10.25560/87649
Singer, Raffael
Imperial College London
On tilts of certain infinite level shimura varieties
Imperial College London
2020
Buzzard, Kevin
Engineering and Physical Sciences Research Council
2021-02
2020-09
2021-04-16
Doctor of Philosophy (PhD)
10044/1/87649
Creative Commons Attribution NonCommercial Licence
https://creativecommons.org/licenses/by-nc/4.0/
Scholze's Siegel modular varieties $\cX_{\Gamma(p^\infty)}$ over a mixed characteristic perfectoid field are known to be perfectoid, yet their tilt remains somewhat elusive. We give two different methods to describe the tilt of spaces related to $\cX_{\Gamma(p^\infty)}$. The first, following an idea of Lurie, uses Drinfeld level structures to construct integral perfectoid models of infinite level Shimura curves. For the second method we show how to recover the Tate module of the universal family of abelian varieties from its anticanonical part together with the Weil pairing. This allows us to extend Scholze's description of the tilt of $\cX_{\Gamma_0(p^\infty)}(\epsilon)_a$ to level $\Gamma(p^\infty)$.