arXiv · math/0004072
This paper has been superseded by math.NT/0205008
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Amilcar Pacheco. 2002-05-01. Idempotent relations and generalized p-ranks of curves. https://arxiv.org/abs/math/0004072
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The analytic Emerton-Gee stack is a rigid analytic moduli stack of $(φ,Γ)$-modules over the Robba ring, and is expected to play a fundamental role in the categorical $p$-adic local Langlands correspondence for locally analytic representations. In this paper, we establish the fundamental geometric properties of the analytic Emerton-Gee stack: it is a rigid analytic Artin stack, normal, a local complete intersection, and equidimensional of the expected dimension. As an application, our normality result removes the reducedness hypothesis in the paper of Le, Strauch, and Wang, thereby yielding an unconditional proof of the conjecture on the global sections of the analytic Emerton-Gee stack formulated by Emerton, Gee, and Hellmann.
Let $S=(s_1,s_2,\cdots)$ be a strictly increasing sequence of positive integers. For any real number $α>0$, let $S_α=(\lfloorαs_1\rfloor,\lfloorαs_2\rfloor,\cdots)$. We prove that if $s_{n+1}\le 2s_n$ for every positive integer $n$, then there exists $0<α<1$ such that $S_α$ is strongly complete. In addition, this gives a negative answer to a problem of Ma and Chen in [Acta Arith. 203 (2022), 307--318]. We also prove that the same result is true if $s_{n+1}\le 2s_n$ for all sufficiently large $n$.
The Coleman power series defined on a Lubin--Tate tower of extensions over $K$ are compatible with respect to two formal group laws: the multiplicative formal group law and some Lubin--Tate formal group law defined over $\mathcal{O}_K$. We ask if it is possible to generalize Coleman series in order to find series which are compatible with respect to any two Lubin--Tate formal group laws in the same way. We provide a precise formulation of this question and a partial answer towards the classification of all such power series that involves the eigenspaces of Coleman's trace operator. Some additional eigenspaces of Coleman's trace operator are also introduced. These new eigenspaces are used to construct certain fixed points of the $ψ$-operator on the family of trianguline dihedral $(φ, Γ)$-modules.