Torsion of genus 2 Jacobians over ℚ

1. Inputs

We use the following results on elliptic curves.

1. Mazur, Theorem (8). The torsion subgroup of \(E(\Q)\), for an elliptic curve \(E/\Q\), is \(\Z/m\) with \(m\leq10\) or \(m=12\), or \(\Z/2\times\Z/2\nu\) with \(\nu\leq4\). In particular, no elliptic curve over \(\Q\) has a rational point of order \(11\).

2. Kenku–Momose, 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

   \[ \begin{aligned}(m,N)\notin\{&(1,11),(1,13),(1,14),(1,15),\\ &(1,16),(1,18),(2,10),(2,12)\},\end{aligned} \]

   then \(Y_1(m,N)(k)=\emptyset\) for every quadratic field \(k\). We apply this with \((m,N)=(1,22)\). The curve \(X_1(22)\) has genus \(1+180/12-20/2=6\): the group \(\pm\Gamma_1(22)\) has index \(180\) in \(\mathrm{SL}_2(\Z)\), and \(X_1(22)\) has \(20\) cusps and no elliptic points. Hence no elliptic curve over a quadratic field has a rational point of order \(22\). This agrees with the complete list of torsion groups over quadratic fields, as restated in Le Fourn–Najman, Theorem 2.1.

3. Le Fourn–Najman, Theorem 1.1. A \(\Q\)-curve is an elliptic curve over a number field that is isogenous over \(\overline{\Q}\) to all of its \(\Gal(\overline{\Q}/\Q)\)-conjugates; curves with complex multiplication are allowed. If \(E\) is a \(\Q\)-curve over a quadratic field \(K\), then \(E(K)\) has no point of order \(11\). Indeed, no group in the list of their Theorem 1.1 has an element of order \(11\). The key case is Le Fourn–Najman, Proposition 3.2, which excludes the torsion group \(\Z/11\); by Le Fourn–Najman, Theorem 2.1, it is the only torsion group over a quadratic field that contains a point of order \(11\).

4. Bourdon–Clark, Theorem 4.1. Let \(\mathcal O\) be an order of conductor \(f\) in an imaginary quadratic field \(M\) (denoted \(K\) there), and put \(w=\#\mathcal O^\times\). Let \(M(f)\) be the ring class field of conductor \(f\). It equals \(M(j(E))\) for every elliptic curve \(E\) with complex multiplication by \(\mathcal O\). By part (a) of the theorem, there is a positive integer \(T(\mathcal O,1,N)\) with the following property. For every \(d\geq1\), there exist a number field \(F\supseteq M(f)\) with \([F:M(f)]=d\) and an \(\mathcal O\)-CM elliptic curve \(E/F\) with an \(F\)-rational point of order \(N\) if and only if \(T(\mathcal O,1,N)\mid d\). Parts (d) and (e) compute this integer for a prime \(N=\ell\geq5\); we take \(a=0\) and \(b=1\) there. Then \(T(\mathcal O,1,\ell)=\widetilde T/w\), where \(\widetilde T=\ell^2-1\) if \(\ell\nmid f\) and \(\ell\) is inert in \(M\) (part (d)(i)). In each of the remaining cases (d)(ii)–(v), \(\widetilde T=\ell-1\).

   For \(\ell=11\), we have \(\widetilde T\in\{120,10\}\). If \(w\neq2\), then \(\mathcal O=\Z[i]\) or \(\mathcal O=\Z[\zeta_3]\). Then \(f=1\), and \(11\) is inert in \(M\), because \(11\equiv3\pmod4\) and \(11\equiv2\pmod3\). These two orders give \(T=120/4=30\) and \(T=120/6=20\). In all other cases \(w=2\) and \(T\in\{60,5\}\). Every value is divisible by \(5\). Consequently: if an elliptic curve \(E\) with complex multiplication by an order in \(M\) is defined over a number field \(F\supseteq M(j(E))\) and has an \(F\)-rational point of order \(11\), then \(5\mid[F:M(j(E))]\).

We also use two standard facts about \(p\)-adic representations.

5. Lattice independence. Let \(T_1,T_2\) be stable \(\Z_p\)-lattices in a finite-dimensional \(\Q_p\)-representation of a group. Then \(T_1/pT_1\) and \(T_2/pT_2\) have the same composition factors. Indeed, the characteristic polynomial of each group element on \(T_i/pT_i\) is the reduction of its characteristic polynomial on the rational representation, and the Brauer–Nesbitt theorem applies. The statement also applies to isomorphic representations, after transporting a lattice through the isomorphism.

