Torsion of genus 2 Jacobians over ℚ

1. Notation and standard facts

Fix an embedding \(\overline{\Q}\subset\C\). For a number field \(K\subset\overline{\Q}\), write \(G_K=\Gal(\overline{\Q}/K)\), and let \(c\in G_{\Q}\) be complex conjugation. Let \(p\) be an odd prime, and put

with their actions of \(G_{\Q}\). Let \(\rho:G_{\Q}\to\GL(T)\) be the action on \(T\), and \(\chi:G_{\Q}\to\F_p^\times\) the mod-\(p\) cyclotomic character. Since \(\chi\) is surjective, it has order \(p-1\); in particular \(\chi\neq\mathbf 1\). For a finite-dimensional representation \(M\) of \(G_{\Q}\) over \(\F_p\), we write \(M^{\mathrm{ss}}\) for its semisimplification and \(M^*\) for its dual.

Let \(D=\End(A_{\overline{\Q}})\otimes\Q\). The group \(G_{\Q}\) acts on \(D\) by \(\Q\)-algebra automorphisms, \({}^\sigma\varphi=\sigma\circ\varphi\circ\sigma^{-1}\). An element of \(D\) fixed by \(G_K\) is defined over \(K\). Let \(L\) be the smallest field over which all geometric endomorphisms of \(A\) are defined. It is a finite Galois extension of \(\Q\), and \(\Gamma=\Gal(L/\Q)\) acts faithfully on \(D\). We use the following standard facts.

1. (F1) Tate modules. The map \(\End(A_{\overline{\Q}})\otimes\Z_p\to\End_{\Z_p}(T)\) is injective (Milne, Theorem I.10.15, applied over \(\overline{\Q}\)). Hence \(D\otimes_{\Q}\Q_p\to\End_{\Q_p}(V)\) is injective. It is \(G_{\Q}\)-equivariant when \(G_{\Q}\) acts on \(\End_{\Q_p}(V)\) by conjugation: \(T_p({}^\sigma\varphi)=\rho(\sigma)\,T_p(\varphi)\,\rho(\sigma)^{-1}\).

2. (F2) Lattice independence. Let \(W\) be a finite-dimensional \(\Q_p\)-vector space with a continuous action of \(G_{\Q}\). For a \(G_{\Q}\)-stable \(\Z_p\)-lattice \(\Lambda\subset W\), the semisimplification of \(\Lambda/p\Lambda\) depends only on the isomorphism class of \(W\). Indeed, the characteristic polynomials on \(\Lambda/p\Lambda\) are the reductions of those on \(W\), and the Brauer–Nesbitt theorem applies. In particular, a lattice may be transported through an isomorphism of representations.

3. (F3) Polarization. Every abelian variety over \(\Q\) has a polarization over \(\Q\). Combined with the Weil pairing (Milne, Section I.13), it gives a nondegenerate alternating \(G_{\Q}\)-equivariant pairing \(V\times V\to\Q_p(1)\), hence an isomorphism

   \[ V\simeq V^*\otimes_{\Q_p}\Q_p(1). \]

   The polarization is used only on the rational Tate module, so its degree may be divisible by \(p\).

4. (F4) Idempotents over \(\Q\). If \(D\) contains an idempotent \(e\neq0,1\) fixed by \(G_{\Q}\), then \(A\) is not simple over \(\Q\). Indeed, \(e\) is defined over \(\Q\), so \(ne\in\End(A)\) for some integer \(n\geq1\). Its image is an abelian subvariety of \(A\) over \(\Q\). It is nonzero because \(e\neq0\). It is proper because \(ne\circ n(1-e)=0\) and \(n(1-e)\neq0\), so \(ne\) is not an isogeny.

5. (F5) Complex conjugation. Since \(A\) is defined over \(\R\), complex conjugation of coordinates gives an antiholomorphic involution \(\mathcal C\) of \(A(\C)\) which is a group automorphism. Write \(A(\C)=\Lie(A_{\C})/\Lambda\) with \(\Lambda=H_1(A(\C),\Z)\), so that \(H=\Lambda\otimes\R\) is identified with \(\Lie(A_{\C})\), and let \(J\) be the complex structure on \(H\). The map \(\mathcal C\) lifts to an \(\R\)-linear involution \(c_H\) of \(H\), its differential at \(0\), and

   \[ c_HJ=-Jc_H, \]

