Optimal Quotients

Let $K$ be as in Section 2, let $J$ be an abelian variety equipped with a symmetric principal polarization $\theta_J$ (see Definition 5.1). For example, $J$ could be the Jacobian of a curve equipped with the canonical principal polarization arising from the $\theta$-divisor.

Definition 3.1 (Optimal quotient)   An optimal quotient of $J$ is an abelian variety $A$ and a smooth surjective morphism $\pi:J\rightarrow A$ whose kernel is connected (i.e., an abelian variety).

Remark 3.2   Any connected scheme of finite type over a field is geometrically connected if it contains a rational point (e.g., if it is a group scheme). See [8, IV$_2$, §4.5.13].

Let $\pi:J\rightarrow A$ be an optimal quotient. Denote by $J^{\vee}$ and $A^{\vee}$ the abelian varieties dual to $J$ and $A$, respectively. Upon composing the dual of $\pi$ with $\theta_J^{\vee}=\theta_J$, we obtain a map

$$A^{\vee}\xrightarrow{\pi^{\vee}} J^{\vee}\xrightarrow{\theta_J} J.$$

Proposition 3.3   The map $\theta_J\circ \pi^{\vee} \colon{} A^{\vee}\rightarrow J$ is a closed immersion.

Proof. Since $\theta_J$ is an isomorphism, we want to prove that $\pi^{\vee}$ is a closed immersion. It is a general fact that duals to surjections of abelian varieties with abelian variety kernel are closed immersions, but for lack of an adequate reference we recall the proof. Since a monomorphism between smooth finite type group schemes over a field is necessarily a closed immersion, it suffices to show that the commutative proper group scheme $\ker(\pi^{\vee})$ vanishes. Since a non-zero commutative proper group scheme $G$ over a field $F$ necessarily has a non-zero finite subgroup scheme $G[n]$ for some $n$ (since either $(G_{/\overline{F}})_{\rm {red}}^0$ is an abelian variety or else $G$ is finite and non-zero), it suffices to show that $\ker(\pi^{\vee})[n]$ vanishes for all positive integers $n$. In other words, it suffices to show that the induced map $A^{\vee}[n] \rightarrow J^{\vee}[n]$ is a closed immersion for all $n$.

Since Cartier duality interchanges faithfully flat maps and closed immersions, and the scheme-theoretic Weil pairing identifies the Cartier dual of the map induced by $\pi^{\vee}$ on $n$-torsion with $\pi:J[n] \rightarrow A[n]$, we just have to show that these latter maps are faithfully flat for all integers $n$. Using the short exact sequence

$$0 \rightarrow \ker(\pi) \rightarrow J \rightarrow A \rightarrow 0$$

in the abelian category of fppf abelian sheaves over $\Spec(K)$, the snake lemma gives an exact sequence

$$0 \rightarrow \ker(\pi)[n] \rightarrow J[n] \rightarrow A[n] \rightarrow 0$$

because $n:\ker(\pi) \rightarrow \ker(\pi)$ is a faithfully flat map (hence fppf surjective), as $\ker(\pi)$ is an abelian variety. This gives an isomorphism of group schemes

$$J[n]/\ker(\pi)[n] \simeq A[n]$$

compatible with the maps from $J[n]$, whence $J[n] \rightarrow A[n]$ is faithfully flat.

$\qedsymbol$

proof

Since tex2html_wrap_inline$&thetas#theta;_J$ is an isomorphism and tex2html_wrap_inline$&pi#pi;^&or#vee;$ is a homomorphism of algebraic groups over a field, it suffices to prove that tex2html_wrap_inline$&pi#pi;^&or#vee;$ is a closed immersion. Since the dual of tex2html_wrap_inline$&pi#pi;^&or#vee;$ is tex2html_wrap_inline$(&pi#pi;^&or#vee;)^&or#vee;=&pi#pi;$ and tex2html_wrap_inline$&pi#pi;$ is surjective, the map tex2html_wrap_inline$&pi#pi;^&or#vee;$ must have finite kernel. Thus tex2html_wrap_inline$A^&or#vee; &rarr#rightarrow; C=(&pi#pi;^&or#vee;)$ is an isogeny. Let tex2html_wrap_inline$G$ denote the kernel of this isogeny, and dualize. By [, §11] we have the following two commutative diagrams: displaymath G[r] & A^&or#vee;@-»[r] [dr]_&pi#pi;^&or#vee; & C[d]
&& J^&or#vee;             dualize              A & C^&or#vee;[l] & G^&or#vee;[l]
& J,[u]_&phiv#varphi;[ul]^&pi#pi; where tex2html_wrap_inline$G^&or#vee;$ is the Cartier dual of tex2html_wrap_inline$G$. Since tex2html_wrap_inline$G^&or#vee;$ is finite, tex2html_wrap_inline$(&phiv#varphi;)$ is of finite index in tex2html_wrap_inline$(&pi#pi;)$. Since tex2html_wrap_inline$(&pi#pi;)$ is an abelian variety, as a group it is divisible. But a divisible group has no nontrivial finite-index subgroups (divisibility is a property inherited by quotients, and nonzero finite groups are not divisible). Thus tex2html_wrap_inline$(&phiv#varphi;)=(&pi#pi;)$, so tex2html_wrap_inline$G^&or#vee;=0$. It follows that tex2html_wrap_inline$G=0$.

Henceforth we will abuse notation and denote the injection $A^{\vee}\rightarrow J$ by $\pi^{\vee}$. We define $\theta_A$ to be the composite $\pi\circ \pi^{\vee}$, so the kernel of $\theta_A$ equals the scheme-theoretic intersection of $A^{\vee}$ and $B=\ker(\pi)$, as depicted in the following diagram:

$$\xymatrix{ A^{\vee}\cap B\ar[r]\ar[d] & B\ar[d] \\ A^{\vee}\ar@{^(->}[r]^{\pi^{\vee}}\ar[dr]_{\theta_A} & J\ar[d]^\pi\\ & A. }$$

Since $\theta_A$ is a polarization (due to how its definition uses the polarization $\theta_J$) the degree of $\theta_A$ is a perfect square (see [16, §16, p. 150]).

Definition 3.4 (Degree)   Define the degree of $A$ as a quotient of $J$ to be the integer

$$m_A=\sqrt{\char93 \ker(\theta_A)}.$$