# zbMATH — the first resource for mathematics

Isocrystals with additional structure. (English) Zbl 0597.20038
From the author’s introduction: ”Let k be an algebraically closed field of characteristic $$p>0,$$ and let K be the fraction field of the Witt ring W(k). The Frobenius automorphism of k induces an automorphism $$\sigma$$ of K. An isocrystal is a finite dimensional vector space V over K together with a $$\sigma$$-semilinear bijection $$\Phi: V\to V.$$ Let V be an n- dimensional vector space over $${\mathbb{Q}}_ p$$ and let $$G=GL(V)$$. For any element $$b\in G(K)$$ we get an isocrystal $$(V_ K,\Phi)$$, where $$V_ K=V\otimes_{{\mathbb{Q}}_ p}K$$ and $$\Phi =b\circ (id_ V\otimes \sigma)$$. If b’ is $$\sigma$$-conjugate to b (in other words, if $$b'=gb\sigma (g)^{-1}$$ for some $$g\in G(K))$$, then the two isocrystals we get are isomorphic. This construction yields a bijection from the set of $$\sigma$$-conjugacy classes in G(K) to the set of isomorphism classes of n-dimensional isocrystals.”
The paper under review studies the set B(G) of $$\sigma$$-conjugacy classes in G(K) for arbitrary connected reductive groups G over $${\mathbb{Q}}_ p$$. The results about B(G) are useful in studying the points mod p on Shimura varieties.
Reviewer: H.-J.Bartels

##### MSC:
 20G05 Representation theory for linear algebraic groups 20G25 Linear algebraic groups over local fields and their integers 13K05 Witt vectors and related rings (MSC2000) 14L35 Classical groups (algebro-geometric aspects)
Full Text:
##### References:
 [1] A. Borel : Automorphic L-functions, Automorphic Forms, Representations and L- functions , Proc. Sympos. Pure Math., vol. 33, part 2, Amer. Math. Soc., Providence, R.I., 1979, pp. 27-61. · Zbl 0412.10017 [2] P. Berthelot and A. Ogus : F-isocrystals and de Rham cohomology. I , Invent. Math. 72 (1983), 159-199. · Zbl 0516.14017 [3] R. Kottwitz : Stable trace formula: cuspidal tempered terms , preprint. · Zbl 0576.22020 [4] R. Kottwitz : Shimura varieties and twisted orbital integrals , preprint. · Zbl 0533.14009 [5] M. Kneser : Galoiskohomologie halbeinfacher algebraischer Gruppen über p-adischen Körpern. I . Math. Zeit. 88 (1965) 40-47; II, Math. Zeit. 89 (1965) 250-272. · Zbl 0143.04702 [6] R.P. Langlands : On the zeta-functions of some simple Shimura varieties , Can. J. math. 31 (1979) 1121-1216. · Zbl 0444.14016 [7] N. Saavedra Rivano : Catégories Tannakiennes , Lecture Notes in Mathematics 265, Springer-Verlag, Heidelberg (1972). · Zbl 0241.14008 [8] R. Steinberg : Regular elements of semisimple algebraic groups , Publ. Math. IHES 25 (1965) 49-80. · Zbl 0136.30002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.