   because \(\mathcal C\) is antiholomorphic. Each \(\varphi\in D\) acts on \(H\) by an \(\R\)-linear map \(\rho_H(\varphi)\) commuting with \(J\): this is the analytic representation, equivalently the rational representation on \(H_1(A(\C),\Q)\) tensored with \(\R\). The resulting map \(\rho_H:D\otimes_{\Q}\R\to\End_{\R}(H)\) is injective. Finally, for \(\varphi\in\End(A_{\overline{\Q}})\) we have \({}^c\varphi=\mathcal C\circ\varphi\circ\mathcal C\) on \(A(\C)\), since both sides are morphisms (the right-hand side is \(\varphi\) with complex-conjugated coefficients) that agree on \(A(\overline{\Q})\). Therefore, by linearity,

   \[ \rho_H({}^c\varphi)=c_H\,\rho_H(\varphi)\,c_H^{-1}\qquad(\varphi\in D). \]

2. Consequences of rational torsion of rank two

In this section \(p\) is an odd prime and \(\dim_{\F_p}A(\Q)[p]\geq2\). Simplicity over \(\Q\) and geometric splitting are not used here.

Proof. The subspace \(A(\Q)[p]=A[p]^{G_{\Q}}\) is a trivial submodule of dimension at least \(2\). Thus \(\mathbf 1\) occurs in \(A[p]^{\mathrm{ss}}\) with some multiplicity \(m\geq2\).

Transport the lattice \(\operatorname{Hom}_{\Z_p}(T,\Z_p)\otimes_{\Z_p}\Z_p(1)\) into \(V\) through the isomorphism of (F3). Its reduction is \(A[p]^*\otimes\chi\). By (F2),

The multiplicity of \(\chi\) on the right equals the multiplicity of \(\mathbf 1\) in \((A[p]^{\mathrm{ss}})^*\), which is \(m\). Hence \(\chi\) occurs at least twice in \(A[p]^{\mathrm{ss}}\). Since \(\chi\neq\mathbf 1\) and \(\dim A[p]=4\), it follows that \(m=2\) and \(A[p]^{\mathrm{ss}}\simeq\mathbf 1^2\oplus\chi^2\).

For the second claim, \(\End_{\F_p}(A[p])\simeq A[p]\otimes A[p]^*\). In the Grothendieck group of finite-dimensional \(\F_p[G_{\Q}]\)-modules, the class of a module is the sum of the classes of its composition factors. Tensor product over \(\F_p\) is exact in each variable, so the class of \(M\otimes N\) depends only on the classes of \(M\) and \(N\), bilinearly. Since \((A[p]^*)^{\mathrm{ss}}\simeq\mathbf 1^2\oplus\chi^{-2}\), we get

This determines the composition factors. ∎

This statement concerns composition factors. It does not assert that \(A[p]\) or its endomorphism module is semisimple. Only \(\chi\neq\mathbf 1\) was used, so Lemma 1 holds for every odd \(p\).

Proof. The module \(B_{\Z_p}\) is a submodule of the finite free \(\Z_p\)-module \(\End_{\Z_p}(T)\), hence finite free. It spans \(B\), because \(p^kX\in\End_{\Z_p}(T)\) for every \(X\in B\) and \(k\) large. It is \(G_{\Q}\)-stable because \(B\) and \(\End_{\Z_p}(T)\) are. It is saturated in \(\End_{\Z_p}(T)\): if \(X\in B_{\Z_p}\) and \(X\in p\End_{\Z_p}(T)\), then \(p^{-1}X\in B\cap\End_{\Z_p}(T)=B_{\Z_p}\). Thus \(B_{\Z_p}\cap p\End_{\Z_p}(T)=pB_{\Z_p}\), which is the injectivity of the reduction map.

