Vaidehee Thatte

Research Activities

I am primarily interested in number theory, arithmetic algebraic geometry, commutative algebra, and valuation theory; more specifically, in ramification theory. I have an emerging interest in model theory. I am also excited to work on topics in arithmetic statistics.

Ramification theory serves the dual purpose of a diagnostic tool and treatment by helping us locate, measure, and treat the anomalous behavior of mathematical objects. The majority of my current work and future projects aim to develop ramification theory for arbitrary valuation fields, extending the classical theory of complete discrete valuation fields with perfect residue fields.

The study of (wild) ramification leads to a better understanding of the many interesting challenges arising in the general case and provides us with ways to overcome them. One such example is the mysterious phenomenon of the defect (or ramification deficiency) that is unique to the positive residue characteristic case and is the main obstacle in several open problems in mathematics (e.g. local uniformization).

Publications

Click on the collapsible text for a description.

Degree p Extensions and "Best f". [Tha 24] Appendix to [KT 24]. Tunisian Journal of Mathematics 6-4 (2024), pp. 633-646. DOI 10.2140/tunis.2024.6.589. (Pre-print)

Degree p extensions are the building blocks of the general theory. In this appendix to [KT 24], 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 already given in [Tha 16, Tha 18]; we provide the details here for the sake of completeness. As seen in these earlier works, the 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 provide a detailed account of results for degree p Artin–Schreier defect extensions of higher rank valuation rings, mentioned in [Tha 18, §8].

Upper Ramification Groups for Arbitrary Valuation Rings. [KT 24] Joint with Kazuya Kato. Tunisian Journal of Mathematics 6-4 (2024), pp. 589-633. DOI 10.2140/tunis.2024.6.589. (Pre-print)

Abstract: 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.

An explicit self-duality. [SPEC] Stacks Project Expository Collection, Joint with Nikolas Kuhn, Devlin Mallory, Kirsten Wickelgren. Cambridge University Press (2022). (arXiv).

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.

Local Oort Groups and the Isolated Differential Data Criterion. [DDKOT] Joint with Huy Dang, Soumyadip Das, Kostas Karagiannis, Andrew Obus. Journal de Théorie des Nombres de Bordeaux vol. 34 [1](2022). (arXiv).

Abstract: 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.

Ramification Theory for Degree p Extensions of Arbitrary Valuation Rings in Mixed Characteristic (0, p). [Tha 18] Journal of Algebra 507 (2018), pp. 225-248. (arXiv)

Abstract: 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 [Tha 16], 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 [Tha 16].

Ramification Theory for Artin-Schreier Extensions of Valuation Rings. [Tha 16] Journal of Algebra 456 (2016), pp. 355-389. (arXiv)

Abstract: 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.

In preparation

Conductor-Discriminant Formula for Arbitrary Valuation Fields - I.

Consider a degree p Galois extension L/K of Henselian valued fields, with B/A the corresponding extension of valuation rings. In the classical case the different ideal D of B with respect to A equals the annihilator of the B-module Ω of relative differential 1-forms over A. However, we saw in [Tha 16] that this is not true in general. In this paper, we extend the relevant results of [Tha 16] to Artin-Schreier defect extensions with valuation of rank >1 and subsequently to the mixed characteristic case, in the spirit of the conductor-discriminant formula.

Ramification Breaks for Artin-Schreier-Witt Extensions of Arbitrary Valuation Fields.

On failures of the Hasse norm principle with Anand Deopurkar, Rachel Newton and Rosa Winter.

On ramification filtrations with Yuri Yatagawa.

In early stages

On wild Brauer groups and Brauer-Manin obstructions with Seoyoung Kim.

Recent invited academic visits

TIFR, Mumbai 2024-25.
IISER, Pune January 5 - 11, 2025.
Tokyo Institute of Technology (now Institute of Science Tokyo) September 11 - October 8, 2024.

Invited academic visits before 2024.

Tokyo Institute of Technology March 17-April 10, 2023.
IHES (Institut des Hautes Études Scientifiques) October 27 - December 21, 2022.
PIMS, Vancouver, BC March 26-29, 2018.

Invited talks: conferences & workshops

British Mathematical Colloquium, University of Manchester. June 17-20, 2024.
Number Theory in Tokyo, Tokyo Institute of Technology, Japan. March 20-24, 2023.
Special session AMS Spring Southeastern Sectional Meeting University of Virginia, Charlottesville. March 2022. Canceled by AMS due to COVID-19.
AMS Fall Western Sectional SS "Arithmetic Geometry". October 23, 2021.
Singularities and Arithmetics, Tohoku University, Japan. Feb 17 - 20, 2020.
Wild Ramification and Irregular Singularities, Warsaw, (IMPAN), Sep 23–27, 2019.
PIMS Focus Periods, Vancouver, BC Two lectures (Short Term Visitor), March 26-29, 2018.
CMS Summer SS "Number Theory" June 1-4, 2018 (canceled).
CMS Summer SS "Cohomology - a link between numbers and geometry" June 1-4, 2018 (canceled).
Indian Women and Mathematics, Indian Institute of Science, Bangalore. July 13-15, 2017.

