arXiv subjects

Rustam Steingart

Publications and source records attributed to Rustam Steingart.

7 recordsLinked to original sources

arXiv2025

On equivariant vector bundles on the Fargues--Fontaine curve over a finite extension

Let $K/E/\mathbb{Q}_p$ be a tower of finite extensions with $E$ Galois. We relate the category of $G_K$-equivariant vector bundles on the Fargues--Fontaine curve with coefficients in $E$ with $E$-$G_K$-$B$-pairs and describe crystalline and de Rham objects in explicit terms. When $E$ is a proper extension, we give a new description of the category in terms of compatible tuples of $\mathbf{B}_e$-modules, which allows us to compute Galois cohomology in terms of an explicit \v{C}ech complex which can serve as a replacement of the fundamental exact sequence.

math.NT↗

arXiv2025

Partial Bloch--Kato Selmer groups of $B$-pairs as delta functors

In this article we revisit the partial Selmer groups introduced by Ding in cohomological degree one. On the subcategory of partially de Rham positive $B$-pairs we extend them to higher cohomological degree and show that the resulting groups form a cohomological delta functor satisfying a variant of the Euler--Poincar\'e characteristic formula and Tate duality.

math.NT↗

arXiv2024

On the higher analytic vectors of $\mathbf{B}_e$

We prove that the first derived analytic vectors of the subring of Fontaine's period ring $\mathbf{B}_e$ stable under the kernel of the cyclotomic character are non-zero. Subsequently we compute their analytic cohomology. We also give a description of the cokernel of the restriction of a variant of the Bloch-Kato exponential map for $\mathbb{Q}_p(n)$ to analytic vectors in terms of derived analytic vectors. In order to achieve the above, we relate pro-analytic vectors with derived analytic vectors in condensed mathematics for regular LF-spaces.

math.NT↗

arXiv2024

$\epsilon$-isomorphisms for rank one $(\varphi,\Gamma)$-modules over Lubin-Tate Robba rings

Inspired by Nakamura's work (arXiv:1305.0880) on $\epsilon$-isomorphisms for $(\varphi,\Gamma)$-modules over (relative) Robba rings with respect to the cyclotomic theory, we formulate an analogous conjecture for $L$-analytic Lubin-Tate $(\varphi_L,\Gamma_L)$-modules over (relative) Robba rings for any finite extension $L$ of $\mathbb{Q}_p.$ In contrast to Kato's and Nakamura's setting, our conjecture involves $L$-analytic cohomology instead of continuous cohomology within the generalized Herr complex. Similarly, we restrict to the identity components of $D_{cris}$ and $D_{dR},$ respectively. For rank one modules of the above type or slightly more generally for trianguline ones, we construct $\epsilon$-isomorphisms for their Lubin-Tate deformations satisfying the desired interpolation property.

math.NT↗

arXiv2023

Comparisons of Lie algebra cohomologies of $(\varphi,\Gamma)$-modules

We generalise a result of Fourquaux and Xie thereby completely determining the relationship between $\mathbb{Q}_p$ and $L$-analytic Lie algebra cohomology of analytic $(\varphi_L,\Gamma_L)$-modules. We use the results to conclude that for $L\neq \mathbb{Q}_p,$ there exist examples of \'etale $(\varphi_L,\Gamma_L)$-modules over Robba rings whose $\mathbb{Q}_p$-analytic cohomology does not arise as a base change of Galois cohomology.

math.NT↗

arXiv2022

Iwasawa cohomology of analytic $(\varphi_L,\Gamma_L)$-modules

We show that the coadmissibility of the Iwasawa cohomology of an $L$-analytic Lubin-Tate $(\varphi_L,\Gamma_L)$-module $M$ is necessary and sufficient for the existence of a comparison isomorphism between the former and the analytic cohomology of its Lubin-Tate deformation, which, roughly speaking, is given by the base change of $M$ to the algebra of $L$-analytic distributions. We verify that coadmissibility is satisfied in the trianguline case and show that it can be ``propagated'' to a reasonably large class of modules, provided it can be proven in the \'etale case.

math.NT↗

arXiv2022

Finiteness of analytic cohomology of Lubin-Tate $(\varphi_L,\Gamma_L)$-modules

We prove finiteness and base change properties for analytic cohomology of families of $L$-analytic $(\varphi_L,\Gamma_L)$-modules parametrised by affinoid algebras in the sense of Tate. For technical reasons we work over a field $K$ containing a period of the Lubin-Tate group, which allows us to describe analytic cohomology in terms of an explicit generalized Herr complex.

math.NT↗