6. Determinant. Every abelian variety over \(\Q\) has a polarization defined over \(\Q\). It induces a nondegenerate alternating \(G_{\Q}\)-equivariant pairing \(V_p(A)\times V_p(A)\to\Q_p(1)\). Hence the determinant of \(G_{\Q}\) acting on \(V_p(A)\) is the square of the \(p\)-adic cyclotomic character. Therefore, for every \(G_{\Q}\)-stable lattice \(T'\subset V_p(A)\), the determinant of \(G_{\Q}\) on \(T'/pT'\) is \(\chi_p^2\). The kernel of \(\chi_p^2\) is \(G_{\Q(\zeta_p+\zeta_p^{-1})}\). For \(p=11\), this field has degree \(5\) over \(\Q\).

2. Coprime torsion in an elliptic isogeny class

Lemma 1. Let \(E_1,E_2\) be \(K\)-isogenous elliptic curves over a number field \(K\). Suppose that \(E_1(K)\) contains a point of prime order \(r\), and \(E_2(K)\) contains a point of a distinct prime order \(s\). Then their \(K\)-isogeny class contains an elliptic curve with a \(K\)-rational point of order \(rs\).

Proof. Choose a \(K\)-isogeny \(f:E_1\to E_2\). Let \(N_s\) be the \(s\)-primary subgroup scheme of \(\ker(f)\), and factor \(f\) as

The degree of \(q\) is a power of \(s\), so \(q\) maps the given point of order \(r\) to a point of order \(r\) in \(E'(K)\). The degree of \(f'\) is prime to \(s\), so \(f'\) induces a Galois-equivariant isomorphism \(E'[s]\simeq E_2[s]\). Thus \(E'(K)\) also contains a point of order \(s\). The sum of the two points has order \(rs\). ∎

3. Elliptic subvarieties and the endomorphism field

Let \(A/\Q\) be an abelian surface. Put \(D=\End^0(A_{\overline{\Q}})=\End(A_{\overline{\Q}})\otimes\Q\), and let \(L\) be the smallest field over which all endomorphisms of \(A_{\overline{\Q}}\) are defined. It is a finite Galois extension of \(\Q\). The group \(\Gamma=\Gal(L/\Q)\) acts faithfully on \(D\) by \(\Q\)-algebra automorphisms, \(\varphi\mapsto\varphi^\sigma\). For \(\sigma\in G_{\Q}\), \(\varphi\in\End(A_{\overline{\Q}})\), and \(x\in A(\overline{\Q})\), we have \(\sigma(\varphi(x))=\varphi^\sigma(\sigma(x))\).

Lemma 2. (a) Every abelian subvariety \(B\) of \(A_{\overline{\Q}}\) is the image of an endomorphism \(\varepsilon\) of \(A_{\overline{\Q}}\) with \(\varepsilon|_B=[n]\) for some \(n\geq1\). In particular, \(B\) is defined over \(L\). (b) If \(B_1,B_2\) are abelian subvarieties of \(A_{\overline{\Q}}\), then every homomorphism \(B_1\to B_2\) over \(\overline{\Q}\) is defined over \(L\).

Proof. (a) By Poincaré's complete reducibility theorem, there is an abelian subvariety \(B'\) with \(B+B'=A_{\overline{\Q}}\) and \(B\cap B'\) finite. The addition map \(\mu:B\times B'\to A_{\overline{\Q}}\) is an isogeny. Choose an isogeny \(\nu\) in the other direction and \(n\geq1\) with \(\mu\circ\nu=[n]\) and \(\nu\circ\mu=[n]\). Put \(\varepsilon=\iota_B\circ\mathrm{pr}_1\circ\nu\), where \(\iota_B\) is the inclusion. For \(b\in B\) we get \(\varepsilon(b)=\mathrm{pr}_1(\nu(\mu(b,0)))=nb\). Thus \(\varepsilon(A_{\overline{\Q}})=B\). Since \(\varepsilon\) is defined over \(L\), every \(\sigma\in G_L\) satisfies \(\sigma(B)=\varepsilon^\sigma(A_{\overline{\Q}})=B\). By Galois descent, \(B\) is defined over \(L\).

(b) Let \(\varepsilon_1\) be as in (a) for \(B_1\), and let \(\iota_2\) be the inclusion of \(B_2\). For a homomorphism \(\varphi:B_1\to B_2\), the endomorphism \(\psi=\iota_2\circ\varphi\circ\varepsilon_1\) of \(A_{\overline{\Q}}\) is fixed by \(G_L\). Since \(\iota_2\) and \(\varepsilon_1\) are defined over \(L\), every \(\sigma\in G_L\) satisfies \(\iota_2\circ\varphi^\sigma\circ\varepsilon_1=\psi^\sigma=\iota_2\circ\varphi\circ\varepsilon_1\). As \(\varepsilon_1\) maps onto \(B_1\) and \(\iota_2\) is injective, \(\varphi^\sigma=\varphi\). ∎

The case of an elliptic curve over \(\Q\). Suppose that \(P\in A(\Q)\) has order \(11\), and that \(A\) contains an elliptic curve \(E\) defined over \(\Q\). Put \(F=A/E\), an elliptic curve over \(\Q\). Either \(P\in E(\Q)\), or \(P\) maps to a point of order \(11\) in \(F(\Q)\). Both possibilities contradict Mazur's theorem. Hence \(A(\Q)[11]\neq0\) implies that \(A\) is simple over \(\Q\).

