Research
Preprints
Note: The PDF files hosted here are maintained regularly and are usually more up-to-date than the corresponding versions on the arXiv.
Preprint
On canonicity for integral models for Shimura varieties with hyperspecial level (joint with Alex Youcis)
A characterization (and construction) of integral canonical models at places of good reduction that covers pre-abelian and exceptional types (the latter for some ineffectively large primes).
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We give a characterization---and in some cases, a new construction---of integral canonical models of Shimura varieties that uses the notion of an aperture appearing in work of the first author with Gardner on some conjectures of Drinfeld. This applies to Shimura varieties of pre-abelian type at primes of hyperspecial level, recovering and extending previous work of Kisin, Kim--Madapusi and Imai--Kato--Youcis, but also to exceptional Shimura varieties for large enough primes. The characterization in the exceptional case is \emph{a priori} different from the one recently shown by Bakker--Shankar--Tsimerman, and recovers many of their results, such as the existence of prime-to-$p$ Hecke operators, the non-emptiness of the $\mu$-ordinary stratum and the theory of the canonical lift. In fact, we give a uniform proof of the non-emptiness of \emph{all} possible Newton strata, and---in the pre-abelian case---of the non-emptiness of Ekedahl--Oort strata and central leaves as well. An important ingredient in the proofs is a generalization of Tate's full faithfulness theorem for $p$-divisible groups to the context of apertures. This leads to a mapping property for the integral canonical model that characterizes maps into it from all normal, flat and excellent schemes over $\mathbb{Z}_{(p)}$.
@article{MadYoucis,
title={On canonicity for integral models of Shimura varieties with hyperspecial level},
author={Keerthi Madapusi and Alex Youcis},
year={2026},
eprint={2604.06442},
archivePrefix={arXiv},
primaryClass={math.NT},
url={https://arxiv.org/abs/2604.06442},
}
Submitted
Perfect $F$-gauges and finite flat group schemes (joint with Shubhodip Mondal)
An $F$-gauge theoretic classification of finite flat group schemes over general $p$-adic bases, leading to many applications, and recovering many earlier known classifications.
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We show an equivalence of categories, over general $p$-adic bases, between finite locally $p^n$-torsion commutative group schemes and $\mathbb{Z}/p^n\mathbb{Z}$-modules in perfect $F$-gauges of Tor amplitude $[-1,0]$ with Hodge-Tate weights $0,1$. By relating fppf cohomology of group schemes and syntomic cohomology of $F$-gauges, we deduce some consequences: These include the representability of relative fppf cohomology of finite flat group schemes under proper smooth maps of $p$-adic formal schemes, as well as a reproof of a purity result of Česnavičius-Scholze. We also give a general criterion for a classification in terms of objects closely related to Zink's windows over frames and Lau's divided Dieudonné crystals, and we use this to recover several known classifications, and also give some new ones.
@ARTICLE{Madapusi2025-ay,
title = "{Perfect $F$-gauges and finite flat group schemes}",
author = "Madapusi, Keerthi and Mondal, Shubhodip",
journal = "arXiv [math.NT]",
month = "1~" # sep,
year = 2025,
url = "http://arxiv.org/abs/2509.01573"
}
Submitted
An algebraicity conjecture of Drinfeld and the moduli of $p$-divisible groups (joint with Zachary Gardner)
Proving certain conjectures of Drinfeld leading to a good definition of stacks of `$p$-divisible groups with $G$-structure' even when this cannot make literal sense. These objects that we have termed apertures are important for a bunch of global applications that will be considered in works in progress.
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We use the newly developed stacky prismatic technology of Drinfeld and Bhatt-Lurie to give a uniform, group-theoretic construction of smooth stacks $\mathrm{BT}^{G,\mu}_{n}$ attached to a smooth affine group scheme $G$ over $\mathbb{Z}_p$ and $1$-bounded cocharacter $\mu$, verifying a recent conjecture of Drinfeld. This can be viewed as a refinement of results of B\"ultel-Pappas, who gave a related construction using $(G,\mu)$-displays defined via rings of Witt vectors. We show that, when $G = \mathrm{GL}_h$ and $\mu$ is a minuscule cocharacter, these stacks are isomorphic to the stack of truncated $p$-divisible groups of height $h$ and dimension $d$ (the latter depending on $\mu$). This gives a generalization of results of Ansch\"utz-Le Bras, yielding a linear algebraic classification of $p$-divisible groups over very general $p$-adic bases, and verifying another conjecture of Drinfeld.
