ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11, 14 Tsuji, Takeshi, Research Institute for Mathematical Sciences, Kyoto University, Japan p-adic Hodge Theory in the semi-stable reduction case

I will give a brief survey of recent developments of p-adic Hodge theory in the semi-stable reduction case. For a proper smooth algebraic variety X over the complex number field, Hodge theory relates the singular cohomology H^{∗}(Xan, ℚ) of the analytic space Xan associated to X with the de Rham cohomology HdR^{∗}(X/ℂ). padic Hodge theory is its analogue for a proper smooth algebraic variety X over a p-adic field (more precisely, a complete discrete valuation field of mixed characteristic (0, p) with perfect residue field, e. g. a finite extension of ℚ_{p}). The p-adic étale cohomology Hét^{∗}(X\overline{K}, ℚ_{p}), which is a ℚ_{p}-vector space of finite dimension endowed with a continuous linear action of the absolute Galois group of K, plays the role of the singular cohomology. The de Rham cohomology HdR^{∗}(X/K) is still defined in the p-adic case and is a K-vector space of finite dimension. The p-adic Hodge Theory dates back to the works of J. Tate and A. Grothendieck ([9], [8]) on an abelian variety with good reduction and the conjecture of Tate ([9]) (the Hodge-Tate conjecture=CHT): an analogue of the Hodge decomposition. By establishing a functor connecting p-adic étale and de Rham cohomologies endowed with various additional structures (called a “mysterious functor” by A. Grothendieck) , J. -M. Fontaine (and U. Jannsen in the semi-stable case) formulated several refined conjectures: the de Rham conjecture=CdR, the crystalline conjecture=Ccrys and the semi-stable conjecture=Cst ([4], [5], [6]). These conjectures were studied by many mathematicians such as M. Raynaud, J. -M. Fontaine, G. Laffaille, W. Messing, S. Bloch, K. Kato, G. Faltings and O. Hyodo, and the conjectures CHT, CdR and Ccrys were already solved without any restriction by G. Faltings in the end of 1980s. There were two approaches to the conjectures: the method of syntomic cohomology and p-adic vanishing cycles by Fontaine-Messing and Kato, and the method of almost étale extensions by Faltings. (Recently another method was found by W. Niziol). For Cst, we needed the assumption dim X < (p − 1)/2 (K. Kato by the first method) or that X has a model with at most a semi-stable curve singularity (G. Faltings by the second one) at that time. In these four years, the restriction was removed first by myself by the Kato’s method (for p ≥ 3 in 1994 and for p = 2 in 1996) [10] and then by G.

Faltings by his method (1996) [7] together with its generalization to nonconstant coefficients and the relative case. Also a beautiful theory for ptorsion cohomologies in the semi-stable case was established by C. Breuil (1996 [1]) by overcoming the difficulty in treating the cohomologies degreewise in a previous work of Faltings. Thanks to the alteration of de Jong [3], we can now derive easily all the other conjectures from Cst and further I proved, by the method of [2], that the p-adic étale cohomology of a proper variety is always a potentially semi-stable representation even if it has a singularity. I will survey these developments in my talk.

References

[1] Breuil, C., Cohomologie étale de p-torison et cohomologie cristalline en réduction semi-stable, to appear in Duke Math. J.

[2] Deligne, P., Théorie de Hodge III, Publ. Math. IHES 44 (1974), 5-77.

[3] de Jong, A. J., Smoothness, semi-stability and alterations, Publ. Math. IHES 83 (1996), 51–93.

[4] Fontaine, J. -P., Sur certains types de représentations p-adiques de groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate, Ann. of Math. 115 (1982), 529–577.

[5] Fontaine, J. -P., Le corps des périodes p-adiques, Périodes p-adiques, Séminaire de Bures, 1988, Astérisque 223 (1994), 59–111.

[6] Fontaine, J. -P., Représentations p-adiques semi-stables, Périodes padiques, Séminaire de Bures, 1988, Astérisque 223 (1994), 113-183.

[7] Faltings, G., Almost étale extensions, Max-Planck-Institut für Mathematik Preprint Series 1998.

[8] Grothendieck, A., Groupes de Barsotti-Tate et cristaux, Actes Congrès Int. Math. Nice 1970, I, Gauthier-Villars, Paris, 1971.

[9] Tate, J., p-divisible groups, in Proc. of a conf. on local fields, Driebergen (1967), 158–183.

[10] Tsuji, T. p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, preprint 1997.