Isocrystal - Knowlpedia

Definition

Let $k$ be a perfect field of characteristic $p$ , let $K_0=W(k)[1/p]$ , and let $\sigma$ be the Frobenius automorphism of $K_0$ . An isocrystal over $k$ is a finite-dimensional $K_0$ -vector space $D$ equipped with a bijective $\sigma$ -semilinear map

\varphi:D\longrightarrow D, \qquad \varphi(av)=\sigma(a)\varphi(v).

Equivalently, it is an $F$ -crystal after inverting $p$ ; lattices and $p$ -power torsion are forgotten.

Slopes

When $k$ is algebraically closed, the Dieudonné–Manin classification decomposes an isocrystal, up to isomorphism, into isoclinic pieces indexed by rational Newton slopes. The multiset of slopes, with multiplicities, forms its Newton polygon.

For example, the one-dimensional isocrystal with $\varphi=p^m\sigma$ has slope $m$ . Rational slopes with nontrivial denominators occur in higher dimension.

G-isocrystals

For a reductive group $G$ , a $G$ -isocrystal can be encoded by a tensor functor from $\operatorname{Rep}(G)$ to isocrystals. After a trivialization, its Frobenius is represented by an element $b\in G(\breve F)$ , and changing the trivialization changes $b$ by sigma-conjugacy. The resulting classes form the Kottwitz set $B(G)$ .

References

Jean Dieudonné, “Groupes de Lie et hyperalgèbres de Lie sur un corps de caractéristique $p>0$ ,” Commentarii Mathematici Helvetici 28 (1954), 87–118.
Robert E. Kottwitz, “Isocrystals with additional structure,” Compositio Mathematica 56 (1985), 201–220. Numdam.