Splitting types. Now let \(A\) be geometrically split, so that \(A_{\overline{\Q}}\) is isogenous to \(E_1\times E_2\). Exactly one of the following holds.

• (N) \(E_1\) and \(E_2\) are not isogenous over \(\overline{\Q}\).
• (S) \(E_1\) and \(E_2\) are isogenous to an elliptic curve \(E\) without complex multiplication. Then \(D\simeq M_2(\Q)\).
• (CM) \(E_1\) and \(E_2\) are isogenous to an elliptic curve \(E\) with complex multiplication by an order in an imaginary quadratic field \(M\). Then \(D\simeq M_2(M)\), with centre \(M\).

In cases (S) and (CM), every elliptic subvariety and every elliptic quotient of \(A_{\overline{\Q}}\) is isogenous to \(E\): a nonzero homomorphism between such a curve and \(E^2\) gives a nonzero homomorphism between that curve and \(E\). Sections 4–8 show that \(A(\Q)[11]=0\) in case (S), and Section 9 shows the same in case (CM). Section 10 treats case (N).

4. The endomorphism field of a non-CM square

Assume case (S) and fix an isomorphism \(D\simeq M_2(\Q)\). By the Skolem–Noether theorem, each \(\gamma\in\Gamma\) acts on \(D\) as conjugation by a matrix \(S_\gamma\in\GL_2(\Q)\), unique up to \(\Q^\times\). Since the action is faithful, \(\gamma\mapsto[S_\gamma]\) embeds \(\Gamma\) into \(\PGL_2(\Q)\).

Lemma 3. The group \(\Gamma\) is cyclic of order \(1,2,3,4\), or \(6\), or dihedral of order \(4,6,8\), or \(12\). We write these groups as

where the subscript denotes the group order. The group \(D_4\) is the Klein four group.

Proof. A finite subgroup of \(\PGL_2(\R)\) fixes a point of the hyperbolic plane, and its action on the tangent plane at that point embeds it in \(O(2)\). Hence it is cyclic or dihedral.

Let \([S]\in\PGL_2(\Q)\) have finite order \(n>2\). If \(S\) had a repeated eigenvalue \(\lambda\), then \(S=\lambda(I+N)\) with \(N\) nilpotent and nonzero, and \(S^n=\lambda^n(I+nN)\) would not be scalar. Hence \(S\) has distinct eigenvalues \(\lambda_1,\lambda_2\). The order of \([S]\) equals the order of \(\zeta=\lambda_1/\lambda_2\), so \(\zeta\) is a primitive \(n\)-th root of unity, and

The number \(\zeta+\zeta^{-1}\) has degree \(\varphi(n)/2\) over \(\Q\), so \(\varphi(n)=2\) and \(n\in\{3,4,6\}\). A cyclic group of order \(n\), and a dihedral group of order \(2n\), contain elements of order \(n\). This gives the list. ∎

If \(\Gamma\in\{C_3,C_4,C_6,D_6,D_8,D_{12}\}\), let \(\langle\rho\rangle\) be its rotation subgroup, of order \(n\in\{3,4,6\}\). For a cyclic group it is the whole group; for a dihedral group it is the cyclic subgroup of index \(2\). Every element of \(\Gamma\) conjugates \(\rho\) to \(\rho\) or to \(\rho^{-1}\).

From now on until the end of Section 8, set \(p=11\), \(k=\Q_{11}\), \(\mathcal O=\Z_{11}\), \(T=T_{11}(A)\), and \(V=V_{11}(A)\). The isomorphism \(D\simeq M_2(\Q)\) extends to \(D\otimes_{\Q}k\simeq M_2(k)\), which acts on \(V\). For \(\sigma\in G_{\Q}\) with image \(\gamma\in\Gamma\), \(x\in D\otimes_{\Q}k\), and \(v\in V\), we have

5. A residual tensor lemma

Lemma 4. Assume case (S), and suppose that \(A(\Q)\) contains a point \(P\) of order \(11\). Suppose that there is an \(\mathcal O\)-lattice \(\Lambda\subset k^2\) with the following two properties.

1. For every \(\gamma\in\Gamma\), we have \(S_\gamma\Lambda=\lambda\Lambda\) for some \(\lambda\in k^\times\).
2. Some \(\gamma_0\in\Gamma\) has a representative \(S\in k^\times S_{\gamma_0}\) with \(S\Lambda=\Lambda\) whose reduction \(\overline S\in\GL(\Lambda/11\Lambda)\) has no eigenvector.

Then \(\Q(\zeta_{11}+\zeta_{11}^{-1})\subseteq L\). In particular, \(5\) divides \([L:\Q]\).

