Serre–Tate, as stated in Section 1 of the proof for order 63: let \(K\) be a finite extension of \(\Q_p\) with residue field \(k\), and \(A\) an abelian variety over \(K\). (a) Good reduction is detected by the inertia action on \(T_\ell(A)\), \(\ell\neq p\), and is preserved by \(K\)-isogenies [ST, §1, Theorem 1 and Corollary 2]. (b) If \(A\) has good reduction \(\widetilde A\) and \(m\) is prime to \(p\), then \(A(K)[m]\) injects into \(\widetilde A(k)\) [ST, Lemma 2]. (c) If \(A\) has potentially good reduction, then it has good reduction over a finite extension of \(K\) with residue field \(k\) [ST, §2]. For an abelian surface \(B\) over \(\F_q\), \(\#B(\F_q)\le(1+\sqrt q)^4\) by Weil's theorem.

• (I3) The case distinction. Let \(A/\Q\) be a geometrically split abelian surface. Exactly one of the following holds: (N) \(A_{\bar\Q}\) is isogenous to \(E_1\times E_2\) with \(E_1\), \(E_2\) not isogenous; (CM) \(A_{\bar\Q}\) is isogenous to \(E^2\) with \(E\) a CM curve; (S) \(A_{\bar\Q}\) is isogenous to \(E^2\) with \(E\) without CM. In case (S), \(D=\End^0(A_{\bar\Q})\cong M_2(\Q)\); \(L\) is the smallest field over which all geometric endomorphisms of \(A\) are defined, a finite Galois extension of \(\Q\); and \(\Gamma=\Gal(L/\Q)\) acts faithfully on \(D\), the element \(g\) acting by \(\varphi\mapsto g(\varphi)=R_g\varphi R_g^{-1}\) with \(R_g\in D^\times\) unique up to \(\Q^\times\). Thus \(\Gamma\) embeds in \(\PGL_2(\Q)\), and \(\End^0_{\Q}(A)=D^\Gamma\). An element \(r\in\Gamma\) of order \(n\in\{3,4,6\}\) has a unique representative \(R\) with \(\tr R=1\), and \(R^2-R+1/c=0\) with \(c=1,2,3\) respectively. (Sections 2 and 5 of the proof for order 63.)

• (I4) [P63, Lemma 2] Let \(X\) be an abelian surface over a number field \(k\), \(E\subset X\) an elliptic curve defined over \(k\), and \(Q\in X(k)\) of prime order \(q\). Then either \(Q\in E(k)\), or the image of \(Q\) in \((X/E)(k)\) has order \(q\).

• (I5) [P63, Lemma 3] In case (S), every elliptic curve in \(A_{\bar\Q}\) is defined over \(L\), and any two of them are \(L\)-isogenous.

• (I6) [P63, Proposition 8] Let \(A\) be of type (S) and simple over \(\Q\), and suppose that \(A(\Q)\) has a point of order \(7\). Then \(\Gamma\) is cyclic of order \(2\), \(3\) or \(6\). The standing hypothesis of that proof is a point of order \(63\), but the proof of Proposition 8 (its Section 7, together with \(\Gamma\neq C_1\) from Section 5) uses only simplicity over \(\Q\) and the point of order \(7\), through Lemmas 6 and 7 there with \(p=7\) and their common input, Lemma 5, which uses a polarization over \(\Q\) of arbitrary degree.

• (I7) Lattices and pairings. For a continuous representation of a profinite group on a finite-dimensional \(\Q_\ell\)-vector space, the composition factors of the reduction of a stable lattice do not depend on the lattice (Brauer–Nesbitt); the same holds for two isomorphic representations, after transporting a lattice through the isomorphism. For an isogeny \(u:E\to E'\) of elliptic curves and the Weil pairings \(e\) on \(V_\ell\), \(e(ux,uy)=\deg(u)\,e(x,y)\); this extends to quasi-isogenies, with \(\deg(u/m)=\deg(u)/m^2\). Moreover \(e(gx,gy)=\chi_\ell(g)\,e(x,y)\) for \(g\in G_{\Q}\), where \(g:V_\ell(E)\to V_\ell(E^g)\).

