Statement of the Theorem

We now state our main result, supressing some of the definitions of the terms used until later (see Section 6 below for a more complete statement and the proof). Let $K$ be as above, and suppose $\pi:J\rightarrow A$ is an optimal quotient. Assume that $J$ is equipped with a symmetric principal polarization $\lambda$, in the sense of Definition 5.1. For example, the $\theta$ polarization of the Jacobian of a curve is a symmetric principal polarization. Also assume that $J$ has semistable reduction, and that $A$ has purely toric reduction.

We express the component group of $A$ in terms of the monodromy pairing associated to $J$. Let $m_A=\sqrt{\deg(\theta_A)}$, where $\theta_A:A^{\vee}\rightarrow A$ is induced by the canonical principal polarization of $J$ arising from the $\theta$-divisor. Let $X_J$ be the character group of the toric part of the closed fiber of the Néron model of $J$. Let  $\mathcal{L}$ be the saturation of the image of $X_A$ in $X_J$. The monodromy pairing induces a map $\alpha :X_J\rightarrow \Hom(\mathcal{L},\mathbf{Z})$. Let $\Phi_X$ be the cokernel of $\alpha$ and $m_X=[\alpha (X_J):\alpha (\mathcal{L})]$ be the order of the finite group $\alpha (X_J)/\alpha (\mathcal{L})$. The main result of this paper is that

$$\frac{\char93 \Phi_A}{m_A} = \frac{\char93 \Phi_X}{m_X},$$

and this is recorded as Theorem 6.1 below.

Using the snake lemma, one sees that $\Phi_X$ is isomorphic to the image of the natural map $\Phi_J \rightarrow \Phi_A$, and the above formula implies that the cokernel of the map $\Phi_J \rightarrow \Phi_A$ has order $m_A/m_X$. A non-obvious consequence of this is that $m_X \mid m_A$.

In the context of modular forms, if the optimal quotient $J\rightarrow A$ arises from a newform on $\Gamma_0(N)$, then the quantities $m_A$, $m_X$ and $\Phi_X$ can be explicitly computed, hence we can compute $\char93 \Phi_A$. Note that the authors have not computed the structure of $\Phi_A$ as a group.