arXiv subjects

Finn Wiersig

Publications and source records attributed to Finn Wiersig.

5 recordsLinked to original sources

arXiv2026

Galois and Pro-étale Cohomology of Overconvergent de Rham Period Rings

Motivated by the theory of p-adic differential equations and p-adic geometric representation theory, we introduce overconvergent variants of Fontaine's classical period rings. In particular, we study the positive overconvergent de Rham period ring, which is the stalk of the structure sheaf of the analytic Fargues-Fontaine curve at infinity. Our main results include the computation of the Galois cohomology of these overconvergent period rings, as well as the cohomology of the associated period sheaves and period structure sheaves.

math.NT↗

arXiv2026

A fully faithful p-adic Riemann-Hilbert functor for coadmissible D-cap-modules

This article establishes a Riemann-Hilbert correspondence in rigid-analytic geometry. We construct an explicit solution functor and prove that it is fully faithful on Ardakov-Wadsley's coadmissible D-cap-modules. For vector bundles with flat connection, our functor is canonically identified with Scholze's horizontal sections functor.

math.AG↗

arXiv2026

The p-adic Cauchy Theorem and Overconvergent Period Sheaves

The classical p-adic Cauchy theorem asserts that formal solutions of ordinary p-adic differential equations are convergent. In this article we establish a geometric analogue of this result for arbitrary smooth rigid-analytic varieties. More precisely, we show that the horizontal sections functor defined using the overconvergent de Rham period structure sheaf agrees with Scholze's horizontal sections functor defined using OBdR. Equivalently, every formal solution arising from Scholze's construction is already overconvergent. As an application, we identify Scholze's horizontal sections functor with the de Rham functor for D-cap-modules on vector bundles with flat connection.

math.NT↗

arXiv2023

No short polynomials vanish on bounded rank matrices

We show that the shortest nonzero polynomials vanishing on bounded-rank matrices and skew-symmetric matrices are the determinants and Pfaffians characterising the rank. Algebraically, this means that in the ideal generated by all $t$-minors or $t$-Pfaffians of a generic matrix or skew-symmetric matrix one cannot find any polynomial with fewer terms than those determinants or Pfaffians, respectively, and that those determinants and Pfaffians are essentially the only polynomials in the ideal with that many terms. As a key tool of independent interest, we show that the ideal of a sufficiently general $t$-dimensional subspace of an affine $n$-space does not contain polynomials with fewer than $t+1$ terms.

math.AC↗

arXiv2021

Short polynomials in determinantal ideals

We show that a determinantal ideal generated by $t$-minors does not contain any nonzero polynomials with $t!/2$ or fewer terms. Geometrically this means that any nonzero polynomial vanishing on all matrices of rank at most $t-1$ has more than $t!/2$ terms.

math.AC↗