P-adic Hodge theory

introduction

The Hodge decomposition of the cohomology of smooth projective complex varieties is a fundamental tool in the study of their geometry. Over an arbitrary base field of characteristic zero, 'etale cohomology with \(\Q_p\)-coefficients is a good substitute for singular cohomology with complex coefficients, but in general no analogue of the Hodge decomposition is known. However, owing to a fundamental insight of Tate, we know that over a \(p\)-adic base field a version of Hodge decomposition can indeed be constructed. Moreover, the cohomology groups involved carry an action of the Galois group of the base field, whose interaction with the Hodge decomposition can be analyzed by methods inspired by the study of the monodromy action in the complex case. This has deep consequences for the study of varieties of arithmetic interest, and can even be used to prove some purely geometric statements.

The first proof of the \(p\)-adic Hodge decomposition is due to Faltings [faltings1]; several other proofs have been given since. Perhaps the latest one to date is a wonderful proof by Beilinson [Bei] which is the closest to geometry. It can be hoped that its groundbreaking new ideas will lead to important applications; some of them already appear in the recent construction of \(p\)-adic realizations of mixed motives by D\'eglise and Niziol. Moreover, one of the key tools in Beilinson's approach is Illusie's theory of the derived de Rham complex which has also reappeared during the recent development of derived algebraic geometry. Beilinson's work may thus also be viewed as a first bridge between this emerging field and \(p\)-adic Hodge theory.

articles

Peter Scholze, Canonical q-deformations in arithmetic geometry, arXiv:1606.01796 [math.AG], June 06 2016, http://arxiv.org/abs/1606.01796
Laurent Berger, Lubin's conjecture for full \(p\)-adic dynamical systems, arXiv:1603.03631[math.NT], March 11 2016, http://arxiv.org/abs/1603.03631v2
Rebecca Bellovin, \(p\)-adic Hodge theory in rigid analytic families, 10.2140/ant.2015.9.371, http://dx.doi.org/10.2140/ant.2015.9.371, Algebra Number Theory 9 (2015) 371-433, http://arxiv.org/abs/1306.5685v2
[Bei] A. Beilinson, \(p\)-adic periods and derived de Rham cohomology, J. Amer. Math. Soc.} 25 (2012), no. 3, 715-738.
[faltings1] Faltings, \(p\)-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), 255-299.

expositions

Tamás Szamuely, Gergely Zábrádi, The p-adic Hodge decomposition according to Beilinson, arXiv:1606.01921 [math.NT], June 06 2016, http://arxiv.org/abs/1606.01921
Bhatt, Bhargav, Matthew Morrow, and Peter Scholze. “Integral \(p\)-Adic Hodge Theory - Announcement.” arXiv:1507.08129 [math], July 29, 2015. http://arxiv.org/abs/1507.08129.

메타데이터

위키데이터

ID : Q7116911

Spacy 패턴 목록

[{'LOWER': 'p'}, {'OP': '*'}, {'LOWER': 'adic'}, {'LOWER': 'hodge'}, {'LEMMA': 'theory'}]