About

Mark Kisin is a professor in the Department of Mathematics at the University of Chicago. His research involves the analysis of local deformation rings in the context of Galois representations, particularly focusing on p-adic Galois representations and their associated modules. His work includes the construction and study of auxiliary schemes related to the deformation theory of finite at group schemes, and the description of their local structures in terms of Shimura varieties and Hilbert modular varieties. Kisin's contributions extend to the development of methods for analyzing the components of deformation rings, including the use of Grassmannian calculations and the study of singularities within these schemes. His research also involves the examination of the relationships between Galois modules, their functors, and the associated categories, with an emphasis on the properties of these modules under various functorial operations. His work is fundamental in understanding the structure of deformation spaces and their applications in number theory and arithmetic geometry.

Research topics

• Pure mathematics
• Mathematics
• Statistics

Selected publications

• Integral models of Shimura varieties with parahoric level structure, II

  Forum of Mathematics Pi · 2026-01-01 · 1 citations

  preprintOpen access1st authorCorresponding

  Abstract We construct integral models of Shimura varieties of abelian type with parahoric level structure over odd primes. These models are étale locally isomorphic to corresponding local models.

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• Strongly compatible systems associated to semistable abelian varieties

  ArXiv.org · 2025-05-04

  preprintOpen access1st authorCorresponding

  We prove a motivic refinement of a result of Weil, Deligne and Raynaud on the existence of strongly compatible systems associated to abelian varieties. More precisely, given an abelian variety $A$ over a number field $\mathrm{E}\subset \mathbb C$, we prove that after replacing $\mathbb E$ by a finite extension, the action of $\mathrm{Gal}(\overline{\mathrm E}/\mathrm E)$ on the $\ell$-adic cohomology $\mathrm H^1_{\mathrm{\acute{e}t}}(A_{\overline{\mathrm E}},\mathbb Q_\ell)$ gives rise to a strongly compatible system of $\ell$-adic representations valued in the Mumford--Tate group $\mathbf G$ of $A$. This involves an independence of $\ell$-statement for the Weil--Deligne representation associated to $A$ at places of semistable reduction, extending previous work of ours at places of good reduction.

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• Independence of $\ell$ for Frobenius conjugacy classes attached to abelian varieties

  Annals of Mathematics · 2025-11-01

  article1st authorCorresponding

  Let $A$ be an abelian variety over a number field $\mathrm{E}\subset \mathbb{C}$, and let $\mathbf{G}$ denote the Mumford--Tate group of $A$. After replacing $\mathrm{E}$ by a finite extension, the action of the absolute Galois group $\mathrm{Gal}(\overline{\mathrm{E}}/\mathrm{E})$ on the $\ell$-adic cohomology $\mathrm{H}^1_{\mathrm{\acute{e}t}}(A_{\overline{\mathrm{E}}},\mathbb{Q}_\ell)$ factors through $\mathbf{G}(\mathbb{Q}_\ell)$. We show that for $v$ an odd prime of $\mathrm{E}$ where $A$ has good reduction, the conjugacy class of Frobenius $\mathrm{Frob}_v$ in $\mathbf{G}(\mathbb{Q}_\ell)$ is independent of $\ell$. Along the way, we prove that under certain hypotheses, every point in the $\mu$-ordinary locus of the special fiber of Shimura varieties has a special point lifting it.

  PublisherDOI

• Essential dimension via prismatic cohomology

  Duke Mathematical Journal · 2024-10-15

  articleOpen access
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• Finiteness of reductions of Hecke orbits

  Journal of the European Mathematical Society · 2023-12-16 · 1 citations

  articleOpen access1st authorCorresponding

  We prove two finiteness results for reductions of Hecke orbits of abelian varieties over local fields: one in the case of supersingular reduction and one in the case of reductive monodromy. As an application, we show that only finitely many abelian varieties on a fixed isogeny leaf admit CM lifts, which in particular implies that in each fixed dimension g only finitely many supersingular abelian varieties admit CM lifts. Combining this with the Kuga–Satake construction, we also show that only finitely many supersingular K 3 surfaces admit CM lifts. Our tools include p -adic Hodge theory and group-theoretic techniques.