2. Elliptic curves over a quadratic field

Proof. The \(G_K\)-module \(E'[7]\) has the trivial character as a composition factor, and since its determinant is the cyclotomic character \(\bar\chi_7\), its composition factors are \(\mathbf 1\) and \(\bar\chi_7|_{G_K}\). By (I7), applied to the isomorphism \(V_7(E')\cong V_7(E'')\) induced by the \(K\)-isogeny, so are those of \(E''[7]\). Let \(C\subset E''[7]\) be a \(G_K\)-stable line. If \(G_K\) acts trivially on \(C\), put \(E'''=E''\). Otherwise \(G_K\) acts on \(C\) by \(\bar\chi_7\), hence trivially on \(E''[7]/C\); put \(E'''=E''/C\); it contains the subgroup \(E''[7]/C\) of order \(7\), whose points are \(K\)-rational because \(G_K\) acts trivially on it. In both cases \(E'''(K)\) has a point of order \(7\), and the \(K\)-isogeny \(E''\to E'''\) has degree \(1\) or \(7\), so it maps \(E''[2^\infty]\) isomorphically and \(G_K\)-equivariantly onto \(E'''[2^\infty]\). If \(E''(K)[2^\infty]\) had order at least \(4\), it would contain \(\Z/4\) or \(\Z/2\times\Z/2\), and \(E'''(K)\) would contain a subgroup isomorphic to \(\Z/28\) or to \(\Z/2\times\Z/14\), contradicting (I1). ∎

Proof.

1. A sequence. Choose \(\tilde\sigma\in G_{\Q}\) restricting to \(\sigma\) on \(K\). The curve \(E^\sigma=\tilde\sigma(E)\subset A_K\) is defined over \(K\) and different from \(E\), so \(E\cap E^\sigma\) is finite. Let \(\pi:A_K\to Y:=A_K/E^\sigma\) be the quotient map; it is defined over \(K\), and its restriction to \(E\) has the finite kernel \(E\cap E^\sigma\), so it is a \(K\)-isogeny \(E\to Y\). If \(P\in A(\Q)\) and \(\pi(P)=0\), then \(P\in E^\sigma\), hence \(P=\tilde\sigma(P)\in\tilde\sigma(E^\sigma)=E\), and \(P\in E(K)\). So the sequence

   \[ 0\to A(\Q)[2^\infty]\cap E(K)\to A(\Q)[2^\infty]\xrightarrow{\ \pi\ }Y(K)[2^\infty] \]

   is exact, and \(\#A(\Q)[2^\infty]\)\({}\le\#E(K)[2^\infty]\cdot\#Y(K)[2^\infty]\).

2. A point of order \(7\). Let \(P_7\in A(\Q)\) have order \(7\). By (I4) over \(K\), applied to \(E^\sigma\subset A_K\), either \(P_7\in E^\sigma\), and then \(P_7\in E(K)\) by step 1, or \(\pi(P_7)\in Y(K)\) has order \(7\). So \(E(K)\) or \(Y(K)\) has a point of order \(7\).

3. Since \(E\) and \(Y\) are \(K\)-isogenous, Lemma 1 gives \(\#E(K)[2^\infty]\le2\) and \(\#Y(K)[2^\infty]\le2\), and step 1 gives \(\#A(\Q)[2^\infty]\le4\). ∎

Type (N). Let \(A\) be of type (N) and simple over \(\Q\). Then \(A_{\bar\Q}\) contains exactly two elliptic curves; \(G_{\Q}\) exchanges them, since otherwise they would be defined over \(\Q\); so they are defined over a quadratic field \(K\), and \(\sigma\) exchanges them (Section 3 of the proof for order 63). Proposition 2 applies.

Type (S) with \(\Gamma=C_2\). Let \(A\) be of type (S), simple over \(\Q\), with \(\Gamma\cong C_2\). Then \(K:=L\) is quadratic, and every elliptic curve \(E\subset A_{\bar\Q}\) is defined over \(K\) by (I5). If \(E^\sigma=E\), then \(E\) would be stable under \(G_{\Q}\), hence an elliptic curve in \(A\) defined over \(\Q\), and \(A\) would not be simple over \(\Q\). So \(E^\sigma\neq E\), and Proposition 2 applies.

3. The CM case: a point count at \(3\)

Let \(A\) be of type (CM), and let \(P_7\in A(\Q)\) have order \(7\). The CM curve \(E\) has integral \(j\)-invariant and hence potentially good reduction at every prime [ST, §2]. Over a finite extension of \(\Q_3\), \(A\) becomes isogenous to \(E^2\) and \(E\) acquires good reduction; by (I2)(a), \(A\) acquires good reduction there. By (I2)(c), there is a finite extension \(K'\) of \(\Q_3\) with residue field \(\F_3\) over which \(A\) has good reduction \(\widetilde A\). The subgroup of \(A(\Q)\subseteq A(K')\) generated by \(P_7\) and \(A(\Q)[2^\infty]\) has order \(7\cdot\#A(\Q)[2^\infty]\), which is prime to \(3\); by (I2)(b) it injects into \(\widetilde A(\F_3)\). Hence

because \(16\sqrt3<28\) is equivalent to \(768<784\). So \(\#A(\Q)[2^\infty]\le4\). This case does not use simplicity over \(\Q\).