Proof. Let \(\mathcal R=\End_{\mathcal O}(\Lambda)\simeq M_2(\mathcal O)\), a maximal order in \(M_2(k)\). By (i), \(S_\gamma\mathcal RS_\gamma^{-1}=\End_{\mathcal O}(S_\gamma\Lambda)=\mathcal R\) for every \(\gamma\). Put \(T'=\mathcal RT\). This is a finitely generated \(\mathcal O\)-submodule of \(V\) containing \(T\), hence a full lattice, and it is \(G_{\Q}\)-stable by (1).

Let \(e_{ij}\) be the matrix units of \(\mathcal R\simeq M_2(\mathcal O)\) with respect to a basis of \(\Lambda\), and put \(W_0=e_{11}T'\). Then \(T'=e_{11}T'\oplus e_{22}T'\), and \(e_{21}:e_{11}T'\to e_{22}T'\) is an isomorphism with inverse \(e_{12}\). Hence \(T'\simeq\Lambda\otimes_{\mathcal O}W_0\) as \(\mathcal R\)-modules, with \(W_0\) free of rank \(2\). Reducing modulo \(11\), we obtain

where \(\mathcal R/11\mathcal R=\End(U)\) acts on the first factor.

By (i), each \(\sigma\in G_{\Q}\), with image \(\gamma\in\Gamma\), has a representative \(S_\sigma\in k^\times S_\gamma\) with \(S_\sigma\Lambda=\Lambda\). It is unique up to \(\mathcal O^\times\), so its reduction \(\overline S_\sigma\in\GL(U)\) is well defined up to \(\F_{11}^\times\). By (1), \(\sigma(xv)=(\overline S_\sigma x\overline S_\sigma^{-1})\sigma(v)\) for \(x\in\End(U)\) and \(v\in\overline V\).

The modules \(\overline V\) and \(A[11]=T/11T\) are reductions of two \(G_{\Q}\)-stable lattices in \(V\). By lattice independence, they have the same composition factors. The point \(P\) spans a trivial submodule of \(A[11]\), so \(\overline V\) has a trivial \(G_{\Q}\)-composition factor.

Consider the group

which acts on \(\overline V\) through \(\sigma\), and on \(U\) through \(\beta\). Its composition factors on \(\overline V\) are the same as those of \(G_{\Q}\). For \((\sigma,\beta)\in\widetilde G\), the map \((\beta^{-1}\otimes1)\circ\sigma\) on \(\overline V\) commutes with \(\End(U)\otimes1\). Hence it equals \(1\otimes w(\sigma,\beta)\) for a unique \(w(\sigma,\beta)\in\GL(\overline W_0)\). Then \(\sigma\) acts on \(\overline V\) as \(\beta\otimes w(\sigma,\beta)\), and \(w\) is a homomorphism. Writing \(W\) for \(\overline W_0\) with this action, we get an isomorphism of \(\widetilde G\)-modules

The module \(U\) is irreducible. Indeed, for any \(\sigma_0\in G_{\Q}\) with image \(\gamma_0\), the element \((\sigma_0,\overline S)\) lies in \(\widetilde G\), and \(\overline S\) has no eigenvector by (ii). This is the point where the projective Galois action on \(U\) is used, and not only the structure of an order.

The module \(W\) is also irreducible. Otherwise its composition factors are two characters \(\psi_1,\psi_2\), and the composition factors of \(U\otimes W\) are the irreducible two-dimensional modules \(U\otimes\psi_1\) and \(U\otimes\psi_2\). This contradicts the trivial composition factor of \(\overline V\).

The subgroup \(H=\{(h,1):h\in G_L\}\) is normal in \(\widetilde G\) and acts trivially on \(U\). The restriction \(W|_H\) is semisimple by Clifford's theorem. Explicitly, the \(\widetilde G\)-translates of a simple \(H\)-submodule of \(W\) are simple \(H\)-submodules, and their sum is a nonzero \(\widetilde G\)-submodule, hence equal to \(W\). Now \(\overline V|_H\simeq W|_H\oplus W|_H\) has a trivial composition factor. Thus the semisimple module \(W|_H\) contains a trivial submodule, so \(W^H\neq0\). Since \(H\) is normal, \(W^H\) is a \(\widetilde G\)-submodule, so \(W^H=W\). Hence \(G_L\) acts trivially on \(U\otimes W=\overline V\).

