Keerthi Madapusi Pera
Department of Mathematics
University of Chicago,
5734 S University Ave,
Chicago, IL 60616, U.S.A.
Email: keerthi [at] math [dot] uchicago [dot] edu
I am an assistant professor at the department of mathematics at the University of Chicago, which is where I also got my PhD. My advisor was Mark Kisin, who is now at Harvard. I grew up in
Publications and pre-prints:
We use E. Lau's classification of 2-divisible groups using Dieudonne displays to construct integral canonical models for Shimura varieties of abelian type at 2-adic places where the level is hyperspecial. We apply this to prove the Tate conjecture for K3 surfaces in characteristic 2.
Let M be the Shimura variety associated with the group of spinor similitudes of a quadratic space over the rational numbers of signature (n,2). We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points on M to the central derivatives of certain L-functions. As an application of this result, we prove an averaged version of Colmez's conjecture on the Faltings heights of CM abelian varieties.
Let M be the Shimura variety associated to the group of spinor similitudes of a rational quadratic space of signature (n, 2). We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of special divisors and CM points on M to the central derivatives of certain L-functions. Each such L-function is the Rankin-Selberg convolution associated with a cusp form of half-integral weight n/2 + 1, and the weight n/2 theta series of a positive definite quadratic space of rank n. When n = 2 the Shimura variety M is a classical quaternionic Shimura curve, and our result is a variant of the Gross-Zagier theorem on heights of Heegner points.
We show that the classical Kuga-Satake construction gives rise, away from characteristic 2, to an open immersion from the moduli of primitively polarized K3 surfaces (of any fixed degree) to a certain normal integral model for a Shimura variety of orthogonal type. This allows us to attach to every polarized K3 surface in odd characteristic an abelian variety such that divisors on the surface can be identified with certain endomorphisms of the attached abelian variety. Using a result of Kisin, we can then prove the Tate conjecture for K3 surfaces over finitely generated fields of odd characteristic. We also show that the moduli stack of primitively polarized K3 surfaces of fixed degree 2d is quasi-projective and, when d is not divisible by p^2, is geometrically irreducible in characteristic p. We indicate how the same method applies to prove the Tate conjecture for co-dimension 2 cycles on cubic fourfolds.
We construct regular integral canonical models for Shimura varieties attached to Spin groups at (possibly ramified) primes $p>2$ where the level is not divisible by $p$. We exhibit these models as schemes of 'relative PEL type' over integral canonical models of larger Spin Shimura varieties with good reduction at $p$. Work of Vasiu-Zink then shows that the classical Kuga-Satake construction extends over the integral model and that the integral models we construct are canonical in a very precise sense. We also construct good compactifications for our integral models. Our results have applications to the Tate conjecture for K3 surfaces, as well as to Kudla's program of relating intersection numbers of special cycles on orthogonal Shimura varieties to Fourier coefficients of modular forms.
We construct projective toroidal compactifications for integral models of Shimura varieties of Hodge type that parameterize isogenies of abelian varieties with additional structure. We also construct integral models of the minimal (Satake-Baily-Borel) compactification. Our results essentially reduce the problem to understanding the integral models themselves. As such, they cover all previously known cases of PEL type, as well as all cases of Hodge type involving parahoric level structures. At primes where the level is hyperspecial, we show that our compactifications are canonical in a precise sense. We also provide a new proof of Y. Morita's conjecture on the everywhere good reduction of abelian varieties whose Mumford-Tate group is anisotropic modulo center. Along the way, we demonstrate an interesting rationality property of Hodge cycles on abelian varieties with respect to p-adic analytic uniformizations.
This is a completely rewritten version. If you are looking for the old version of the paper, you can find it here
Thesis: Toroidal Compactifications of Integral Canonical Models of Shimura varieties of Hodge type.
The results of my thesis have been superseded by those in the paper 'Toroidal compactifications...' above. If you would still like to see it, click here.
Some (very incomplete) notes of mine:
The first two were written back when I was an innocent first year student, who believed that the only way to learn something was to write it all up in gory detail. I still find parts of them useful, so I’ve put them up for general consumption. Caveat: citation in these notes is quite poor, but obviously stuff has made it into them from all over the place. And nothing in them is original. There are some mysterious references in these notes to other notes that I have written. Those notes can be found here.
You might have known me under the last name Madapusi Sampath (or simply Madapusi). I'm in the process of getting my name officially to Madapusi Pera, a combination of my last name and my wife's.