The proofs use deformation techniques from derived algebraic geometry, combined with an animated variant of Lau's theory of higher frames and displays, and---with a view towards applications to the study of local and global Shimura varieties---actually prove representability results for a wide range of stacks whose tangent complexes are $1$-bounded in a suitable sense. As an immediate application, we prove algebraicity for the stack of perfect $F$-gauges of Hodge-Tate weights $0,1$ and level $n$.
@ARTICLE{gmm,
title = "{An algebraicity conjecture of Drinfeld and the moduli of
$p$-divisible groups}",
author = "Gardner, Zachary and Madapusi, Keerthi",
journal = "arXiv preprint arXiv:2412.10226",
month = "13~" # dec,
year = 2024
}
In Progress
Work in Progress
$p$-isogenies with $\mathcal{G}$-structure: Rapoport--Zink spaces and Igusa stacks (joint with Si Ying Lee)
A new approach to isogenies in mixed characteristic via apertures and the Vinberg monoid. Avoids entirely the usage of abelian varieties or $p$-divisible groups, and allows one to define integral models for $p$-Hecke correspondences even for exceptional Shimura varieties
Draft in progress
Project description
We give a new approach to the definition of $p$-isogenies between objects with $\mathcal{G}$-structure via the Vinberg monoid. We combine this with work of Gardner--Madapusi to construct Rapoport-Zink spaces associated with minuscule cocharacters of reductive group schemes over $\mathbb{Z}_p$. We show that these methods also yield a robust theory of Hecke correspondences on integral canonical models of Shimura varieties in the sense of the work of Madapusi--Youcis. As a consequence, we deduce the existence of formal Igusa stacks and Rapoport-Zink uniformizations for such canonical models.
Work in Progress
The Langlands--Rapoport--$\tau$ conjecture for integral canonical models (joint with Si Ying Lee and Alex Youcis)
This project gives a uniform proof of the Langlands--Rapoport--$\tau$ conjecture (as formulated by Kisin-Shin-Zhu) for all integral canonical models, including those of pre-abelian type and of exceptional type for $p$ sufficiently large. The slogan is that the structure of the space of isogenies is enough to detect all the relevant rational structures, including $\ell$-independence of Frobenius realizations as well as the reductive group of self-quasi-isogenies of a mod-$p$ point.
Draft in progress
Project description
We apply our group-theoretic construction of spaces of isogenies on integral canonical models of Shimura varieties to give a construction of a reductive group whose $\mathbb{Q}$-points are the group of self-quasi-isogenies of a mod-$p$ point of the model. Along the way, we give a proof of the $\ell$-independence of the realizations of the Frobenius quasi-isogeny that makes no appeal to the theory of companions. With the construction of the reductive group in hand, we show that the methods of Kisin--Shin--Zhu in the Hodge type case apply quite generally to give a uniform proof of their Langlands--Rapoport--$\tau$ conjecture for all integral canonical models, including those of exceptional type for large enough $p$.
Work in Progress
Generalized Fontaine--Laffaille theory (joint with Shubhodip Mondal)
A description of $F$-gauges with Hodge--Tate weights in $[0,p-2]$ in terms of `crystalline' data that works for arbitrary $p$-adic formal schemes.
Draft in progress
Project description
We prove a version of Fontaine-Laffaille theory for arbitrary $p$-adic formal schemes. That is, we show that prismatic $F$-gauges over such formal schemes with Hodge-Tate weights in $\{0,\ldots,p-2\}$ can be described in terms of modules over $p$-completed derived de Rham cohomology with additional structure. This generalizes results of Terentiuk-Vologodsky-Xu, who considered the case of the formal scheme $\mathrm{Spf}~W(\kappa)$ for a perfect field $\kappa$. We give applications to the classification of $p$-adic local systems over $p$-adic formal schemes. The proofs use descent to reduce to the case of semiperfect animated commutative $\mathbb{F}_p$-algebras, where we use frame theoretic methods appearing in the work of Gardner-Madapusi. The key computation is that of the syntomic cohomology of $\mathbb{F}_p/{}^{\mathbb{L}}p$.
