# $\sigma$-conjugate iff $p$-adically close

First some notations. Let $p$ be a prime, $k$ a perfect field of characteristic $p$, $W=W(k)$ the ring of Witt vectors over $k$, $\sigma : W \rightarrow W$ the Frobenius, $R$ a commutative $\mathbb{Z}_p$-algebra, $R_W = R \otimes_Z W$, $\phi$ the $R$-linear extension of $\sigma$ to $R_W$. Also denote by $\phi$ the endomorphism of $R_W((u))$ given by $\phi(\sum_i a_i u^i)=\sum_i \phi(a_i) u^{pi}$.

• $LG(R) = GL_d(R_W((u)))$
• $LG^+(R) = GL_d(R_W[[u]])$
• $LG^{\leq m}(R) = \{ A \in GL_d(R_W((u))) \mid A, A^{-1} \in u^{-m}M_d(R_W[[u]]) \}$

Proposition 2.2 in Pappas, Rapoport - $\phi$-modules and coefficient spaces, available here states the following

Suppose $n > \frac{2m}{p-1}$

1. For each $g\in U_n(R)$, $A\in LG^{\leq m}(R)$ there is a unique $H\in U_n(R)$ such that $g^{-1} A \phi(g) = H^{-1}A$.
2. Conversely for each $A\in LG^{\leq m}(R)$, $H\in U_n(R)$ there is a unique $g\in U_n(R)$ such that $g^{-1} A \phi(g) = H^{-1}A$.

In the footnote the authors state that the analogous fact in classical Dieudonné theory is also true. However it seems that the proof given there does not work in that case, the problem being that the Frobenius on the Witt vectors leaves the uniformizer $p$ fixed whereas in the situation of Proposition 2.2 above the uniformizer $u$ is raised to the $p$-th power by $\phi$. In the first statement this forces me to assume $m$ to be $0$, but the comparison of the coefficients in uniqueness part of 2. doesn't seem to be working at all.

How can one prove this result in the classical case?

EDIT: Scholze uses a similar thing in 'The Langlands-Kottwitz Method and Deformation Spaces of p-Divisible Groups'. This is Lemma 4.4 there. But he does not get explicit bounds.

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Conjectures on explicit bounds (at least for $G=GL_n$, and $\sigma$-conjugacy classes corresponding to $p$-divisible groups) are known as Traverso's conjecture. There is a bunch of papers by Vasiu (and coauthors) on this topic. Their arguments are making heavy use of (truncated) $p$-divisible groups. For general groups and general $\sigma$-conjugacy classes, I couldn't find a reference when writing the paper. Rapoport told me that Kottwitz was aware of this when writing his classical papers on isocrystals with additional structure, but he did not include that result. – Peter Scholze Jan 2 '13 at 10:17
Also, one should get explicit bounds from the method of proof in my paper. I think Rapoport had given this problem as a Master's Thesis, but I am not sure what became of it. – Peter Scholze Jan 2 '13 at 10:19
Also beware that contrary to the Pappas-Rapoport paper, one has to assume that $k$ is algebraically closed in the classical case. – Peter Scholze Jan 2 '13 at 10:20