By the Jordan–Hölder theorem, the composition factors of \(B_{\Z_p}/pB_{\Z_p}\) are among those of \(\End_{\F_p}(A[p])\), which are listed in Lemma 1. By (F2), \(\Lambda/p\Lambda\) and \(B_{\Z_p}/pB_{\Z_p}\) have the same semisimplification. ∎

Proof. Suppose such a \(\delta\) exists. By (F1), its image in \(\End_{\Q_p}(V)\) is nonzero, and \(B=\Q_p\delta\) is a \(G_{\Q}\)-stable line on which \(G_{\Q}\) acts through \(\varepsilon\). The lattice \(\Z_p\delta\) reduces to the character

which is nontrivial because \(-1\neq1\) in \(\F_p\). By Lemma 2, \(\overline\varepsilon\) is one of \(\mathbf 1\), \(\chi\), \(\chi^{-1}\). It is not \(\mathbf 1\). It has order \(2\), whereas \(\chi\) and \(\chi^{-1}\) have order \(p-1\geq4\). This is a contradiction. ∎

From now on, \(A\) is simple over \(\Q\) and geometrically split, \(p\geq5\), and we suppose, to reach a contradiction, that \(\dim_{\F_p}A(\Q)[p]\geq2\). Exactly one of the following holds: \(A_{\overline{\Q}}\) is isogenous to \(E_1\times E_2\) with \(E_1,E_2\) not isogenous; \(A_{\overline{\Q}}\) is isogenous to \(E^2\) with \(E\) a CM elliptic curve; \(A_{\overline{\Q}}\) is isogenous to \(E^2\) with \(E\) without CM. We treat these cases in Sections 3, 4, and 5–6.

3. Nonisogenous geometric factors

Suppose that \(A_{\overline{\Q}}\) is isogenous to \(E_1\times E_2\), where \(E_1\) and \(E_2\) are not geometrically isogenous. Since \(\operatorname{Hom}(E_1,E_2)=0\),

where each factor is \(\Q\) or an imaginary quadratic field. The only idempotents of this product of two fields other than \(0\) and \(1\) are \(e_1=(1,0)\) and \(e_2=(0,1)\). The group \(G_{\Q}\) acts by algebra automorphisms, so it permutes \(\{e_1,e_2\}\). If it fixed \(e_1\), then \(A\) would not be simple over \(\Q\), by (F4). Hence the permutation action defines a nontrivial character \(\varepsilon:G_{\Q}\to\{\pm1\}\), with \(\varepsilon(\sigma)=-1\) exactly when \(\sigma\) exchanges \(e_1\) and \(e_2\). The element

satisfies \({}^\sigma\delta=\varepsilon(\sigma)\delta\). This contradicts Corollary 3.

This argument also covers elliptic factors with complex multiplication; only the two idempotents are used. Simplicity over \(\Q\) is used to show that \(\varepsilon\) is nontrivial.

4. The square of a CM elliptic curve

Suppose that \(A_{\overline{\Q}}\) is isogenous to \(E^2\), where \(E\) has complex multiplication by an imaginary quadratic field \(M\). Then \(D\simeq M_2(M)\), and its centre \(Z=Z(D)\) is an imaginary quadratic field isomorphic to \(M\). Choose \(\delta\in Z\) with \(\delta^2=-d\) for a rational number \(d>0\). Since \(G_{\Q}\) acts on \(D\) by algebra automorphisms, it preserves \(Z\) and acts on it by field automorphisms. Hence there is a character \(\varepsilon:G_{\Q}\to\{\pm1\}\) with

We show that \(\varepsilon(c)=-1\). Choose an isogeny \(f:A_{\C}\to E_{\C}^2\). Conjugation by \(f\) identifies \(D\) with \(\End^0(E^2)=M_2(\End^0(E))\). The field \(\End^0(E)\) acts on the one-dimensional space \(\Lie(E_{\C})\) through a field embedding into \(\C\), so the centre of \(M_2(\End^0(E))\) acts on \(\Lie(E_{\C}^2)=\Lie(E_{\C})^2\) by complex scalars. Transporting through the \(\C\)-linear isomorphism \(df:\Lie(A_{\C})\to\Lie(E^2_{\C})\), we find a field embedding \(\iota:Z\to\C\) such that

where the right-hand side means multiplication by the complex number \(\iota(z)\) on \(H=\Lie(A_{\C})\). By (F5), and since \(c_H\) is \(\C\)-antilinear,

Here \(\overline{\iota(\delta)}=-\iota(\delta)\) because \(\iota(\delta)^2=-d<0\). Since \(\rho_H\) is injective, \({}^c\delta=-\delta\). Thus \(\varepsilon\) is nontrivial, and Corollary 3 gives a contradiction.

Simplicity over \(\Q\) is not used in this case.