4. Type (S) with \(\Gamma\) cyclic of order \(3\) or \(6\)

In this section \(A\) is of type (S), \(\Gamma=\langle r\rangle\) is cyclic of order \(n\in\{3,6\}\), and \(R\) is the trace-one representative of \(r\), with \(R^2-R+1/c=0\) and \(c=1\) or \(3\). Put \(F=\Q[R]\subset D\). The discriminant of \(t^2-t+1/c\) is \(-3\) or \(-1/3\), so \(F\) is a field isomorphic to \(\Q(\sqrt{-3})\). Since \(R\) is not a scalar, its centralizer in \(D\) is \(F\), so \(\End^0_{\Q}(A)=D^\Gamma=F\). We write \(\bar x\) for the conjugate of \(x\in F\) and \(N(x)=x\bar x\); thus \(\bar R=1-R\) and \(N(R)=\det R=1/c>0\). The two roots \(R,\bar R\) of \(t^2-t+1/c\) are the eigenvalues of \(R\), so \(R/\bar R\) is a primitive \(n\)-th root of unity (Section 5 of the proof for order 63); explicitly \(R/\bar R=(1\pm\sqrt{-3})/2\) when \(n=6\), and \((-1\pm\sqrt{-3})/2\) when \(n=3\).

Proof.

1. The multiplicity space. Choose an elliptic curve \(E\subset A_L\); it is defined over \(L\) by (I5). Put \(H=\Hom^0(E_{\bar\Q},A_{\bar\Q})\), a \(\Q\)-vector space of dimension \(2\). Every element of \(H\) is defined over \(L\): if \(p\in\Hom^0(A_L,E)\) is a quasi-projection defined over \(L\) (that is, \(p\circ i=1\) for the inclusion \(i\); for instance the map induced by a rank-one idempotent \(e\in D\) with image \(E\), which exists by Poincaré reducibility, as the projection along a complement of \(E\), and is defined over \(L\) like every element of \(D\)), then \(h\mapsto h\circ p\) embeds \(H\) \(G_L\)-equivariantly into \(D\), on which \(G_L\) acts trivially. The algebra \(D\) acts on \(H\) by composition, faithfully because \(D\) is simple and the action is unital; so \(D=\End_{\Q}(H)\) as \(\dim_{\Q}H=2\), and \(H\) is a one-dimensional \(F\)-vector space. The evaluation map

   \[ \Phi:V_\ell(E)\otimes_{\Q}H\to V_\ell(A),\qquad v\otimes h\mapsto h(v), \]

   is an isomorphism: for a basis \(h_1,h_2\) of \(H\), a nonzero integer multiple of \((h_1,h_2):E^2\to A_{\bar\Q}\) is a homomorphism, and it is an isogeny (otherwise its image would be an elliptic curve \(E_1\), and \(h_1,h_2\) would be linearly dependent in the one-dimensional space \(\Hom^0(E,E_1)\)); \(\Phi\) is the map it induces, up to that integer. The map \(\Phi\) is \(F\)-linear, \(F\) acting on \(H\). Hence \(V_\ell(A)\cong V_\ell(E)\otimes_{\Q_\ell}(F\otimes\Q_\ell)\) is free of rank \(2\) over \(F\otimes\Q_\ell\). As \(F=\End^0_{\Q}(A)\), the action of \(G_{\Q}\) is \(F\otimes\Q_\ell\)-linear.

2. The action of \(g\in G_{\Q}\). The curve \(E^g\subset A_L\) is \(L\)-isogenous to \(E\) by (I5); choose an invertible \(u\in\Hom^0(E,E^g)\). Define \(B:V_\ell(E)\to V_\ell(E)\) by \(B(v)=u^{-1}(g(v))\) and \(C:H\to H\) by \(C(h)=g(h)\circ u\). For \(v\in V_\ell(E)\) and \(h\in H\),

   \[ g\bigl(h(v)\bigr)=g(h)\bigl(g(v)\bigr)=g(h)\bigl(u(Bv)\bigr)=C(h)(Bv), \]

   so \(g\circ\Phi=\Phi\circ(B\otimes C)\). For \(\varphi\in D\) we have \(C(\varphi\circ h)=g(\varphi)\circ C(h)\); so \(C\in\GL(H)=D^\times\) satisfies \(g(\varphi)=C\varphi C^{-1}\). In particular \(C\) commutes with \(F=D^\Gamma\), so \(C\) is multiplication by an element \(\gamma\in F^\times\), and

   \[ \delta_\ell(g)=\det(B)\,\gamma^2. \]

