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Sandra Rozensztajn

Publications and source records attributed to Sandra Rozensztajn.

9 recordsLinked to original sources
arXiv2023

Super-Hölder vectors and the field of norms

Let E be a field of characteristic p. In a previous paper of ours, we defined and studied super-Hölder vectors in certain E-linear representations of Z_p. In the present paper, we define and study super-Hölder vectors in certain E-linear representations of a general p-adic Lie group. We then consider certain p-adic Lie extensions K_\infty / K of a p-adic field K, and compute the super-Hölder vectors in the tilt of K_\infty. We show that these super-Hölder vectors are the perfection of the field of norms of K_\infty / K. By specializing to the case of a Lubin-Tate extension, we are able to recover E((Y)) inside the Y-adic completion of its perfection, seen as a valued E-vector space endowed with the action of O_K^\times given by the endomorphisms of the corresponding Lubin-Tate group.

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arXiv2022

Decompletion of cyclotomic perfectoid fields in positive characteristic

Let $E$ be a field of characteristic $p$. The group $\mathbf{Z}_p^\times$ acts on $E((X))$ by $a \cdot f(X) = f((1+X)^a-1)$. This action extends to the $X$-adic completion $\tilde{\mathbf{E}}$ of $\cup_{n \geq 0} E((X^{1/p^n}))$. We show how to recover $E((X))$ from the valued $E$-vector space $\tilde{\mathbf{E}}$ endowed with its action of $\mathbf{Z}_p^\times$. To do this, we introduce the notion of super-Hölder vector in certain $E$-linear representations of $\mathbf{Z}_p$. This is a characteristic $p$ analogue of the notion of locally analytic vector in $p$-adic Banach representations of $p$-adic Lie groups.

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arXiv2019

On the locus of $2$-dimensional crystalline representations with a given reduction modulo $p$

We consider the family of irreducible crystalline representations of dimension $2$ of ${\rm Gal}(\overline{\bf Q}_p/{\bf Q}_p)$ given by the $V_{k,a_p}$ for a fixed weight integer $k\geq 2$. We study the locus of the parameter $a_p$ where these representations have a given reduction modulo $p$. We give qualitative results on this locus and show that for a fixed $p$ and $k$ it can be computed by determining the reduction modulo $p$ of $V_{k,a_p}$ for a finite number of values of the parameter $a_p$. We also generalize these results to other Galois types.

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arXiv2018

Reduction of Galois Representations of slope 1

We compute the reductions of irreducible crystalline two-dimensional representations of $G_{\mathbf{Q}_p}$ of slope 1, for primes $p \geq 5$, and all weights. We describe the semisimplification of the reductions completely. In particular, we show that the reduction is often reducible. We also investigate whether the extension obtained is peu or très ramifiée, in the relevant reducible non-semisimple cases. The proof uses the compatibility between the $p$-adic and mod $p$ Local Langlands Correspondences, and involves a detailed study of the reductions of both the standard and non-standard lattices in certain $p$-adic Banach spaces.

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arXiv2017

An algorithm for computing the reduction of $2$-dimensional crystalline representations of $\text{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$

We describe an algorithm to compute the reduction modulo $p$ of a crystalline Galois representation of dimension $2$ of $\text{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$ with distinct Hodge-Tate weights via the semi-simple modulo $p$ Langlands correspondence. We give some examples computed with an implementation of this algorithm in SAGE.

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arXiv2015

Potentially semi-stable deformation rings for discrete series extended types

We define deformation rings for potentially semi-stable deformations of fixed discrete series inertial type in dimension $2$. In the case of representations of the Galois group of $\mathbf{Q}_p$, we prove an analogue of the Breuil-Mézard conjecture for these rings. As an application, we give some results on the existence of congruences modulo $p$ for newforms in $S_k(Γ_0(p))$.

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arXiv2014

Asymptotic values of modular multiplicities for GL_2

We study the irreducible constituents of the reduction modulo p of irreducible algebraic representations V of Res_{K/Q_p} GL_2 for K a finite extension of Q_p. We show that asymptotically, the multiplicity of each constituent depends only on the dimension of V and the central character of its reduction modulo p. As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.

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arXiv2007

Comparaison entre cohomologie cristalline et cohomologie étale $p$-adique sur certaines variétés de Shimura

Let $X$ be an integral model at a prime $p$ of a Shimura variety of PEL type having good reduction, associated to a reductive group $G$. To $\mathbb{Z}_p$ reprsententations of the group $G$ can be associated two kinds of sheaves : crystals on the special fiber of $X$, and locally constant étale sheaves on the generic fiber. We establish a comparison between the cohomology of these two kinds of sheaves.

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