  PublisherDOI

• Modular functions and resolvent problems

  Mathematische Annalen · 2022 · 5 citations

  • Mathematics
  • Pure mathematics
  PublisherOA PDFDOI

• Honda–Tate theory for Shimura varieties

  Duke Mathematical Journal · 2022 · 11 citations

  1st authorCorresponding
  • Mathematics
  • Pure mathematics
  • Statistics

  A Shimura variety of Hodge type is a moduli space for abelian varieties equipped with a certain collection of Hodge cycles. We show that the Newton strata on such varieties are nonempty provided that the corresponding group G is quasisplit at p, confirming a conjecture of Fargues and Rapoport in this case. Under the same condition, we conjecture that every mod p isogeny class on such a variety contains the reduction of a special point. This is a refinement of Honda–Tate theory. We prove a large part of this conjecture for Shimura varieties of PEL type. Our results make no assumption on the availability of a good integral model for the Shimura variety. In particular, the group G may be ramified at p.

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• The stable trace formula for Shimura varieties of abelian type

  arXiv (Cornell University) · 2021-10-11 · 8 citations

  preprintOpen access1st authorCorresponding

  We express the Frobenius-Hecke traces on the compactly supported cohomology of a Shimura variety of abelian type in terms of elliptic parts of stable Arthur-Selberg trace formulas for the endoscopic groups. This confirms predictions of Langlands and Kottwitz at primes where the level is hyperspecial.

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• The essential dimension of congruence covers

  Compositio Mathematica · 2021-10-27 · 6 citations

  preprintOpen access

  Abstract Consider the algebraic function $\Phi _{g,n}$ that assigns to a general $g$ -dimensional abelian variety an $n$ -torsion point. A question first posed by Klein asks: What is the minimal $d$ such that, after a rational change of variables, the function $\Phi _{g,n}$ can be written as an algebraic function of $d$ variables? Using techniques from the deformation theory of $p$ -divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $p$ -dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $p$ -dimension of congruence covers of the moduli space of genus $g$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $M$ is proper . As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety.

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• Independence of $\ell$ for Frobenius conjugacy classes attached to abelian varieties

  arXiv (Cornell University) · 2021-03-17 · 3 citations

  preprintOpen access1st authorCorresponding

  Let $A$ be an abelian variety over a number field $\mathrm E\subset \mathbb C$ and let $\mathbf G$ denote the Mumford--Tate group of $A$. After replacing $\mathrm E$ by a finite extension, the action of the absolute Galois group $\mathrm{Gal}(\overline{\mathrm E}/\mathrm E)$ on the $\ell$-adic cohomology $\mathrm{H}^1_{\mathrm{\acute{e}t}}(A_{\overline{\mathrm E}},\mathbb Q_\ell)$ factors through $\mathbf G(\mathbb Q_\ell).$ We show that for $v$ an odd prime of $\mathrm E$ where $A$ has good reduction, the conjugacy class of Frobenius $\mathrm{Frob}_v$ in $\mathbf G(\mathbb Q_\ell)$ is independent of $\ell$. Along the way we prove that every point in the $μ$-ordinary locus of the special fiber of Shimura varieties has a special point lifting it.

  PublisherOA PDFDOI

Recent grants

• Modularity and p-adic Langlands

  NSF · $252k · 2007–2011

• Arithmetic Geometry

  NSF · $390k · 2016–2021

• Shimura Varieties and Galois representations

  NSF · $305k · 2013–2017

• The Fontaine-Mazur conjecture via p-adic modular forms

  NSF · $107k · 2004–2007

Frequent coauthors

• Benson Farb

  University of Chicago

  9 shared

• Jesse Wolfson

  6 shared

• Matthew Emerton

  University of Chicago

  5 shared

• G Lehrer

  Australian Mathematical Sciences Institute

  5 shared

• Miaofen Chen

  4 shared

• G. Pappas

  Michigan State University

  4 shared

• Eva Viehmann

  University of Münster

  4 shared

• Wei Ren

  4 shared