Seminar : $p$-adic uniformization of Shimura curves

This is the first seminar in the series of seminars about moduli of $p$-divisible groups and Shimura varieties. In this seminar we study moduli of $p$-divisible groups. We start with studying Drinfeld's interpretation of a rigidified moduli problem of $p$-divisible groups.

Literature

[BC] Boutot, Carayol - Uniformisation $p$-adique des courbes de shimura: les theoremes de Cerednik et de Drinfeld, 1991
[Drin] Drinfeld - Coverings of $p$-adic symmetric domains, 1976

Meetings

Thursday, July 4, 2013
14:00-16:00	V3-200	Hendrik	The proof of Drinfeld's Theorem [Abstract] We introduce the moduli problems $\overline{G}$ and $\overline{H}$, where $\overline{H}$ is based on the functor $F$. Drinfeld's theorem says that $\overline{G}$ is representable by the formal scheme $\widehat{\Omega} \hat{\otimes} \hat{\mathbb{Z}_p^{nr}}$. We prove this using a map $\xi : \overline{G} \rightarrow \overline{H}$. First however, in order to define $\xi$, we prove that the sheaf $\eta_M$ and its components are $\pi$-adic sheafs that are constructible.
Tuesday, June 25, 2013
11:00-12:20 + 17:00-18:00	V3-200	Hendrik	Admissible triples [Abstract] We define what an admissible triple $(\eta,T, u)$ is and note that it has much to do with the moduli problem $F$ that we considered before. We discuss the notion of quasi-isogenies of height zero and rigidifications for special formal $O_D$-modules and admissible triples. Let $L$ be an algebraically closed field of characteristic $p > 0$. The main goal of this talk is to establish a correspondence between special formal $O_D$-modules of height $4$ over $L$ equipped with a rigidification and admissible triples over $L$ equipped with a rigidification.
Thursday, June 20, 2013
14:00-15:00	V5-227	Cheng	Moduli problem of abelian surfaces, II [Abstract] We define a similar moduli problem as last time, but now over $\mathbb{Z}_p$-schemes and with a "special" condition on the action. Most of the lecture we spent on proving that there exists a principal polarization that induces the $*$-involution on $O_\Delta$. This step involves the knowledge of the representability of the moduli problem involving $p$-divisible groups. Finally we prove the Cerednik-Drinfeld theorem that gives a $p$-adic uniformisation of the Shimura curve $S_U$.
Thursday, June 14, 2013
15:00-16:00	V5-227	Cheng	Moduli problem of abelian surfaces, I [Abstract] We introduced the moduli problem $M_U$ for abelian surfaces. We discussed the notion of $$-polarizations and showed that there exists a unique principal $$-polarization on an abelian surface defined over a scheme of characteristic zero.
Thursday, June 13, 2013
14:00-15:00	V5-227	Hendrik	The construction of $\eta_M$, II [Abstract] This time we define the map $L_M$ and $\eta_M$. We show invariance of $\eta_M$ under nilpotent base-change. Lastly, we define what a critical index is.
Friday, June 7, 2013
11:00-12:00	V5-227	Hendrik	The construction of $\eta_M$, I [Abstract] Starting from a special graded Cartier module $M$, we set out to construct a graded $\cal{O}[\Pi]$-module $\eta_M$. To do this, we define first a map $\lambda_M$, then $L_M$ and then $\phi_M$!
Wednesday, June 5, 2013
14:00-15:30	V5-227	Hendrik	Cartier modules with actions [Abstract] We start to link the two moduli problems by introducing Cartier modules. We define a $V$-reduced Cartier $\cal{O}$-module, a $V$-basis, discuss the equivalence of Cartier $\mathbb{E}_{\cal{O}}$- modules with an $\cal{O}'$-action and the ones with an $\cal{O'}$-coefficient ring. Then we discuss Cartier modules with an $\cal{O}_D$-action, where $D$ is a quaternion algebra.
Thursday, May 30, 2013
14:00-15:30	V5-227	Hendrik	Functorial description of $\widehat{\Omega}$, II [Abstract] We construct the functor $F$ and show that it is represented by the formal scheme $\widehat{\Omega}$. The functors $F_s$ and $F_{[s,s']}$ from the previous talk are used in an essential way to patch together this big functor $F$.
Thursday, May 17, 2013
14:00-15:30	V5-227	Hendrik	Functorial description of $\widehat{\Omega}$, I [Abstract] We construct the functors $F_s$ and $F_{[s,s']}$ of Deligne and show that they are representable by respectively $\widehat{\Omega_s}$ and $\widehat{\Omega_{[s,s']}}$.

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Seminar : $p$-adic uniformization of Shimura curves

Literature

Meetings

Thursday, July 4, 2013

Tuesday, June 25, 2013

Thursday, June 20, 2013

Thursday, June 14, 2013

Thursday, June 13, 2013

Friday, June 7, 2013

Wednesday, June 5, 2013

Thursday, May 30, 2013

Thursday, May 17, 2013