   By (I7), \(e(Bv,Bw)\)\({}=\deg(u)^{-1}e(g(v),g(w))\)\({}=\deg(u)^{-1}\chi_\ell(g)\,e(v,w)\) for the Weil pairing \(e\) of \(E\), so \(\det B=\chi_\ell(g)/\deg(u)\). Hence

   \[ \delta_\ell(g)=\chi_\ell(g)\,\varepsilon(g),\qquad \varepsilon(g):=\gamma^2/\deg(u)\in F^\times. \]

   The element \(\varepsilon(g)=\delta_\ell(g)/\chi_\ell(g)\) does not depend on the choice of \(u\), and its definition does not involve \(\ell\).

3. \(\varepsilon\) is a character of \(\Gamma\). It is a homomorphism, being a quotient of two homomorphisms to the commutative group \((F\otimes\Q_\ell)^\times\). For \(g\in G_L\) we may take \(u=1\), since \(E^g=E\); then \(C(h)=g(h)=h\) for all \(h\in H\), so \(\gamma=1\) and \(\varepsilon(g)=1\). Hence \(\varepsilon\) factors through \(\Gamma\), and its values are roots of unity in \(F\), of norm \(1\).

4. The value at \(r\). Let \(g\) restrict to \(r\). Both \(C\) and \(R\) induce the action of \(g\) on \(D\), so \(\gamma=\lambda R\) with \(\lambda\in\Q^\times\) (Skolem–Noether). Then \(\varepsilon(g)=\mu R^2\) with \(\mu=\lambda^2/\deg(u)>0\), and \(1=N(\varepsilon(g))=\mu^2N(R)^2\) gives \(\mu=1/N(R)\). So \(\varepsilon(g)=R^2/(R\bar R)=R/\bar R\). ∎

Step 4 uses \(N(R)=1/c>0\). For \(\Gamma=C_2\) the same computation, with \(R^2=d\) and \(N(R)=-d\), gives \(\varepsilon=1\) when \(\End^0_{\Q}(A)=\Q(\sqrt d)\) is real; this is one reason why \(\Gamma=C_2\) is treated by Proposition 2 instead.

Proof. Take \(\ell=2\). Since \(-3\equiv5\pmod8\), the prime \(2\) is inert in \(F\), so \(F\otimes\Q_2\) is the unramified quadratic extension of \(\Q_2\); let \(\mathcal O\) be its ring of integers, with residue field \(\F_4\). The lattice \(T'=\mathcal O\,T_2(A)\subset V_2(A)\) is \(G_{\Q}\)-stable and is a free \(\mathcal O\)-module of rank \(2\). So \(M=T'/2T'\) is a two-dimensional \(\F_4\)-vector space on which \(G_{\Q}\) acts \(\F_4\)-linearly, with \(\F_4\)-determinant the reduction of \(\delta_2=\chi_2\varepsilon\) modulo \(2\mathcal O\). As \(\chi_2\) takes values in \(1+2\Z_2\), this determinant is the reduction \(\bar\varepsilon\) of \(\varepsilon\). By Lemma 3, \(\varepsilon(r)^2\) is a primitive cube root of unity \(\zeta\), and \(\zeta\not\equiv1\pmod{2\mathcal O}\) because \(N(1-\zeta)=3\) is a unit at \(2\). So \(\bar\varepsilon\neq1\).

Suppose that all composition factors of \(A[2]=T_2(A)/2T_2(A)\) are trivial. By (I7), applied to the lattices \(T_2(A)\) and \(T'\), all composition factors of the \(\F_2[G_{\Q}]\)-module \(M\) are trivial. In a basis adapted to a composition series, the image of \(G_{\Q}\) in \(\GL_{\F_2}(M)\) consists of unipotent upper triangular matrices, so it is a \(2\)-group. Its image under the \(\F_4\)-determinant is then a \(2\)-group in \(\F_4^\times\), which has order \(3\); so \(\bar\varepsilon=1\), a contradiction.

If \(A(\Q)[2]\) had rank at least \(3\), then \(G_{\Q}\) would act trivially on a three-dimensional subspace of \(A[2]\) and, since \(\GL_1(\F_2)=1\), on the one-dimensional quotient; all composition factors of \(A[2]\) would be trivial. ∎

Proposition 4 uses neither a point of order \(7\) nor simplicity over \(\Q\).

