Statement and Proof of the Main Theorem

Let $K$ be as in Section 2, and let $\pi:J\rightarrow A$ be an optimal quotient. Assume that $J$ is equipped with a symmetric principal polarization $\lambda$, that $J$ has semistable reduction, and that $A$ has purely toric reduction. Let $X_A$, $X_{A^{\vee}}$, and $X_J$ denote the character groups of the toric parts of the closed fibers of the abelian varieties $A$, $A^{\vee}$, and $J$, respectively.

Let $\pi:J\rightarrow A$ be an optimal quotient, and let $\theta:A^{\vee}\rightarrow A$ denote the induced polarization. Let $\pi_*$, $\pi^*$, $\theta_*$, and $\theta^*$ be the maps induced on character groups by the various functorialities, as indicated in the following two key diagrams:

$$\xymatrix @=3pc{A^{\vee} \ar@{^(->}[r]^{\pi^{\vee}}\ar[dr]_{\thet... ...*} & X_J \ar@{-»}[d]^{\pi_*} \\ & X_{A^{\vee}}.\ar@/^1.5pc/[ul]^{\theta_*}}$$

The surjectivity of $\pi_*$ is proved in Theorem 8.2. The injectivity of $\pi^*$ follows because

$$\theta_*\pi_*\pi^*=\theta_*\theta^*=\deg(\theta)\neq 0,$$

and multiplication by a nonzero integer on a free abelian group is injective.

Let $\mathcal{L}$ be the saturation of $\pi^* X_A$ in $X_J$; thus $\pi^* X_A$ is a finite-index subgroup of  $\mathcal{L}$ and the quotient $X_J/\mathcal{L}$ is torsion free. Let

$$\alpha :X_J\rightarrow \Hom(\pi^* X_A,\mathbf{Z})$$

be the map defined by the monodromy pairing restricted to $X_J\times \pi^* X_A$. For $L$ of finite index in $\mathcal{L}$, define the degree of $L$ to be

$$m_L = [\alpha (X_J):\alpha (L)],$$

and the component group of $L$ to be

$$\Phi_L = \coker( X_J \rightarrow \Hom(L,\mathbf{Z})).$$

When $L=\mathcal{L}$ and $A$ is fixed, for simplicity we write $m_X=m_\mathcal{L}$ and $\Phi_X=\Phi_\mathcal{L}$.

Recall that $\Phi_A$ is the component group of $A$ and $m_A$ is the square root of the degree of the induced map $A^{\vee}\rightarrow A$.

Theorem 6.1   For any subgroup $L$ of finite index in $\mathcal{L}$, the following relation holds:

$$\frac{\char93 \Phi_A}{m_A} = \frac{\char93 \Phi_L}{m_L}.$$