5. The square of a non-CM elliptic curve: the endomorphism field

Suppose finally that \(A_{\overline{\Q}}\) is isogenous to \(E^2\), where \(E\) has no complex multiplication. Then

By Skolem–Noether, every \(\Q\)-algebra automorphism of \(M_2(\Q)\) is inner. The faithful action of \(\Gamma\) on \(D\) therefore identifies \(\Gamma\) with a finite subgroup of \(\operatorname{Aut}(M_2(\Q))=\PGL_2(\Q)\).

Proof. This is a special case of Beauville, Introduction and Proposition 1.1. For completeness: a finite subgroup of \(\PGL_2(\R)\) acts by isometries on the hyperbolic plane, so it fixes a point. Its action on the tangent plane at that point embeds it in \(O(2)\), so it is cyclic or dihedral. 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 \(\zeta\). Thus

which forces \(\varphi(n)=2\), that is, \(n\in\{3,4,6\}\). ∎

Proof. Since \(p\geq5\) and \(|\Gamma|\) divides \(24\), the prime \(p\) does not divide \(|\Gamma|\).

Let \(B\subseteq\End_{\Q_p}(V)\) be the image of \(D\otimes_{\Q}\Q_p\), which is isomorphic to it by (F1), and let \(B_{\Z_p}\) be the lattice of Lemma 2. The group \(G_{\Q}\) acts on \(B\) through \(\Gamma\), and this action is faithful because the action on \(D\) is. Put \(\overline B=B_{\Z_p}/pB_{\Z_p}\).

The action of \(\Gamma\) on \(\overline B\) is faithful. Let \(g\in\Gamma\) act trivially on \(\overline B\), and suppose \(g\neq1\). Choosing a basis of \(B_{\Z_p}\), write \(g=1+p^kY\) with \(k\geq1\) and \(Y\) a matrix over \(\Z_p\) that is nonzero modulo \(p\). Let \(m\) be the order of \(g\); it divides \(|\Gamma|\), so \(p\nmid m\). Since \(2k\geq k+1\),

which is impossible because \(m\) is a unit in \(\Z_p\). Hence \(g=1\).

\(\Gamma\) is abelian, and \(\overline B\) is a sum of characters. By Maschke's theorem, \(\overline B\) is a semisimple \(\F_p[\Gamma]\)-module, so it is the direct sum of its composition factors. By Lemma 2, each of these is one of \(\mathbf 1\), \(\chi\), \(\chi^{-1}\). Thus

where each \(\psi_i\) is trivial on \(G_L\). Since \(\Gamma\) acts faithfully, it embeds in \((\F_p^\times)^4\), so \(\Gamma\) is abelian.

The cyclotomic field. If \(\Gamma\) were trivial, every element of \(D\) would be defined over \(\Q\), including the idempotent \(\left(\begin{smallmatrix}1&0\\0&0\end{smallmatrix}\right)\). This contradicts simplicity over \(\Q\), by (F4). Hence \(\Gamma\neq1\), and by faithfulness some \(\psi_i\) is nontrivial, so it is \(\chi\) or \(\chi^{-1}\). This character is trivial on \(G_L\), so \(G_L\subseteq\ker\chi=G_{\Q(\zeta_p)}\), that is, \(\Q(\zeta_p)\subseteq L\). Hence \(\Gamma\) has a cyclic quotient \(\Gal(\Q(\zeta_p)/\Q)\) of order \(p-1\geq4\).

The list. The abelian groups in Lemma 4 are \(C_1,C_2,C_3,C_4,C_6\) and \(D_4=C_2\times C_2\). Their cyclic quotients have orders dividing \(1,2,3,4,6\) and \(2\), respectively. A cyclic quotient of even order \(p-1\geq4\) occurs only for \(\Gamma=C_4\) with \(p=5\) and for \(\Gamma=C_6\) with \(p=7\). In both cases \([L:\Q]=|\Gamma|=p-1=[\Q(\zeta_p):\Q]\), so \(L=\Q(\zeta_p)\). ∎

6. Complex conjugation and cyclic endomorphism fields

Let \(\operatorname{sgn}:\PGL_2(\R)\to\{\pm1\}\) be the sign of the determinant of a representative. It is well defined, since \(\det(\lambda M)=\lambda^2\det M\), and it is a homomorphism. Every isomorphism \(D\otimes_{\Q}\R\simeq M_2(\R)\) gives an isomorphism \(\operatorname{Aut}(D\otimes_{\Q}\R)\simeq\PGL_2(\R)\), unique up to conjugation, so the sign of an element of \(\Gamma\subseteq\operatorname{Aut}(D)\subseteq\operatorname{Aut}(D\otimes_{\Q}\R)\) is well defined.