By fact (f) of Section 1, the determinant of \(G_{\Q}\) on \(\overline V\) is \(\chi_{11}^2\). Therefore \(\chi_{11}^2|_{G_L}=1\), and \(\Q(\zeta_{11}+\zeta_{11}^{-1})\subseteq L\). This field has degree \(5\). ∎

6. Rotations of order 3, 4, and 6

Assume case (S), let \(P\in A(\Q)\) have order \(11\), and suppose that \(\Gamma\in\{C_3,C_4,C_6,D_6,D_8,D_{12}\}\), with rotation \(\rho\) of order \(n\in\{3,4,6\}\). Choose a representative \(R\in\GL_2(\Q)\) of \(\rho\). If \(\operatorname{tr}R=0\), the Cayley–Hamilton theorem gives \(R^2=-\det R\), so \(\rho^2=1\). Hence \(\operatorname{tr}R\neq0\), and we normalize \(R\) to have trace \(1\). By the proof of Lemma 3, \(1/\det R=2+\zeta_n+\zeta_n^{-1}\), which equals \(1,2,3\) for \(n=3,4,6\). Thus the characteristic polynomial of \(R\) is

with discriminant \(-3\), \(-1\), \(-1/3\), respectively. Modulo \(11\), these are \(8\), \(10\), and \(7\), since \(3^{-1}\equiv4\). The nonzero squares modulo \(11\) are \(1,3,4,5,9\). Hence none of the three discriminants is a square modulo \(11\). Since \(1/c\) is a unit, the polynomial is irreducible modulo \(11\) with unit discriminant. So \(k'=k[R]\) is the unramified quadratic extension of \(k\), and \(\mathcal O[R]\) is its ring of integers.

Stable lattices. The plane \(k^2\) is a one-dimensional \(k'\)-vector space. An \(\mathcal O[R]\)-stable lattice is a finitely generated \(\mathcal O[R]\)-submodule spanning it, hence of the form \(\mathcal O[R]u\). Since \(11\) is a uniformizer of \(\mathcal O[R]\), any two such lattices differ by a power of \(11\). Fix one, \(\Lambda\).

Property (i). Let \(\gamma\in\Gamma\). Then \(S_\gamma RS_\gamma^{-1}\) is a scalar multiple of \(R\) or of \(R^{-1}=(1-R)/\det R\). It has trace \(1\), so it equals \(R\) or \(1-R\). Hence \(S_\gamma\) normalizes \(\mathcal O[R]\). Thus \(S_\gamma\Lambda\) is \(\mathcal O[R]\)-stable, and so it is a power of \(11\) times \(\Lambda\).

Property (ii). Take \(\gamma_0=\rho\) and \(S=R\). Then \(R\Lambda\subseteq\Lambda\), and \(\det R=1/c\) is a unit, so \(R\Lambda=\Lambda\). The reduction \(\overline R\) has the irreducible characteristic polynomial \(X^2-X+1/c\) over \(\F_{11}\), so it has no eigenvector.

Lemma 4 now gives \(5\mid|\Gamma|\). This is impossible for groups of order \(3,4,6,8,12\). Hence \(\Gamma\in\{C_1,C_2,D_4\}\).

7. The Klein four group

Assume case (S), let \(P\in A(\Q)\) have order \(11\), and suppose that \(\Gamma=D_4\simeq C_2\times C_2\). The map

is well defined, because rescaling \(S_\gamma\) by \(t\) multiplies the determinant by \(t^2\). It is a homomorphism. Since \(|\Gamma|=4\), its kernel contains a nonidentity element \(r\). Choose a representative \(R\in\GL_2(\Q)\) of \(r\). Since \(r^2=1\), we have \(R^2=aI\) with \(a\in\Q^\times\). As \(R\) is not scalar, its minimal and characteristic polynomials equal \(X^2-a\). Hence \(\operatorname{tr}R=0\) and \(\det R=-a\), so \(v_{11}(a)\) is even. Replacing \(R\) by \(11^{-v_{11}(a)/2}R\), we may assume that \(a\) is an \(11\)-adic unit. Put

Since \(\Gamma\) is abelian, \(S_\gamma RS_\gamma^{-1}=\mu R\) with \(\mu\in\Q^\times\) for every \(\gamma\in\Gamma\). Squaring gives \(aI=\mu^2aI\), so \(\mu=\pm1\). We write \(\mu=\epsilon(\gamma)\). This defines a homomorphism \(\epsilon:\Gamma\to\{\pm1\}\) with \(\epsilon(r)=1\). In particular, every \(S_\gamma\) normalizes \(\mathcal B\).

The nonsplit case. Suppose that \(a\) is not a square modulo \(11\). Then \(X^2-a\) is irreducible modulo \(11\) with unit discriminant \(4a\), and \(\mathcal B\) is the ring of integers of the unramified quadratic extension of \(k\). The argument of Section 6 applies verbatim. All \(\mathcal B\)-stable lattices are powers of \(11\) times one of them, \(\Lambda\). Each \(S_\gamma\) maps \(\Lambda\) to a \(\mathcal B\)-stable lattice. The representative \(R\) of \(r\) satisfies \(R\Lambda=\Lambda\), since \(\det R=-a\) is a unit. Its reduction has the irreducible characteristic polynomial \(X^2-\overline a\). Lemma 4 gives \(5\mid4\), a contradiction.

