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Effective methods in *p*-adic cohomology

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March 15-19, 2010

Mathematical Institute, University of Oxford

This five day workshop was organised by Kiran Kedlaya and Alan Lauder, with the assistance of Sebastian Pancratz, and was supported by a grant from the European Research Council. It took place in the Mathematical Institute, University of Oxford, with accommodation for participants being provided by Hertford College. The conference dinner was held at St John's College.

The workshop drew together a broad spectrum of researchers at different stages in their careers and in different parts of mathematics, all of whom have a common interest in effective theorems and explicit calculations in rigid and crystalline cohomology.

The themes of the workshop included

- Applications to point counting on curves over finite fields
- Experimental investigations of L-functions of curves over function fields
*p*-adic cohomological calculations in mathematical physics, including investigation of singular varieties and degenerations- Computation of Coleman integrals,
*p*-adic heights on abelian varieties over number fields, and*p*-adic regulators in K-theory - Effective theorems on
*p*-adic differential equations - Computation of Picard-Fuchs differential systems

Here is the programme of the workshop.

Here are handwritten notes (taken by George Walker) for some of the talks along with some slides and an abstract.

- J. Balakrishnan: Local heights on elliptic curves Slides
- F. Baldassarri: Convergence polygon of a connection and differential
Grothendieck-Ogg-Shafarevich formula for coverings of
*p*-adic analytic curves Abstract - L. Berger: Computing the reduction of some
*p*-adic representations Notes - P. Berthelot: Rational points for regular models of algebraic varieties of Hodge level at least 1 Notes
- A. Besser: Deformation techniques for computation of Coleman integrals Notes
- D. Haessig: Relative Dwork cohomology of non-degenerate exponential sums Notes
- H. Hubrechts: Solving
*p*-adic differential equations in point counting algorithms Slides - R. Kloosterman: Average ranks of elliptic 3-folds and zeta functions of singular hypersurfaces Notes
- K. Kedlaya: Getting precise about precision Slides
- A. Lauder: Degenerations and limit Frobenius structures in rigid cohomology Notes
- B. Le Stum: Towards an overconvergent Deligne-Kashiwara correspondence Notes Slides
- A. Mellit: Computing Frobenius traces in a non-hypergeometric family Slides
- S. Pancratz: Fast reduction in the algebraic de Rham cohomology of projective hypersurfaces Slides
- F. Rodriguez-Villegas: Hypergeometric motives Notes
- K. Samol, Dwork congruences for reflexive polytopes Notes
- I. Shapiro: Frobenius maps on quintic 3-folds Notes
- N. Tsuzuki: Logarithmic growth and Frobenius slopes Notes
- D. Wan: Counting points on hypersurfaces - algorithms and complexity Notes
- J.D. Yu: Ordinary crystals with logarithmic poles Notes

Here is a list of participants:

J. Balakrishnan (MIT), F. Baldassarri (Padova),
L. Berger (Lyon), P. Berthelot (Rennes), A. Besser (Ben-Gurion), V. Busch (Hamburg), W. Castryck (Leuven),
B. Chiarellotto (Padova), P. Candelas (Oxford), A. Castano Dominguez (Seville), R. Crew (Florida), G. Chatel (INSA-Rennes), C. Davies (Bonn), R. de Jeu (VU-Amsterdam), R. Gerkmann (Mainz), M. Gros (Rennes), D. Haessig (Rochester), D. Harvey (Courant), D. Holmes (Warwick), H. Hubrechts (Leuven), K. Kedlaya (MIT), R. Kloosterman (Humboldt-Berlin), A. Lauder (Oxford), B. Le Stum (Rennes), A. Mellit (Bonn), S. Muller (Bayreuth), S. Pancratz (Oxford), D. Pigeon (Caen), A. Rojas Leon (Seville), K. Samol (Oxford), V. Settimi (Padova), I. Shapiro (Max-Planck), B. Smith (LIX), J. Stienstra (Utrecht), N. Suwa (Chuo), D. Testa (Oxford), N. Tsuzuki (Tohoku), J. Tuitman (Leuven), G. Walker (Bristol), D. Wan (UC Irvine), C. Wuthrich (Nottingham), J.D. Yu (NU Taiwan).

The workshop lectures were open to all.