arXiv subjects

Vaidehee Thatte

Publications and source records attributed to Vaidehee Thatte.

7 recordsLinked to original sources

arXiv2026

The Hasse norm principle for $A_4$-quartic extensions of global function fields

For a finite extension of global fields $K/k$, the norm map $N_{K/k} : K^\times \to k^\times$ extends to a map on id\`ele groups. The Hasse norm principle holds if every element of $k^\times$ that is a norm everywhere locally is also a norm globally. In this paper, we study the statistics of the Hasse norm principle in a setting that is out of reach in the number field context, namely that of $A_4$-quartic extensions. We show that failures of the Hasse norm principle are generally rare for $A_4$-quartic extensions of global function fields $\mathbb{F}_q(t)$. We achieve this by introducing a decorated Hurwitz space parametrising the failures of the Hasse norm principle and then using the Chebotarev density theorem to estimate their frequency.

math.NT↗

arXiv2024

Degree $p$ Extensions of Arbitrary Valuation Rings and "Best $f$"

We prove the explicit characterization of the so-called "best f" for degree $p$ Artin-Schreier and degree $p$ Kummer extensions of Henselian valuation rings in residue characteristic $p$. This characterization is mentioned briefly in [Th16, Th18]. Existence of best $f$ is closely related to the defect of such extensions and this characterization plays a crucial role in understanding their intricate structure. We also treat degree $p$ Artin-Schreier defect extensions of higher rank valuation rings, extending the results in [Th16], and thus completing the study of degree $p$ extensions that are the building blocks of the general theory.

math.AC↗

arXiv2021

An explicit self-duality

We provide an exposition of the canonical self-duality associated to a presentation of a finite, flat, complete intersection over a Noetherian ring, following work of Scheja and Storch.

math.AC↗

arXiv2019

Local Oort groups and the isolated differential data criterion

It is conjectured that if k is an algebraically closed field of characteristic p > 0, then any branched G-cover of smooth projective k-curves where the "KGB" obstruction vanishes and where a p-Sylow subgroup of G is cyclic lifts to characteristic 0. Obus has shown that this conjecture holds given the existence of certain meromorphic differential forms on P_1^k with behavior determined by the ramification data of the cover. We give a more efficient computational procedure to compute these forms than was previously known. As a consequence, we show that all D_25- and D_27-covers lift to characteristic zero.

math.AG↗

arXiv2019

Upper Ramification Groups for Arbitrary Valuation Rings

T. Saito established a ramification theory for ring extensions locally of complete intersection. We show that for a Henselian valuation ring $A$ with field of fractions $K$ and for a finite Galois extension $L$ of $K$, the integral closure $B$ of $A$ in $L$ is a filtered union of subrings of $B$ which are of complete intersection over $A$. By this, we can obtain a ramification theory of Henselian valuation rings as the limit of the ramification theory of Saito. Our theory generalizes the ramification theory of complete discrete valuation rings of Abbes-Saito. We study "defect extensions" which are not treated in these previous works.

math.NT↗

arXiv2017

Ramification theory for degree $p$ extensions of arbitrary valuation rings in mixed characteristic $(0,p)$

We previously obtained a generalization and refinement of results about the ramification theory of Artin-Schreier extensions of discretely valued fields in characteristic $p$ with perfect residue fields to the case of fields with more general valuations and residue fields. As seen in VT16, the "defect" case gives rise to many interesting complications. In this paper, we present analogous results for degree $p$ extensions of arbitrary valuation rings in mixed characteristic $(0,p)$ in a more general setting. More specifically, the only assumption here is that the base field $K$ is henselian. In particular, these results are true for defect extensions even if the rank of the valuation is greater than $1$. A similar method also works in equal characteristic, generalizing the results of VT16.

math.NT↗

arXiv2014

Ramification Theory for Artin-Schreier Extensions of Valuation Rings

The goal of this paper is to generalize and refine the classical ramification theory of complete discrete valuation rings to more general valuation rings, in the case of Artin-Schreier extensions. We define refined versions of invariants of ramification in the classical ramification theory and compare them. Furthermore, we can treat the defect case.

math.NT↗