Work in Progress
Relative apertures and logarithmic Dieudonné theory (joint with Kentaro Inoue, Teruhisa Koshikawa and Alex Youcis)
A logarithmic variant of my work with Gardner developed by taking a relative approach and applying it to Olsson's log stack. This yields a logarithmic Dieudonné theory over general $p$-complete fs log bases
Draft in progress
Project description
We construct stacks of apertures relative to a $\delta$-stack and prove representability results for them. Taking the $\delta$-stack to be Olsson's stack $\mathcal{L}og$ of fine log structures, we obtain the notion of an $n$-truncated logarithmic $(\mathcal{G},\mu)$-aperture. We show that such apertures are parameterized by a smooth formal algebraic stack $\mathrm{BT}^{\mathcal{G},\mu,\log}_n$ over $\mathcal{L}og$. We combine this with the prismatic Dieudonné theory established by Gardner--Madapusi to obtain a logarithmic prismatic Dieudonné theory over general $p$-adic formal bases.
Work in Progress
Integral canonical models for toroidal compactifications of Shimura varieties (joint with Kentaro Inoue, Teruhisa Koshikawa and Alex Youcis)
The convex hull of my paper with Youcis and the work in progress on logarithmic apertures, giving what I believe is the first notion of an integral canonical model for the compactification of a Shimura variety.
Draft in progress
Project description
We use the notion of a logarithmic aperture to give a definition of integral canonical models of toroidal compactifications of Shimura varieties at an unramifeid prime $p$. We verify that such models exist for Shimura varieties of pre-abelian type when $p>2$ and in general for $p$ sufficiently large. In the process, we also consider integral canonical models for mixed Shimura varieties modeled after the work in Madapusi--Youcis on pure Shimura varieties.
Work in Progress
Derived special cycles on Shimura varieties
Giving a general construction of special cycles on Shimura varieties via the systematic use of prismatic methods and derived algebraic geometry. There's a version of this paper on the arXiv, but it's not done the right way, and can safely be ignored for the time being.
Draft in progress
Project description
I employ methods from derived algebraic geometry to give a uniform moduli-theoretic construction of special cycle classes on integral canonical models of many Shimura varieties of pre-abelian type, including unitary, quaternionic, and orthogonal Shimura varieties, away from primes of bad reduction. All desired properties of these classes, even for those corresponding to degenerate Fourier coefficients under the Kudla correspondence, follow naturally from the construction, which realizes the classes as virtual fundamental classes associated with derived special cycles. Notably, under certain natural conditions, the construction gives a functional on the full space of spherical vectors in the Schwartz space of a Hermitian module, and is therefore amenable to a study under the action of the spherical Hecke algebra. This appears to be new already in the case of modular curves. I formulate Kudla’s modularity conjectures in this general framework, and give some preliminary evidence towards their validity.
An important ingredient of the construction is provided by recent work with Youcis on a syntomic characterization of integral canonical models.
Along the way, I also generalize the Tate conjecture for special homomorphisms to situations where the extremal Hodge number is not necessarily one.
Published Papers
Published
Connected components of special cycles on Shimura varieties
A technical paper required for applications to my paper with Ben Howard on Kudla's conjecture. Much of its content was worked out a while ago (see old preprint), but we needed an extension to higher codimensions, so I also took the opportunity to update and correct the older work.
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Journal Page
Very old version
View abstract
I use methods of Chai-Hida and ordinary -Hecke correspondences to study the set of irreducible components of special fibers of special cycles of sufficiently low codimension in integral models of GSpin Shimura varieties, and apply this to prove irreducibility results for the special fibers of the moduli of polarized K3 surfaces. These results are also applied in joint work with Howard on the modularity of generating series of higher codimension cycles on GSpin Shimura varieties.
@ARTICLE{Madapusi2026-rb,
title = "{Connected components of special cycles on Shimura varieties}",
author = "Madapusi, Keerthi",
journal = "Manuscripta Math.",
publisher = "Springer Science and Business Media LLC",
volume = 177,
number = 4,
pages = 56,
month = "20~" # aug,
year = 2026,
url = "http://dx.doi.org/10.1007/s00229-026-01736-0"
}
Published
Kudla's modularity conjecture on integral models of orthogonal Shimura varieties (joint with Ben Howard)
This concerns the definition and modularity of generating series of cycles of higher codimension.
