Serre-Tate Theory 1

10 Jul

A Mind for Madness

Today we’ll try to answer the question: What is Serre-Tate theory? It’s been a few years, but if you’re not comfortable with formal groups and $latex {p}&fg=000000$-divisible groups, I did a series of something like 10 posts on this topic back here: formal groups, p-divisible groups, and deforming p-divisible groups.

The idea is the following. Suppose you have an elliptic curve $latex {E/k}&fg=000000$ where $latex {k}&fg=000000$ is a perfect field of characteristic $latex {p>2}&fg=000000$. In most first courses on elliptic curves you learn how to attach a formal group to $latex {E}&fg=000000$ (chapter IV of Silverman). It is suggestively notated $latex {\widehat{E}}&fg=000000$, because if you unwind what is going on you are just completing the elliptic curve (as a group scheme) at the identity.

Since an elliptic curve is isomorphic to it’s Jacobian $latex {Pic_E^0}&fg=000000$ there is a conflation that happens. In general, if you have a…

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