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jeunes en arithmétique et variétés algébriques
Program by Yves André

There is a certain flexibility in the choice of the material, because this workshop does not aim at presenting (as usual) the proof of one theorem followed by some applications, but instead a whole colourful landscape with two sides (complex and $p$-adic) and the many passes from one to the other. The talks should therefore concentrate on motivations (analogies, historical sources, examples - as in real history, they should come before the definitions), constructions and principles. Some talks are planned like long walks, others stop at a theorem-shelter; detours and personal viewpoints are encouraged.

Timetable

1. Period mappings: the elliptic case - and hypergeometric functions.
Javier Fresán

[Y], and [G] for the historical viewpoint (from Gauss to Riemann and Schwarz).

2. Period mappings: the case of abelian varieties with prescribed endomorphisms.
Yohan Brunebarbe

Period domains and mappings, Gauss-Manin connection, Hodge filtration. [CMP] [A, II1]

3. Moduli spaces of abelian varieties with prescribed endomorphisms.
Victoria Cantoral Farfán

Shimura varieties of PEL type, Example: Shimura curves. [CMP] [A, II1] [Cl]

4. Abelian varieties and $p$-divisible groups.
Diego Izquierdo

Serre-Tate theorem. [A, II2]

5. Dieudonné modules of $p$-divisible groups.
Ramla Abdellatif

Grothendieck-Messing theorem. [A, II3]

6. Moduli spaces of $p$-divisible groups.
Brian Lawrence

Rapoport-Zink spaces. [A, II4] [RZ]

7. $p$-adic period domains/mappings.
Salim Tayou

Relation to the Gauss-Manin connection. [A, II5, 6] [RZ]

8. $p$-adic local (sometimes global) uniformization of Shimura varieties.
Giacomo Graziani

Rapoport-Zink theorem. [A, II7] [RZ]

9. Example: $p$-adic uniformization of Shimura curves and period mappings.
Yunqing Tang

Cherednik-Drinfeld theorem, relation to the Gauss-Manin connection. [A, II 7.4, III 4.7]

10. Tempered coverings and fundamental group.
Pedro Ángel Castillejo

[A, III 2] [L]

11. Orbifolds and uniformizing differential equations (complex and $p$-adic).
Daniele Turchetti

[A, III 4]

12. (complex and $p$-adic) triangle groups.
Peter Jossen

[A, III 5, 6]

13. $p$-divisible groups over $O_C$: isotriviality.
Sergey Gorchinskiy

[SW 5.1.4]

14. $p$-divisible groups over $O_C$: classification.
Paul Ziegler

[SW 5.2.1]

Rigid connections, $F$-isocrystals and integrality
On a smooth complex projective variety, Carlos Simpson conjectures that a rigid integrable connection is motivic. This in particular implies that the monodromy is integral. We prove the integrality conjecture when the connection defines a smooth moduli point. To this aim, we prove that the mod p reduction of a rigid integrable connection has the structure of an isocrystal with Frobenius structure. We also prove that rigid integrable flat connections with vanishing $p$-curvatures are unitary. This allows one to prove new cases of Grothendieck’s $p$-curvature conjecture. Joint work with Michael Groechenig.