The split case. Suppose that \(a\) is a nonzero square modulo \(11\). By Hensel's lemma, \(a=\alpha^2\) with \(\alpha\in\mathcal O^\times\). Put

These are orthogonal idempotents in \(\mathcal B\) with \(e_1+e_2=1\) and \(R=\alpha(e_1-e_2)\), so \(\mathcal B\simeq\mathcal O\times\mathcal O\). In a basis of \(k^2\) consisting of eigenvectors of \(R\), we have \(e_1=E_{11}\) and \(e_2=E_{22}\). Put \(X=E_{12}\) and \(Y=E_{21}\) in this basis. Then

The character \(\epsilon\) is nontrivial. Otherwise every \(S_\gamma\) commutes with \(R\), and is therefore diagonal in the eigenbasis. The image of the diagonal torus in \(\PGL_2(k)\) is isomorphic to \(k^\times\), via \(\operatorname{diag}(x,y)\mapsto x/y\). The map \(\Gamma\to\PGL_2(\Q)\to\PGL_2(k)\) is injective. Thus \(\Gamma\) would embed in \(k^\times\). Finite subgroups of the multiplicative group of a field are cyclic, but \(\Gamma\) is not.

Hence \(\ker\epsilon=\{1,r\}\). Let \(K=L^{\ker\epsilon}\), a quadratic field. For \(\gamma\in\ker\epsilon\), \(S_\gamma\) commutes with \(e_1\) and \(e_2\). For \(\gamma\notin\ker\epsilon\), \(S_\gamma RS_\gamma^{-1}=-R\), so \(S_\gamma\) exchanges \(e_1\) and \(e_2\).

Two blocks. Put \(T'=\mathcal BT\). It is a full lattice in \(V\), and it is \(G_{\Q}\)-stable by (1). We have

By (1), \(G_K\) preserves \(T_1\) and \(T_2\), and every element of \(G_{\Q}\setminus G_K\) maps \(T_1\) onto \(T_2\) and \(T_2\) onto \(T_1\). Put \(V_i=T_i\otimes_{\mathcal O}k=e_iV\). The maps \(X:V_2\to V_1\) and \(Y:V_1\to V_2\) are inverse isomorphisms, since \(YX=e_2\) and \(XY=e_1\). Thus each \(V_i\) has dimension \(2\), and each \(T_i\) is free of rank \(2\).

The blocks differ by a character. Let \(\sigma\in G_K\) have image \(\gamma\in\ker\epsilon\). Since \(S_\gamma\) commutes with \(e_1,e_2\), we have \(S_\gamma XS_\gamma^{-1}\in e_1M_2(k)e_2=kX\). Write \(S_\gamma XS_\gamma^{-1}=\eta(\sigma)X\). This defines a character \(\eta:G_K\to k^\times\), which factors through \(\Gal(L/K)=\{1,r\}\). For \(r\) we compute

using \(X=e_1Xe_2\). Hence \(\eta\) is the quadratic character of \(L/K\), with values \(\pm1\). It is trivial on \(G_L\), and its reduction \(\overline\eta\) modulo \(11\) is nontrivial.

By (1), \(\sigma(Xv)=\eta(\sigma)X\sigma(v)\) for \(\sigma\in G_K\) and \(v\in V_2\). Thus \(X\) is a \(G_K\)-isomorphism \(\eta\otimes V_2\to V_1\). Put \(M_i=T_i/11T_i\). The lattice \(XT_2\subset V_1\) is \(G_K\)-stable, and its reduction is isomorphic to \(\overline\eta\otimes M_2\). By lattice independence in \(V_1\),

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

Both blocks contain the trivial character. By lattice independence, \(\overline V=T'/11T'\) has a trivial \(G_{\Q}\)-composition factor, because \(A[11]\) does. As a \(G_K\)-module, \(\overline V=M_1\oplus M_2\), so some \(M_i\) has a trivial \(G_K\)-composition factor. Let \(\tau\in G_{\Q}\setminus G_K\). Then \(t\mapsto\tau(t)\) maps \(T_1\) onto \(T_2\), and \(\tau(\sigma t)=(\tau\sigma\tau^{-1})\tau(t)\). So \(M_2\) is the \(\tau\)-conjugate of \(M_1\). A composition factor \(\psi\) of one block gives the composition factor \(\sigma\mapsto\psi(\tau^{-1}\sigma\tau)\) of the other. The trivial character is unchanged, so both blocks have a trivial composition factor. Write \(M_1^{\mathrm{ss}}=\mathbf1\oplus\psi_1\) and \(M_2^{\mathrm{ss}}=\mathbf1\oplus\psi_2\). The relation above gives

Since \(\overline\eta\neq\mathbf1\), we get \(\overline\eta\psi_2=\mathbf1\), so \(\psi_2=\overline\eta\), and then \(\psi_1=\overline\eta\). Hence

The determinant contradiction. On \(G_K\), the determinant of \(\overline V\) is \(\det M_1\cdot\det M_2=\overline\eta^2=1\). By fact (f) of Section 1, \(\chi_{11}^2|_{G_K}=1\). This puts \(\Q(\zeta_{11}+\zeta_{11}^{-1})\), a field of degree \(5\), inside the quadratic field \(K\). This is a contradiction.