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We construct a family of special cycle classes on the regular integral model of an orthogonal Shimura variety, and show that these cycle classes appear as Fourier coefficients of a Siegel modular form. Passing to the generic fiber of the Shimura variety recovers a result of Bruinier and Raum, originally conjectured by Kudla
@article{Howard_Madapusi_2025, title={Kudla’s modularity conjecture on integral models of orthogonal Shimura varieties}, volume={161}, DOI={10.1017/S0010437X26102966}, number={12}, journal={Compositio Mathematica}, author={Howard, Benjamin and Madapusi, Keerthi}, year={2025}, pages={3380–3454}}
Published
Honda-Tate theory for Shimura varieties of Hodge type (joint with Mark Kisin and Sug Woo Shin)
Non-emptiness of Newton strata and construction of CM lifts on Shimura varieties.
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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 $p$.
@ARTICLE{Kisin2022-eo,
title = "{Honda--Tate theory for Shimura varieties}",
author = "Kisin, Mark and {Madapusi Pera}, Keerthi and Shin, Sug Woo",
journal = "Duke Math. J.",
publisher = "Duke University Press",
volume = 171,
number = 7,
pages = "1559--1614",
year = 2022,
}
Published
Arithmetic of Borcherds products (joint with Ben Howard)
This concerns the definition and modularity of arithmetic divisors on orthogonal Shimura varieties.
Journal Page
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We compute the divisors of Borcherds products on integral models of orthogonal Shimura varieties. As an application, we obtain an integral version of a theorem of Borcherds on the modularity of a generating series of special divisors.
@INCOLLECTION{Howard2020-fa,
title = "{Arithmetic of Borcherds products}",
author = "Howard, Benjamin and {Madapusi Pera}, Keerthi",
booktitle = "{Arithmetic divisors on orthogonal and unitary Shimura varieties}",
volume = 421,
pages = "187--297",
series = "Ast\'{e}risque",
year = 2020
}
Published
Compatible systems of Galois representations associated to the exceptional group $E_6$ (joint with George Boxer, Frank Calegari, Matthew Emerton, Brandon Levin, Stefan Patrikis)
A paper that arose from a conversation at the end of a talk by Stefan to which at least two of the coauthors contributed.
Forum of Math. Sigma, Vol. 7 (2019)
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We construct, over any CM field, compatible systems of $\ell$-adic Galois representations that appear in the cohomology of algebraic varieties and have (for all $\ell$) algebraic monodromy groups equal to the exceptional group of type $E_6$.
@article{bcelmp_2019,
title={Compatible systems of Galois representations associated to the exceptional group $E_{6}$},
volume={7},
DOI={10.1017/fms.2018.24},
journal={Forum of Mathematics, Sigma},
author={Boxer, George and Calegari, Frank and Emerton, Matthew and Levin, Brandon and {Madapusi Pera}, Keerthi and Patrikis, Stefan},
year={2019},
pages={e4}}
Published
2-adic integral canonical models (joint with Wansu Kim)
Mildly rejiggering Kisin's construction by bringing in work of Lau and making sure it works when $p=2$. There's an application to the Tate conjecture, but there was an error pointed out to me by Teruhisa Koshikawa, and this led to a later erratum
Forum of Math. Sigma, Vol. 4 (2016); Forum of Math. Sigma, Vol. 8 (2020)
Journal Page
Erratum
View Abstract
We use Lau’s classification of $2$-divisible groups using Dieudonné displays to construct integral canonical models for Shimura varieties of abelian type at $2$-adic places where the level is hyperspecial.
@article{KIM_MADAPUSI PERA_2016,
title={2-adic integral canonical models},
volume={4}, DOI={10.1017/fms.2016.23},
journal={Forum of Mathematics, Sigma},
author={Kim, Wansu and {Madapusi Pera}, Keerthi},
year={2016}, pages={e28}}
@article{2adic_erratum,
title={Erratum to appendix to ‘2-adic integral canonical models’},
volume={8},
DOI={10.1017/fms.2020.2},
journal={Forum of Mathematics, Sigma},
author={{Madapusi Pera}, Keerthi}, year={2020}, pages={e14}}
Published
Faltings heights of abelian varieties with complex multiplication
The main result is a proof of a conjecture of Colmez on the averaged Faltings heights of abelian varieties with CM by a fixed CM field, which in turn has been applied by Tsimerman to complete the proof of the André--Oort conjecture for Siegel modular varieties.
