Perfectoid spaces and the weight-monodromy conjecture, after Peter Scholze
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PERFECTOID SPACES AND THE WEIGHT-MONODROMY CONJECTURE, AFTER PETER SCHOLZE Takeshi Saito
Received: 19 August 2013 / Revised: 7 September 2013 / Accepted: 14 November 2013 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014
Abstract The monodromy weight conjecture is one of the main remaining open problems on Galois representations. It implies that the local Galois action on the -adic cohomology of a proper smooth variety is almost completely determined by the traces. Peter Scholze proved the conjecture in many cases including smooth complete intersections in a projective space, using a new powerful tool in rigid geometry called perfectoid spaces. The main arguments of the proof as well as basic ingredients in the theory of perfectoid spaces are presented. Keywords Perfectoid spaces · The weight-monodromy conjecture · Galois representations · -adic cohomology Mathematics Subject Classification (2000) 11G25 · 14G20 · 14G22 1 Weight-monodromy conjecture The Galois representation associated with the étale cohomology of a variety defined over a number field is a central subject of study in number theory. The Galois action on the Tate module of an elliptic curve and the Galois representation associated with a modular form that played the central role in the proof of Fermat’s last theorem by Wiles and Taylor are typical examples. To simplify the notation, we assume that a proper smooth variety X is defined over the rational number field Q. We fix a prime number and consider the representation of the ¯ absolute Galois group Gal(Q/Q) acting on the -adic étale cohomology H q (XQ¯ , Q ). As is seen in the definition of the Hasse-Weil L-function, a standard method in the study of a Galois representation is to investigate it locally at each prime. Let p be a prime number different from . If X has good reduction at p, the Galois representation H q (XQ¯ , Q ) at p is almost completely understood thanks to the Weil conjecture ¯ p /Qp ) at p proved by Deligne [1]. Namely its restriction to the decomposition group Gal(Q is unramified and the characteristic polynomial det(1 − Fp t : H q (XQ¯ , Q )) of the geometric
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T. Saito ( ) Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan e-mail: [email protected]
T. SAITO
Frobenius is determined by counting the number of points of the reduction of X modulo p defined over Fpn for every n 1. However, for a prime of bad reduction, an important piece, called the weight-monodromy conjecture, is still missing. To state it, let us recall briefly the structure of the absolute Ga¯ p /Qp ). Corresponding to the maximal unramified extension and the maxlois group Gal(Q tr ur 1/m ¯ p, ; p m) ⊂ Q imal tamely ramified extension Qp ⊂ Qur p = Qp (ζm ; p m) ⊂ Qp = Qp (p ¯ the inertia subgroup and its wild part Gal(Qp /Qp ) ⊃ I ⊃ P ⊃ 1 are defined. The quotient ¯ p /Qp )/I is canonically identified with Gal(F¯ p /Fp ) and is topologically generated Gal(Q by the geometric
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