We conclude that \(\Gamma\neq D_4\). No invariant maximal order was assumed in this section.

8. Endomorphism field of degree at most 2

Assume case (S), let \(P\in A(\Q)\) have order \(11\), and suppose that \(\Gamma\in\{C_1,C_2\}\).

If \(\Gamma=C_1\), then \(L=\Q\). Choose an isogeny \(E^2\to A_{\overline{\Q}}\), and let \(B\) be the image of \(E\times0\). By Lemma 2(a), \(B\) is an elliptic curve over \(\Q\) contained in \(A\). This contradicts Section 3.

If \(\Gamma=C_2\), then \(L\) is quadratic. Let \(B\) be as before. By Lemma 2(a), it is an elliptic curve over \(L\) contained in \(A_L\). Put \(C=A_L/B\), an elliptic curve over \(L\). For \(\tau\in G_{\Q}\), the conjugates \(B^\tau\) and \(C^\tau\) are an elliptic subvariety and an elliptic quotient of \(A_{\overline{\Q}}\), because \(A\) is defined over \(\Q\). By Section 3, all of them are isogenous to \(E\) over \(\overline{\Q}\). Hence \(B\) and \(C\) are \(\Q\)-curves over the quadratic field \(L\). The point \(P\) either lies in \(B(L)\) or maps to a point of order \(11\) in \(C(L)\). Both possibilities contradict the theorem of Le Fourn and Najman.

Together with Sections 6 and 7, this proves that \(A(\Q)[11]=0\) in case (S).

Remark. For the statement about points of order \(22\), the input (c) can be replaced here by Kenku–Momose. Let \(B'\) be the image of \(0\times E\). It is an elliptic curve over \(L\), with \(B+B'=A_L\) and \(B\cap B'\) finite. The quotient map therefore restricts to an \(L\)-isogeny \(B'\to C\). By Lemma 2(b), every isogeny \(B\to B'\) over \(\overline{\Q}\) is defined over \(L\). Hence \(B\) and \(C\) lie in one \(L\)-isogeny class. Given a rational point of order \(22\), its multiples of orders \(2\) and \(11\) give \(L\)-rational points of these orders on curves in this class. Lemma 1 then gives an elliptic curve over the quadratic field \(L\) with a rational point of order \(22\), contradicting input (b). The input (c) is used only for the sharper statement that a non-CM geometric square has no rational point of order \(11\).

9. The CM square

Assume case (CM), so that \(D\simeq M_2(M)\), with centre \(M\).

The degree \([L:\Q]\) is prime to \(5\). Suppose not. By Cauchy's theorem, \(\Gamma\) contains an element \(\gamma\) of order \(5\). It acts on the centre \(M\) by a field automorphism. Since \(\Aut(M/\Q)\) has order \(2\), this action is trivial. Thus \(\gamma\) acts on \(D\) by an \(M\)-algebra automorphism. By the Skolem–Noether theorem, it is conjugation by some \(R\in\GL_2(M)\). Since \(\Gamma\) acts faithfully, \([R]\) has order \(5\) in \(\PGL_2(M)\). As in the proof of Lemma 3, the eigenvalues of \(R\) are distinct, and their ratio \(\zeta\) is a primitive fifth root of unity. Hence

Since \(\zeta+\zeta^{-1}\) is a root of \(X^2+X-1\), the element \(t\) is a root of \(X^2-3X+1\), whose discriminant is \(5\). Thus \(\sqrt5\in M\), and \(M=\Q(\sqrt5)\). This contradicts the fact that \(M\) is imaginary quadratic.

The field \(L\) contains the relevant ring class fields. Choose an isogeny \(E^2\to A_{\overline{\Q}}\), and let \(B\) be the image of \(E\times0\). By Lemma 2, \(B\) is an elliptic curve over \(L\), and all its geometric endomorphisms are defined over \(L\). Thus \(\End(B_L)\otimes\Q\simeq M\). The action of \(\End(B_L)\) on the one-dimensional \(L\)-vector space \(\Lie(B)\) gives a ring homomorphism \(\End(B_L)\otimes\Q\to L\). It is injective because \(M\) is a field. Since \(M/\Q\) is Galois, it follows that \(M\subseteq L\). Put \(C=A_L/B\), an elliptic curve over \(L\). It is geometrically isogenous to \(E\), so it also has complex multiplication by an order in \(M\). For \(Z\in\{B,C\}\), we have \(j(Z)\in L\) and \(M\subseteq L\), so

Conclusion. Suppose that \(P\in A(\Q)\) has order \(11\). Either \(P\in B(L)\), or \(P\) maps to a point of order \(11\) in \(C(L)\). In either case, some \(Z\in\{B,C\}\) has an \(L\)-rational point of order \(11\). Input (d) gives \(5\mid[L:M(j(Z))]\), and hence \(5\mid[L:\Q]\). This contradicts the first step. Hence \(A(\Q)[11]=0\) in case (CM).

