ETHZ-UZh-Sbg Arithmetic and Geometry Research Group

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**Workshop on Diophantine problems and p-adic period mappings**

This workshop is organized by J. Ayoub, C. Fuchs, P. Habegger, R. Pink, and G. Wüstholz. The workshop takes place from June 30-July 5, 2019 at the Böglerhof in Alpbach/Tyrol, Austria.

This summer school will continue a tradition of summer schools in Alpbach (see here) in arithmetic and geometry (e.g. 2010, p-adic periods; 2011, motives, periods and transcendence; 2012, multiple zeta-values; 2013, p-adic modular forms; 2014, periods and heights of CM abelian varieties; 2015, special cycles on Shimura varities). PhD students and postdocs present the topics.

We study the very recent paper of Brian Lawrence and Akshay Venkatesh on Diophantine problems and p-adic period mappings.

**Participants:**

Joseph Ayoub (Zurich)

Gabriel Dill (Basel)

Clemens Fuchs (Salzburg)

Ziyang Gao (IMJ-PRG)

Sergey Gorchinskiy (Steklov Math. Inst., NRU HSE, Moscow)

Philipp Habegger (Basel)

Fritz Hörmann (Freiburg)

Emil Jacobsen (Zurich)

Ariyan Javanpeykar (Mainz)

Rafael von Känel (Beijing)

Bruno Klingler (Berlin)

Dmitry Krekov (NRU HSE, Moscow)

Andrew Kresch (Zurich)

Lars Kühne (Basel)

Lorenzo Mantovani (Zurich)

Alberto Merici (Zurich)

Nicolas Müller (Zurich)

Maxim Mornev (Zurich)

Nicola Nesa (Freiburg)

Doosung Park (Zurich)

Richard Pink (Zurich)

Harry Schmidt (Basel)

Paul Steinmann (Zurich)

Yunqing Tang (Princeton)

Gisbert Wüstholz (Zurich)

The participants consist of PhD students and postdocs in arithmetic and geometry from Zurich as well as colleagues from Basel, Freiburg, Lausanne, Leiden, Mainz, Moscow, Princeton, and Salzburg.

**Program:**

The program will start on Sunday evening, the first talk (given virtually online by B. Lawrence) is scheduled for 17:15-18:45. The detailed program can be found here: PDF (written by B. Lawrence).

(Tentative) Schedule:

09:00 - 10:30: First talk

10:30 - 11:00: Coffee break

11:00 - 12:30: Second talk

12:45: Lunch

13:45 - 15:15: Third talk

15:30 - 18:00: Time for informal discussion

19:00: Dinner

**Lecture Notes and Literature:**

Main article (by B. Lawrence and A. Venkatesh): arXiv:1807.02721

[1] D. Arapura: Computation of some Hodge numbers. Available online at PDF.

[2] B. Bakker and J. Tsimerman: The Ax-Schanuel conjecture for variations of Hodge structures. Available online at arXiv:1712.05088.

[3] P. Berthelot: Cohomologie cristalline des schemas de caracteristique p > 0. Springer, Lecture Notes in Mathematics, 1974

[4] P. Berthelot and A. Ogus: Notes on Crystalline Cohomology

[5] B. Bhatt and A. J. de Jong: Crystalline cohomology and de Rham cohomology

[6] O. Brinon and B. Conrad: CMI summer school notes on p-adic Hodge theory. Available online at PDF.

[7] A. Chambert-Loir: Cohomologie cristalline: un survol

[8] G. Faltings: Endlichkeitssätze für abelsche Varietäten über Zahlkörper

[9] B. Farb and D. Margalit: A primer on mapping class groups

[10] J.-M. Fontaine, ed.: Periodes p-adiques. Asterisque 223, 1994

[11] W. Fulton: Hurwitz schemes and irreducibility of moduli of algebraic curves. Annals of math., 1969

[12] B. Lawrence and A. Venkatesh: Diophantine problems and p-adic period mappings. Available online at arXiv:1807.02721.

[13] S. Mochizuki: The geometry of the compactification of the Hurwitz scheme

[14] G. Wüstholz: The finiteness theorems of Faltings. In: Rational points, Springer, 1992

Back to the main page.

This workshop is organized by J. Ayoub, C. Fuchs, P. Habegger, R. Pink, and G. Wüstholz. The workshop takes place from June 30-July 5, 2019 at the Böglerhof in Alpbach/Tyrol, Austria.