Proof. Scale a representative so that \(\det M=-1\). Its characteristic polynomial \(X^2-tX-1\) has discriminant \(t^2+4>0\), so \(M\) has real eigenvalues \(\lambda\neq\mu\) with \(\lambda\mu=-1\). If \(M^n\) is scalar, then \(\lambda^n=\mu^n\). Since \(\lambda\) and \(\mu\) are real of opposite signs, this gives \(\mu=-\lambda\), hence \(\lambda^2=1\) and \(M^2=I\). The class of \(M\) is not trivial because \(\det M<0\), so it has order \(2\).

If \(\Gamma=\langle g\rangle\) has order greater than \(2\), then \(\operatorname{sgn}(g)=+1\) by the first part, and so every power of \(g\) has sign \(+1\). ∎

Proof. Let \(H\), \(J\), \(c_H\) and \(\rho_H\) be as in (F5). The image \(\rho_H(D\otimes_{\Q}\R)\) is a copy of \(M_2(\R)\) inside \(\End_{\R}(H)\simeq M_4(\R)\). Every module over \(M_2(\R)\) is a sum of copies of the standard module \(U=\R^2\). Hence there is a two-dimensional real vector space \(W\) with

The commutant of \(\End_{\R}(U)\otimes1\) in \(\End_{\R}(U\otimes W)\) is \(1\otimes\End_{\R}(W)\). Since \(J\) commutes with \(\rho_H(D)\), we have \(J=1\otimes J_W\) with \(J_W^2=-1\).

By (F5), conjugation by \(c_H\) preserves \(\End_{\R}(U)\otimes1\) and induces on it the action of \(c\) on \(D\otimes_{\Q}\R\). By Skolem–Noether, this automorphism is conjugation by \(M\otimes1\) for some \(M\in\GL(U)\), and the class \([M]\in\PGL_2(\R)\) is the image of \(c\). Then \((M^{-1}\otimes1)c_H\) commutes with \(\End_{\R}(U)\otimes1\), so

The relation \(c_HJ=-Jc_H\) gives \(M\otimes(NJ_W+J_WN)=0\), so \(NJ_W=-J_WN\). In a basis of \(W\) with \(J_W=\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)\), this means

with \(a^2+b^2>0\) because \(N\) is invertible. Since \(c_H^2=1\), we get \((a^2+b^2)M^2\otimes1=1\otimes1\), so \(M^2=sI\) with \(s=(a^2+b^2)^{-1}>0\). If \([M]\) is not trivial, then \(M\) is not scalar, its minimal polynomial is \(X^2-s\), and its eigenvalues are \(\sqrt s\) and \(-\sqrt s\). Thus \(\det M=-s<0\).

Now let \(\Gamma\) be cyclic of order greater than \(2\). By Lemma 6, the image of \(c\) in \(\Gamma\) has sign \(+1\), so it is trivial by the first part. Since \(\Gamma\) acts faithfully on \(D\), the restriction of \(c\) to \(L\) is trivial, so \(L\subset\R\). As \(L/\Q\) is Galois, \(L\) is totally real. ∎

Completion of the proof. In Section 5, \(\Gamma\) is cyclic of order \(p-1\geq4\) and \(L=\Q(\zeta_p)\). By Lemma 7, \(L\) is totally real. But \(\Q(\zeta_p)\) is not real, a contradiction. Together with Sections 3 and 4, this shows that \(\dim_{\F_p}A(\Q)[p]\geq2\) is impossible, and proves the theorem. ∎

7. Where the hypotheses are used

The bound \(p\geq5\). Lemmas 1 and 2 hold for every odd prime. The bound \(p\geq5\) is used in exactly three places.

1. In Corollary 3, which serves Sections 3 and 4: a nontrivial quadratic character is neither \(\chi\) nor \(\chi^{-1}\), because \(\chi\) has order \(p-1\geq4\).

2. In Proposition 5: \(p\nmid|\Gamma|\), so that Maschke's theorem applies and \(\Gamma\) acts faithfully on \(\overline B\).

3. In Proposition 5: the cyclic quotient of order \(p-1\geq4\) leaves only \(\Gamma=C_4\) and \(\Gamma=C_6\), which are cyclic of order greater than \(2\).