Contributed talks

WINGs 2024 April 15-18, 2024.
WINGs 2022 June 6-8, 2022.
Stacks Project ONline Geometry Event (SPONGE) August 3-7, 2020.
CTNT Conference, June 12-14, 2020.
MAAIM, Emory University, GA November 1-3, 2019.
BU-Keio Workshop 2019, MA, June 24-28, 2019.
Ninth Annual Upstate Number Theory Conference, NY, April 27-28, 2019.
Arithmetic of Algebraic Curves, Madison, WI , April 6-8, 2018.
GWAGWMMG, Cambridge, MA , Feb 17-18, 2018.
Journées Arithmétiques, Caen, July 3-7, 2017.
Midwest Number Theory Conference, October 16-18, 2015.

Conferences & workshops (selected)

Workshop on p-adic Arithmetic Geometry (Fall). IAS, Princeton, November 13-17, 2023.
Spring School on non-archimedean geometry and eigenvarieties. Heidelberg, Week 1, March 6-10, 2023.
Women in Arithmetic Geometry Heidelberg, September 26 - 30, 2022.
Women in Numbers Europe - 4, August 29 – September 2, 2022.
Communicating Mathematics, August 8-11, 2022.
Franco-Asian Summer School on Arithmetic Geometry, CIRM, May 30 - June 3, 2022.
Rational points on higher-dimensional varieties, ICMS, April 25-29, 2022.
Arithmetic Geometry - Takeshi 60, September 6-10, 2021.
JMM 2021, co-organizer for Special Session January 2021.
Stacks Project Workshop, MI August 3-7, 2020 .
Western Algebraic Geometry ONline (WAGON), April 18-19, 2020 .
JMM 2020, co-organizer & primary contact for a special session, January 2020.
AMS MRC : Explicit Methods in Arithmetic Geometry in Characteristic p, RI, June 16-22, 2019.
Strength in Numbers, Queen's University, ON (Principal organizer) May 11-12, 2018.

Invited talks: seminars & colloquia

IISER Pune, Colloquium and Seminar talks, January 8 and 9, 2025.
Bhaskaracharya Pratishthana, December 27, 2024.
Oxford Logic Seminar, November 7, 2024.
Kyushu University, October 4, 2024.
Nagoya University, October 3, 2024.
NTT Institute for Fundamental Mathematics. October 1, 2024.
Tokyo Institute of Technology, Arithmetic Geometry Seminar. September 13, 2024.
Columbia University, Automorphic Forms and Arithmetic Seminar. March 1, 2024.
University of Warwick, Algebraic Geometry Seminar. October 18, 2023.
Great Western Number Theory Seminar, University of Reading. September 5, 2023.
Heilbronn Number Theory Seminar, University of Bristol. April 19, 2023.
Nagoya University, April 6, 2023.
l’Institut Henri Poincaré, RéGA (Réseau des étudiants en Géométrie Algébrique). December 7, 2022.
Institut des Hautes Études Scientifiques, Séminaire de Mathématique. November 24, 2022.
University of Oxford, Number Theory Seminar. October 20, 2022.
QMUL Algebra & Number Theory Seminar, April 8, 2022.
Northern Number Theory Seminar, March 30, 2022.
Cambridge Number Theory Seminar, February 15, 2022.
London Number Theory Seminar, December 8, 2021.
VIASM Arithmetic Geometry Online Seminar, Vietnam. November 24, 2021.
University of Sheffield, Number Theory Seminar, November 19, 2021.
University of Warwick, Number Theory Seminar, November 15, 2021.
"What is... a seminar?", Online. October 14, 2021.
POSTECH, Pohang, South Korea. Departmental Colloquium, April 30, 2021.
UC Irvine, Number Theory Seminar, April 29, 2021.
National and Kapodistrian University of Athens, Arithmetic Geometry Seminar, March 5, 2021.
University of Rochester, Number Theory Seminar, February 4, 2021.
Michigan State University, Algebra Seminar, January 27, 2021.
Joint NU/UIC/UofC algebra & geometry seminar, June 17, 2020. Notes
Saitama University, Japan. Seminar Talk, Feb 21, 2020.
Columbia-CUNY-NYU Number Theory Seminar, Feb 21, 2019.
Cornell University Number Theory Seminar, Feb 15, 2019.
CUNY Queens College Colloquium, April 25, 2018.
University of Maryland Algebra and Number Theory Seminar, Feb 19, 2018.
The University of Chicago. Number Theory Seminar, May 30, 2017.
University of Western Ontario. Algebra Seminar, April 12, 2017.
Purdue University. Automorphic Forms and Representation Forms Seminar, April 7, 2016.
University of Arizona. Algebra and Number Theory Seminar, October 20, 2015.
The University of Chicago. Number Theory Seminar, March 10, 2015.

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