Component group of Neron model of a parametrized abelian variety - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T01:01:30Z http://mathoverflow.net/feeds/question/102381 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102381/component-group-of-neron-model-of-a-parametrized-abelian-variety Component group of Neron model of a parametrized abelian variety David Corwin 2012-07-16T19:27:57Z 2012-07-16T21:26:47Z <p>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 $A \cong (\bar{K}^\times \times \bar{K}^\times)/(q_1,q_2)$ 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":</p> <p><code>$$\begin{pmatrix} v(q_{11}) &amp; v(q_{12})\\ v(q_{21}) &amp; v(q_{22}) \end{pmatrix}$$</code></p> <p>has nonzero determinant (for this description, see e.g. Ribet's Ph.D thesis <a href="http://www.jstor.org/stable/10.2307/2373815" rel="nofollow">http://www.jstor.org/stable/10.2307/2373815</a>).</p> <p>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.</p> <p>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.</p> <p>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.</p>