The theorem is false for \(p=3\). The census contains the geometrically split Jacobians 3.3.c, with torsion [3,3], and 3.9.c, with torsion [3,9], both certified simple over \(\Q\). For \(p=3\), the character \(\chi_3\) is itself the quadratic character of \(\Q(\sqrt{-3})\), so the first use of \(p\geq5\) listed above breaks down when \(\varepsilon\) is this character. This is what happens for 3.9.c. The extended genus-\(2\) data of Booker–Sutherland list the geometric endomorphism algebra \(\Q\times\Q\) for it (label 6804.a.1), and its census splitting certificate is defined over \(\Q(\sqrt{-3})\). So the character \(\varepsilon\) of Section 3 is \(\chi_3\).

For 3.3.c, the second use listed above breaks down. With \(Y=2y+x^3\), its model becomes \(Y^2=x^6-52x^3+784\), which has the geometric automorphisms

The first is defined over \(\Q(\zeta_3)\) and the second over \(\Q(\sqrt[3]{784})\), and over no smaller fields. Automorphisms of a curve of genus \(2\) act faithfully on its Jacobian, so \(L\) contains \(\Q(\zeta_3,\sqrt[3]{784})\). This field has Galois group \(S_3\) over \(\Q\), so \(3\) divides \(|\Gamma|\). The same data list \(D=M_2(\Q)\) and Sato–Tate group \(J(E_3)\) for 3.3.c (label 1764.d.2), so it falls under Section 5.

Simplicity over \(\Q\). It is used only twice: in Section 3, to make \(\varepsilon\) nontrivial, and in Proposition 5, to exclude \(\Gamma=1\). The proof therefore shows, without assuming simplicity over \(\Q\), that \(\dim_{\F_p}A(\Q)[p]\leq1\) for \(p\geq5\) whenever \(A_{\overline{\Q}}\) is isogenous to the square of a CM elliptic curve, or to the square of a non-CM elliptic curve with \(\Gamma\neq1\). The case \(\Gamma=1\) genuinely occurs: the census curve 5.5.a, with torsion [5,5], is isogenous over \(\Q\) to the product of the elliptic curves 11a1 and 11a3, which are isogenous over \(\Q\) and have no CM. So \(D=M_2(\Q)\) and all its elements are defined over \(\Q\).

What is not used. The proof uses no bound on the degree of a geometric splitting isogeny and no assumption on the integral endomorphism ring. The polarization may have degree divisible by \(p\). A principal polarization is not assumed. The Weil pairing alone does not exclude rational \(p\)-torsion of rank two, as the examples split over \(\Q\) show. The proof uses the full Galois action on \(A[p]\) and on the geometric endomorphism algebra.

8. Consequence for the census

A group \(A(\Q)_{\mathrm{tors}}\) of one of the forms [5,5], [5,10] or [7,7] has \(\dim_{\F_p}A(\Q)[p]=2\) for \(p=5\) or \(p=7\). Hence the groups [5,5], [5,10] and [7,7] are impossible in the class “geometrically split, simple over \(\Q\)”. No other census group contains \((\Z/p\Z)^2\) for a prime \(p\geq5\).

Simplicity over \(\Q\) is essential. Each of the three groups has a certified example split over \(\Q\): 5.5.a, 5.10.a and 7.7.a. The exclusion therefore concerns the “geometrically split, simple over \(\Q\)” column only. It does not change the combined geometrically split ∞? grade of any of the three groups, and none of them becomes ∅ on the geometrically split side. The theorem says nothing about geometrically simple Jacobians; for example, [5,5] has the geometrically simple example 5.5.b.

The theorem applies to every geometric splitting degree, including degrees divisible by \(p\). It does not require a principal polarization.

References

1. A. Beauville, Finite subgroups of \(\PGL_2(K)\), in Vector bundles and complex geometry, Contemp. Math. 522, Amer. Math. Soc., 2010, 23–29, Introduction and Proposition 1.1. arXiv:0909.3942.

2. J. S. Milne, Abelian varieties, course notes, version 2.00 (2008), Chapter I, Theorem 10.15 and Section 13.

3. A. R. Booker and A. V. Sutherland, Genus 2 curves of small conductor, in preparation; preliminary data in the alpha version of the LMFDB (curves 1764.d.2 and 6804.a.1). Used only for the remark on \(p=3\) in Section 7.