Ann. Math., Vol. 187 (2) (2018)
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Let $M$ be the Shimura variety associated with the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ 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.
@ARTICLE{Andreatta2018-tt,
title = "{Faltings heights of abelian varieties with complex
multiplication}",
author = "Andreatta, Fabrizio and Goren, Eyal Z and Howard, Benjamin and
{Madapusi Pera}, Keerthi",
journal = "Ann. Math.",
publisher = "Annals of Mathematics",
volume = 187,
number = 2,
pages = "391--531",
year = 2018,
url = "http://www.jstor.org/stable/26397727"
}
Published
Height pairings on orthogonal Shimura varieties (joint with Fabrizio Andreatta, Eyal Goren and Ben Howard)
Verification of conjectures of Bruinier--Yang, which can be seen as a generalization of the Gross--Zagier formula to orthogonal Shimura varieties.
Compos. Math., Vol. 153 (3) (2017)
Journal Page
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Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of special divisors and complex multiplication 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 $n=1$, 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.
@ARTICLE{Andreatta2017-gu,
title = "{Height pairings on orthogonal Shimura varieties}",
author = "Andreatta, Fabrizio and Goren, Eyal Z and Howard, Benjamin and
{Madapusi Pera}, Keerthi",
journal = "Compos. Math.",
publisher = "London Mathematical Society",
volume = 153,
number = 3,
pages = "474--534",
month = mar,
year = 2017,
url = "https://www.cambridge.org/core/journals/compositio-mathematica/article/div-classtitleheight-pairings-on-orthogonal-shimura-varietiesdiv/7C8EC526D2AB5BEA4159711695245BEA"
}
Published
The Tate conjecture for K3 surfaces in odd characteristic
An extension of Deligne's Kuga-Satake construction over $\mathbb{Z}[1/2]$, combined with work of Kisin on the Langlands--Rapoport conjecture to prove the Tate conjecture for K3 surfaces.
Invent. Math., Vol. 201 (2) (2015)
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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 regular 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. In turn, this reduces the Tate conjecture for K3 surfaces over finitely generated fields of odd characteristic to a version of the Tate conjecture for certain endomorphisms on the attached Kuga–Satake abelian variety, which we prove. As a by-product of our methods, we also show that the moduli stack of primitively polarized K3 surfaces of 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.
@ARTICLE{mp:tatek3,
title = "{The Tate conjecture for K3 surfaces in odd characteristic}",
author = "{Madapusi}, Keerthi",
journal = "Invent. Math.",
publisher = "Springer Berlin Heidelberg",
volume = 201,
number = 2,
pages = "625--668",
year = 2015
}
Published
Integral canonical models for Spin Shimura varieties
A special case of the general construction by Kisin, extended to certain places of bad reduction. Many subsequent projects, including those about special cycles on orthogonal Shimura varieties, the Colmez conjecture, and the Tate conjecture depend on the methods here.
Compos. Math., Vol. 152 (4) (2016)
Journal Page
View Abstract
We construct regular integral canonical models for Shimura varieties attached to Spin and orthogonal 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 models and that the integral models we construct are canonical in a very precise sense. 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.
@ARTICLE{mp:reg,
title = "{Integral canonical models for {S}pin {S}himura varieties}",
author = "Madapusi Pera, Keerthi",
journal = "Compos. Math.",
volume = 152,
number = 4,
pages = "769--824",
year = 2016
}
Published
Toroidal compactifications of integral models of Shimura varieties of Hodge type
Based on my PhD thesis but is agnostic to singularities.
Ann. Sci. Ec. Norm. Super., Vol. 52 (2) (2019)
Journal Page
View Abstract
We construct toroidal compactifications for integral models of Shimura varieties of Hodge type. 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. 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.
@ARTICLE{mp:toroidal,
title = "{Toroidal compactifications of integral models of Shimura varieties
of Hodge type}",
author = "{Madapusi Pera}, Keerthi",
journal = "Ann. Sci. Ec. Norm. Super.",
volume = 52,
number = 1,
pages = "393--514",
year = 2019
}
PhD Thesis
Doctoral Dissertation
Toroidal compactifications of integral models of Shimura varieties of Hodge type
University of Chicago, 2011 • Advisor: Mark Kisin
The end results of this thesis have been superseded by the paper in Annales d'ENS above, but there are still some auxiliary results here that don't reappear in the published paper.