I am currently a postdoc in the Department of Mathematics at Michigan State University, working in number theory and algebraic geometry. Previously, I was a Harry Bateman Research Instructor at Caltech. During the academic year 2020-2021, I was an AMIAS Member at the Institute for Advanced Study, Princeton, prior to which I was a Ph.D. student at the University of Toronto, where my advisor was Jacob Tsimerman. I am particularly interested in the interactions between algebraic and analytic geometry, in both the complex and non-archimedean settings, in problems of unlikely intersections, functional transcendence, and in the moduli of abelian varieties.
Email address: oswalabh at msu dot edu
Research
A p-adic analogue of Borel's theorem for Shimura varieties of abelian type (joint with Ananth Shankar, Xinwen Zhu and an appendix by Anand Patel), arXiv link.
Abstract: Let S denote a Shimura variety whose connected components have the form D/Gamma, where Gamma is a torsion free arithmetic group acting on the Hermitian symmetric domain D. Borel proved that any holomorphic map from a variety V to S must be algebraic. Using techniques from my thesis, the theory of Rapoport-Zink spaces, and the p-adic logarithmic Riemann-Hilbert correspondence of Diao-Lan-Liu-Zhu, we prove a p-adic analogue of Borel's theorem for Shimura varieties of abelian type.
Lifts of supersingular abelian varieties with small Mumford-Tate groups (joint with Josh Lam). International Mathematics Research Notices. arXiv link
Abstract: We investigate to what extent an abelian variety over a finite field can be lifted to one in characteristic zero with small Mumford-Tate group. We prove that supersingular abelian surfaces, respectively threefolds, can be lifted to ones isogenous to a square, respectively product, of elliptic curves. On the other hand, we show that supersingular abelian threefolds cannot be lifted to one isogenous to the cube of an elliptic curve over the Witt vectors.
Bi-algebraicity in the rank one Riemann--Hilbert correspondence via o-minimality, arXiv link
Abstract: For a smooth, projective, complex algebraic variety X, the Riemann--Hilbert correspondence establishes a complex analytic isomorphism between the `Betti moduli space' of rank n local systems on X and the `de Rham moduli space' of rank n vector bundles with flat connection on X. In the rank one case, C. Simpson precisely characterizes the subvarieties of these moduli spaces that are `bi-algebraic' for this typically transcendental, analytic isomorphism. In this short note, we give a new proof of this characterization of Simpson, using methods from o-minimal geometry. We adapt the o-minimal proof to a p-adic setting, namely that of Mumford curves.
A non-archimedean definable Chow theorem, To appear in Journal of the European Mathematical Society, arXiv Link
Abstract: Peterzil and Starchenko have proved the following surprising generalization of Chow's theorem: A closed analytic subset of a complex algebraic variety that is definable in an o-minimal structure, is in fact an algebraic subset. In this paper, we prove a non-archimedean analogue of this result.
Almost ordinary abelian varieties over finite fields (joint with Ananth Shankar), Journal of the London Mathematical Society, arXiv link
Abstract: We provide a characterization of almost ordinary abelian varieties over finite fields, and use this characterization to provide lower bounds for the sizes of some almost ordinary isogeny classes.