5. Proof of the Theorem and consequence for the census

Proof of the Theorem. Let \(A/\Q\) be simple over \(\Q\) and geometrically split, with a rational point of order \(7\). By (I3), \(A\) is of type (N), (CM) or (S). In type (N), \(\#A(\Q)[2^\infty]\le4\) by Section 2; in type (CM), by Section 3. In type (S), \(\Gamma\) is cyclic of order \(2\), \(3\) or \(6\) by (I6). If \(\Gamma=C_2\), then \(\#A(\Q)[2^\infty]\le4\) by Section 2; if \(\Gamma\) has order \(3\) or \(6\), then \(A(\Q)[2]\) has rank at most \(2\) by Proposition 4. In all cases \(A(\Q)[2]\) has rank at most \(2\). ∎

Proof of the Corollary. The Jacobian of a curve of genus \(2\) over \(\Q\) is an abelian surface over \(\Q\), and \((\Z/2)^3\times\Z/7\) has a point of order \(7\) and \(2\)-rank \(3\). For Jacobians split over \(\Q\), the \(2\)-rank obstruction shows that a rational point of order \(7\) forces \(2\)-rank at most \(2\). ∎

The group [2,2,14] is impossible in the class “geometrically split, simple over \(\Q\)”. It was already impossible in the class “split over \(\Q\)”. Hence the combined geometrically split cell of \([2,2,14]\), whose grade was open (“\(\infty\)?” on the census), becomes impossible (\(\emptyset\)).

The argument places no restriction on the geometric splitting degree, the field of definition of the geometric endomorphisms, or the integral endomorphism ring, and it does not use a principal polarization.

6. Remarks

1. The group \([2,28]\). In types (N) and (CM), and in type (S) with \(\Gamma=C_2\), the proof gives \(\#A(\Q)[2^\infty]\le4\) when \(A(\Q)\) has a point of order \(7\). So a geometrically split abelian surface over \(\Q\), simple over \(\Q\), whose rational torsion contains \(\Z/2\times\Z/28\) is of type (S) with \(\Gamma\) cyclic of order \(3\) or \(6\). These two cases are not settled by this argument. Update, 11 November 2026: the new principal-polarization argument closes both cases and excludes [2,28] for geometrically split Jacobians that are simple over \(\Q\). Its split-over-\(\Q\) case remains open.

2. Modular surfaces. If \(A\) is the abelian surface \(A_f\) attached to a weight-\(2\) newform \(f\) whose coefficient field is \(\End^0_{\Q}(A)\), then \(\delta_\ell\) is the cyclotomic character times the nebentypus of \(f\), so \(\varepsilon\) corresponds to the nebentypus or to its inverse, depending on the identification of \(F\) with the coefficient field; in particular both have order \(|\Gamma|\). For example, \(J_1(13)\) and \(J_1(18)\) are the surfaces of the newforms 13.2.e.a and 18.2.c.a, with nebentypus of order \(6\) and \(3\); their Sato–Tate groups in the LMFDB (curves 169.a.169.1 and 324.a.648.1) are \(E_6\) and \(E_3\), whose component groups \(\Gamma\) are cyclic of order \(6\) and \(3\). This is consistent with Lemma 3, and is not used in the proof.

3. The point of order \(7\) is needed. Without it, rational \(2\)-rank \(3\) occurs on \(\Q\)-simple, geometrically split Jacobians of type (N), of type (CM), and of type (S) with \(\Gamma=C_2\): the LMFDB curves 2560.a.819200.1, 4096.b.65536.1 and 25600.f.512000.1 have rational torsion subgroup \(\Z/2\times\Z/2\times\Z/2\) and Sato–Tate groups \(N(\mathrm{SU}(2)\times \mathrm{SU}(2))\), \(J(C_2)\) and \(J(E_1)\). By Proposition 4, this cannot happen when \(\Gamma\) is cyclic of order \(3\) or \(6\).

References

1. [KM] M. A. Kenku and F. Momose, Torsion points on elliptic curves defined over quadratic fields, Nagoya Math. J. 109 (1988), 125–149, Theorem (0.1). doi:10.1017/S0027763000002816.

2. [P63] Census proof of 21 October 2026, An obstruction to rational \(63\)-torsion on geometrically split abelian surfaces, https://genus-2-torsion.github.io/proofs/qsimple-63-obstruction.html: Lemmas 2, 3 and 5–7, Proposition 8, and Sections 1, 2, 3, 5, 6 and 7.

3. [ST] J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517: Theorem 1 and Corollary 2 (p. 493), Lemma 2 (p. 495), §2 (pp. 496 and 498). doi:10.2307/1970722.