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Let $A$ be an abelian variety of dimension $2$ over a $p$-adic field $K$ with (additive) valuation $v$. Assuming $A$ has multiplicative reduction, the theory of $p$-adic theta functions gives us an isomorphism $(\bar{K}^\times A \cong (\bar{K}^\times \times \bar{K}^\times)/(q_1,q_2)$ where $q_1,q_2=(q_{11},q_{12}),(q_{21},q_{22}) \in \bar{K}^\times \times \bar{K}^\times$ is such that the "valuation matrix":

$$\begin{pmatrix} v(q_{11}) & v(q_{12})v(q_{12})\\ v(q_{21}) & v(q_{22}) \end{pmatrix}$$

has nonzero determinant (for this description, see e.g. Ribet's Ph.D thesis http://www.jstor.org/stable/10.2307/2373815).

In the case of dimension $1$, this is Tate's $p$-adic uniformization of elliptic curves. In that case $v(q)=v(q_{11})$ is negative the valuation of the $j$-invariant of the curve $E$, which is the number of components of the special fiber.

My question is: can we find a similar description of the above matrix in terms of the component group of the special fiber of the Neron model? I'm hoping it will end up being the determinant.

My best guess is to relate the (algebraic) Neron model to the (rigid analytic) parametrized abelian variety using the connection between formal schemes and rigid analysis, but I don't know enough about these topics to do this.

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# Component group of Neron model of a parametrized abelian variety

Let $A$ be an abelian variety of dimension $2$ over a $p$-adic field $K$ with (additive) valuation $v$. Assuming $A$ has multiplicative reduction, the theory of $p$-adic theta functions gives us an isomorphism $(\bar{K}^\times \times \bar{K}^\times)/(q_1,q_2)$ where $q_1,q_2=(q_{11},q_{12}),(q_{21},q_{22}) \in \bar{K}^\times \times \bar{K}^\times$ is such that the "valuation matrix":

$$\begin{pmatrix} v(q_{11}) & v(q_{12})\ v(q_{21}) & v(q_{22}) \end{pmatrix}$$

has nonzero determinant (for this description, see e.g. Ribet's Ph.D thesis http://www.jstor.org/stable/10.2307/2373815).

In the case of dimension $1$, this is Tate's $p$-adic uniformization of elliptic curves. In that case $v(q)=v(q_{11})$ is negative the valuation of the $j$-invariant of the curve $E$, which is the number of components of the special fiber.

My question is: can we find a similar description of the above matrix in terms of the component group of the special fiber of the Neron model? I'm hoping it will end up being the determinant.

My best guess is to relate the (algebraic) Neron model to the (rigid analytic) parametrized abelian variety using the connection between formal schemes and rigid analysis, but I don't know enough about these topics to do this.