This step cannot be replaced by Clark's list, Theorem 5, of torsion orders for abelian surfaces with everywhere potentially good reduction, because that list contains \(22\). The divisibility in input (d) holds for every imaginary quadratic order, including orders whose conductor is divisible by \(11\).

10. Completion of the proof

Let \(A/\Q\) be geometrically split, and suppose that \(P\in A(\Q)\) has order \(22\). Put \(P_2=11P\) and \(P_{11}=2P\), of orders \(2\) and \(11\). By Section 3, \(A\) is simple over \(\Q\). By Sections 6–9, \(A\) is of type (N). These two steps prove the second assertion of the theorem, since they use only \(P_{11}\).

The two elliptic subvarieties. Choose an isogeny \(E_1\times E_2\to A_{\overline{\Q}}\), and let \(B_i\) be the image of \(E_i\). Then \(B_i\) is isogenous to \(E_i\), and \(B_1\cap B_2\) is finite. Let \(B\) be any elliptic subvariety of \(A_{\overline{\Q}}\). By Poincaré's theorem, \(B\) is isogenous to \(E_1\) or to \(E_2\). The quotient \(A_{\overline{\Q}}/B_1\) is an elliptic curve isogenous to \(B_2\), hence to \(E_2\). If \(B\) is isogenous to \(E_1\), the composite \(B\to A_{\overline{\Q}}\to A_{\overline{\Q}}/B_1\) is zero, because \(E_1\) and \(E_2\) are not isogenous. Thus \(B\subseteq B_1\), and \(B=B_1\). Similarly, \(B=B_2\) if \(B\) is isogenous to \(E_2\). Hence \(B_1,B_2\) are the only elliptic subvarieties of \(A_{\overline{\Q}}\).

The group \(G_{\Q}\) permutes \(\{B_1,B_2\}\). Neither curve is defined over \(\Q\), since \(A\) is simple over \(\Q\). Hence the stabilizer of \(B_1\) has index \(2\). It equals \(G_K\) for a quadratic field \(K\). Let \(\sigma\) be the nontrivial automorphism of \(K\). Then \(E=B_1\) is defined over \(K\), and \(B_2=E^\sigma\).

One isogeny class. Put \(F=A_K/E\). The quotient map restricts to a \(K\)-isogeny \(E^\sigma\to F\), because \(E\cap E^\sigma\) is finite. Applying \(\sigma\), we obtain a \(K\)-isogeny \(E\to F^\sigma\).

For each \(\ell\in\{2,11\}\), the exact sequence

shows the following. Either \(P_\ell\in E(K)\), or \(\pi(P_\ell)\) is a point of order \(\ell\) in \(F(K)\). In the second case, \(\pi^\sigma(P_\ell)=\sigma(\pi(P_\ell))\) is a point of order \(\ell\) in \(F^\sigma(K)\), since \(P_\ell\) is rational. Thus the \(K\)-isogeny class of \(E\) contains curves with \(K\)-rational points of orders \(2\) and \(11\). Lemma 1 produces an elliptic curve over the quadratic field \(K\) with a \(K\)-rational point of order \(22\). This contradicts input (b), and proves the theorem. ∎

11. Consequence for the census

The groups [22] and [2,22] are impossible in the class “geometrically split, simple over \(\Q\)”. Both groups were already impossible in the class “split over \(\Q\)”, by Mazur's theorem applied to an elliptic subvariety and its elliptic quotient. Hence both geometrically split example columns are impossible for these two groups, and their combined geometrically split ∞? entry becomes ∅. No other group in the census contains an element of order \(22\).

The point of order \(2\) cannot be omitted. The census group [11] has a geometrically split example that is simple over \(\Q\), namely 11.b. By the theorem, its two geometric elliptic factors are not isogenous over \(\overline{\Q}\).

The argument applies to every geometric splitting degree, including degrees divisible by \(2\) or \(11\). It uses no maximal integral endomorphism ring and no principal polarization.

References

1. A. Bourdon and P. L. Clark, Torsion points and isogenies on CM elliptic curves, J. London Math. Soc. (2) 102 (2020), 580–622, Theorem 4.1(a), (d), (e) (numbering of arXiv:1906.07121v1).

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.

3. 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).

4. S. Le Fourn and F. Najman, Torsion of \(\Q\)-curves over quadratic fields, Math. Res. Lett. 27 (2020), 209–225, Theorem 1.1, Proposition 3.2, and Theorem 2.1 (numbering of arXiv:1803.03460v3).

5. B. Mazur, Modular curves and the Eisenstein ideal, Publications Mathématiques de l'IHÉS 47 (1977), 33–186, Theorem (8).

Related proofs: orders 33 and 34, which use the same residual tensor argument at \(11\), and order 31, which uses a stable split quadratic order.