This summer school will continue a tradition of summer schools in Alpbach (see here) in arithmetic and geometry (e.g. 2010, p-adic periods; 2011, motives, periods and transcendence; 2012, multiple zeta-values; 2013, p-adic modular forms; 2014, periods and heights of CM abelian varieties; 2015, special cycles on Shimura varities). PhD students and postdocs present the topics.

We study the very recent paper of Brian Lawrence and Akshay Venkatesh on Diophantine problems and p-adic period mappings.

Joseph Ayoub (Zurich)

Gabriel Dill (Basel)

Clemens Fuchs (Salzburg)

Ziyang Gao (IMJ-PRG)

Sergey Gorchinskiy (Steklov Math. Inst., NRU HSE, Moscow)

Philipp Habegger (Basel)

Fritz Hörmann (Freiburg)

Emil Jacobsen (Zurich)

Ariyan Javanpeykar (Mainz)

Rafael von Känel (Beijing)

Bruno Klingler (Berlin)

Dmitry Krekov (NRU HSE, Moscow)

Andrew Kresch (Zurich)

Lars Kühne (Basel)

Lorenzo Mantovani (Zurich)

Alberto Merici (Zurich)

Nicolas Müller (Zurich)

Maxim Mornev (Zurich)

Nicola Nesa (Freiburg)

Doosung Park (Zurich)

Richard Pink (Zurich)

Harry Schmidt (Basel)

Paul Steinmann (Zurich)

Yunqing Tang (Princeton)

Gisbert Wüstholz (Zurich)

The participants consist of PhD students and postdocs in arithmetic and geometry from Zurich as well as colleagues from Basel, Freiburg, Lausanne, Leiden, Mainz, Moscow, Princeton, and Salzburg.

The program will start on Sunday evening, the first talk (given virtually online by B. Lawrence) is scheduled for 17:15-18:45. The detailed program can be found here: PDF (written by B. Lawrence).

Sunday, 30.06.2019 (Arrival day): Introduction (welcome talk given virtually online by B. Lawrence) |

Monday: Introduction to p-adic hodge theory I (Doosung Park), II (Maxim Mornev), The S-unit equation (Gabriel Dill) |

Tuesday: Construction of the Kodaira-Parshin family (Nicola Nesa), Friendly places and generic simplicity (Emil Jacobsen), The main argument I (Fritz Hörmann) |

Wednesday: The main argument II (Ziyang Gao), Monodromy I (Alberto Merici) |

Thursday: Monodromy II (Lorenzo Mantovani), Hypersurfaces I: introduction (Bruno Klingler*TBC), Hypersurfaces II: reduction of Theorem 10.1 to Proposition 10.6 (Sergey Gorchinskiy) |

Friday, 05.07.2019: Hypersurfaces III: combinatorics on reductive groups, proof of Proposition 10.6 (Lars Kühne) |

(Tentative) Schedule:

09:00 - 10:30: First talk

10:30 - 11:00: Coffee break

11:00 - 12:30: Second talk

12:45: Lunch

13:45 - 15:15: Third talk

15:30 - 18:00: Time for informal discussion

19:00: Dinner

Main article (by B. Lawrence and A. Venkatesh): arXiv:1807.02721

[1] D. Arapura: Computation of some Hodge numbers. Available online at PDF.

[2] B. Bakker and J. Tsimerman: The Ax-Schanuel conjecture for variations of Hodge structures. Available online at arXiv:1712.05088.

[3] P. Berthelot: Cohomologie cristalline des schemas de caracteristique p > 0. Springer, Lecture Notes in Mathematics, 1974

[4] P. Berthelot and A. Ogus: Notes on Crystalline Cohomology

[5] B. Bhatt and A. J. de Jong: Crystalline cohomology and de Rham cohomology

[6] O. Brinon and B. Conrad: CMI summer school notes on p-adic Hodge theory. Available online at PDF.

[7] A. Chambert-Loir: Cohomologie cristalline: un survol

[8] G. Faltings: Endlichkeitssätze für abelsche Varietäten über Zahlkörper

[9] B. Farb and D. Margalit: A primer on mapping class groups

[10] J.-M. Fontaine, ed.: Periodes p-adiques. Asterisque 223, 1994

[11] W. Fulton: Hurwitz schemes and irreducibility of moduli of algebraic curves. Annals of math., 1969

[12] B. Lawrence and A. Venkatesh: Diophantine problems and p-adic period mappings. Available online at arXiv:1807.02721.

[13] S. Mochizuki: The geometry of the compactification of the Hurwitz scheme

[14] G. Wüstholz: The finiteness theorems of Faltings. In: Rational points, Springer, 1992

Impressum 18.03.2019