K_1 of some noncommutative group rings

A formal power series over a noncommutative Hecke ring and the rationality of the Hecke series for $GSp_4$

The present paper studies Hecke rings derived by the automorphism groups of certain algebras $L_p$ over the ring of $p$-adic integers. Our previous work considered the case where $L_p$ is the Heisenberg Lie algebra (of dimension 3) over the ring of $p$-adic integers. Although this Hecke ring is noncommutative, we showed that a formal power series with coefficients in this Hecke ring satisfies an identity similar to the rationality of the Hecke series for $GL_2$ due to E.~Hecke. In the present paper, we establish an analogous result in the case of the Heisenberg Lie algebra of dimension 5 over …

(ϕ,Γ)-modules over noncommutative overconvergent and Robba rings

We construct noncommutative multidimensional versions of overconvergent power series rings and Robba rings. We show that the category of étale $(φ,Γ)$-modules over certain completions of these rings are equivalent to the category of étale $(φ,Γ)$-modules over the corresponding classical overconvergent, resp. Robba rings (hence also to the category of $p$-adic Galois representations of $\mathbb{Q}_p$). Moreover, in the case of Robba rings, the assumption of étaleness is not necessary, so there exists a notion of trianguline objects in this sense.

Unit L-functions for étale sheaves of modules over noncommutative rings

Let $s\colon X\rightarrow \operatorname{Spec} \mathbb{F}$ be a separated scheme of finite type over a finite field $\mathbb{F}$ of characteristic $p$, let $Λ$ be a not necessarily commutative $\mathbb{Z}_p$-algebra with finitely many elements, and let $\mathcal{F}^\bullet$ be a perfect complex of $Λ$-sheaves on the étale site of $X$. We show that the ratio $L(\mathcal{F}^\bullet,T)/L(R s_!\mathcal{F}^\bullet,T)$, which is a priori an element of $K_1(Λ[[T]])$, has a canonical preimage in $K_1(Λ[T])$. We use this to prove a version of the noncommmutative Iwasawa main conjecture for $p$-adic Lie …

Period Rings with Big Coefficients and Application II

We continue our study on the corresponding noncommutative deformation of the relative $p$-adic Hodge structures of Kedlaya-Liu along our previous work. In this paper, we are going to initiate the study of the corresponding descent of pseudocoherent modules carrying large noncommutative coefficients. And also we are going to more systematically study the corresponding noncommutative geometric aspects of noncommutative deformation of Hodge structures, which will definitely also provide the insights not only for noncommutative Iwasawa theory but also for noncommutative analytic geometry. The nonc…

Period Rings with Big Coefficients and Applications III

We continue our study on the corresponding period rings with big coefficients, with the corresponding application in mind on relative $p$-adic Hodge theory and noncommutative analytic geometry. In this article, we extend the discussion of the corresponding noncommutative descent over étale topology to the corresponding noncommutative descent over pro-étale topology in both Tate and analytic setting.

非交换环上矩阵的类 $\mathrm{SL}_2$ 性质与 Markov 数的推广

$\mathrm{SL}_2$-like Properties of Matrices Over Noncommutative Rings and Generalizations of Markov Numbers

本文研究带反对合非交换环上的 $2\times2$ 矩阵，定义迹与行列式并推广 $\mathrm{SL}_2$ 关系，进而构造量子化矩阵群与 Hopf 代数，并给出非交换 Markov 数。