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Stefan Patrikis

Publications and source records attributed to Stefan Patrikis.

At least 19 recordsLinked to original sources
arXiv2026

Compatibility of $F$-isocrystals on adjoint Shimura varieties

In this article, we extend past results of the last two authors to include compatibility of canonical $\ell$-adic local systems and canonical $F$-isocrystals on adjoint Shimura varieties in the superrigid regime. Our method relies on the crystallinity of canonical $p$-adic local systems due to Esnault--Groechenig as well as Margulis superrigidity and the crystalline-to-étale companion construction of Drinfeld, Abe--Esnault, and Kedlaya.

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arXiv2025

Compatibility of canonical $\ell$-adic local systems on Shimura varieties, II

Let $(G, X)$ be a Shimura datum. In previous work with Christian Klevdal, we showed that the canonical $G(\mathbb{Q}_{\ell})$-valued local systems on Shimura varieties for $G$ form compatible systems after projection to the adjoint group of $G$. In this note, we strengthen this result to prove compatibility for the $G(\mathbb{Q}_{\ell})$-local systems themselves. We also include the crystalline compatibility, extending the adjoint case established in our joint work Jake Huryn, Kiran Kedlaya, and Christian Klevdal.

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arXiv2024

Compatibility of canonical $\ell$-adic local systems on Shimura varieties

For a Shimura variety $(G, X)$ in the superrigid regime and neat level subgroup $K_0$, we show that the canonical family of $\ell$-adic representations associated to a number field point $y \in \mathrm{Sh}_{K_0}(G, X)(F)$, \[ \left\{ ρ_{y, \ell} \colon \mathrm{Gal}(\overline{\mathbb{Q}}/F) \to G^{\mathrm{ad}}(\mathbb{Q}_{\ell}) \right\}_{\ell}, \] form a compatible system of $G^{\mathrm{ad}}(\mathbb{Q}_{\ell})$-representations: there is an integer $N(y)$ such that for all $\ell$, $ρ_{y, \ell}$ is unramified away from $N(y) \ell$, and for all $\ell \neq \ell'$ and $v \nmid N(y)\ell \ell'$, the semisimple parts of the conjugacy classes of $ρ_{y, \ell}(\mathrm{Frob}_v)$ and $ρ_{y, \ell'}(\mathrm{Frob}_v)$ are ($\mathbb{Q}$-rational and) equal. We deduce this from a stronger compatibility result for the canonical $G(\mathbb{Q}_{\ell})$-valued local systems on connected Shimura varieties inside $\mathrm{Sh}_{K_0}(G, X)$. Our theorems apply in particular to Shimura varieties of non-abelian type and represent the first such independence-of-$\ell$ results in non-abelian type.

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arXiv2022

Trianguline lifts of global mod $p$ Galois representations

We show that under a suitable oddness condition, irreducible mod $p$ representations of the absolute Galois group of an arbitrary number field have characteristic zero lifts which are unramified outside a finite set of primes and trianguline at all primes of $F$ dividing $p$. We also prove variants of this result for representations valued in connected reductive groups.

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arXiv2021

Relative deformation theory, relative Selmer groups, and lifting irreducible Galois representations

We study irreducible odd mod $p$ Galois representations $\barρ \colon \mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_p)$, for $F$ a totally real number field and $G$ a general reductive group. For $p \gg_{G, F} 0$, we show that any $\barρ$ that lifts locally, and at places above $p$ to de Rham and Hodge-Tate regular representations, has a geometric $p$-adic lift. We also prove non-geometric lifting results without any oddness assumption.

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arXiv2021

Lifting and automorphy of reducible mod p Galois representations over global fields

We extend the lifting methods of our previous paper to lift reducible odd representations $\barρ:\mathrm{Gal}(\overline{F}/F) \to G(k)$ of Galois groups of global fields $F$ valued in Chevalley groups $G(k)$. Lifting results, when combined with automorphy lifting results pioneered by Wiles in the number field case and the results on the global Langlands correspondence proved by Drinfeld and L. Lafforgue in the function field case, give the only known method to access modularity of mod $p$ Galois representations in both reducible and irreducible cases. In the reducible case this allows one to show that the actual representation, rather than just its semisimplification, arises from reduction of the geometric representation attached to a cuspidal automorphic representation on the dual group of $G$. As a particularly concrete application, we get a version of Serre's modularity conjecture for reducible, odd representations $\barρ: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_2(k)$. This extends earlier results of Hamblen and Ramakrishna in this classical case and proves modularity of infinitely many extensions of fixed characters that are not covered by loc. cit.

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arXiv2020

Potential automorphy of $\mathrm{GSpin}_{2n+1}$-valued Galois representations

We prove a potentially automorphy theorem for suitable Galois representations $Γ_{F^+} \to \mathrm{GSpin}_{2n+1}(\overline{\mathbb{F}}_p)$ and $Γ_{F^+} \to \mathrm{GSpin}_{2n+1}(\overline{\mathbb{Q}}_p)$, where $Γ_{F^+}$ is the absolute Galois group of a totally real field $F^+$. We also prove results on solvable descent for $\mathrm{GSp}_{2n}(\mathbb{A}_{F^+})$ and use these to put representations $Γ_{F^+} \to \mathrm{GSpin}_{2n+1}(\overline{\mathbb{Q}}_p)$ into compatible systems of $\mathrm{GSpin}_{2n+1}(\overline{\mathbb{Q}}_{\ell})$-valued representations.

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arXiv2020

G-rigid local systems are integral

Let $G$ be a reductive group, and let $X$ be a smooth quasi-projective complex variety. We prove that any $G$-irreducible, $G$-cohomologically rigid local system on $X$ with finite order abelianization and quasi-unipotent local monodromies is integral. This generalizes work of Esnault and Groechenig when $G= \mathrm{GL}_n$, and it answers positively a conjecture of Simpson for $G$-cohomologically rigid local systems. Along the way we show that the connected component of the Zariski-closure of the monodromy group of any such local system is semisimple.

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arXiv2020

Lifting $G$-irreducible but $\mathrm{GL}_n$-reducible Galois representations

In recent work, the authors proved a general result on lifting $G$-irreducible odd Galois representations $\mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_{\ell})$, with $F$ a totally real number field and $G$ a reductive group, to geometric $\ell$-adic representations. In this note we take $G$ to be a classical group and construct many examples of $G$-irreducible representations to which these new lifting methods apply, but to which the lifting methods provided by potential automorphy theorems do not.

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arXiv2019

G-Valued Galois Deformation Rings when $\ell \neq p$

For a smooth group scheme $G$ over an extension of $\mathbf{Z}_p$ such that the generic fiber of $G$ is reductive, we study the generic fiber of the Galois deformation ring for a $G$-valued mod $p$ representation of the absolute Galois group of a finite extension of $\mathbf{Q}_\ell$ with $\ell \neq p$. In particular, we show it admits a regular dense open locus, and that it is equidimensional of dimension $\dim G$.

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arXiv2018

Lifting irreducible Galois representations

We study irreducible mod p representations, valued in general reductive groups, of the Galois group of a number field. When the number field is totally real, we show that odd representations satisfying local ramification hypotheses and a certain multiplicity-free condition on the adjoint representation admit geometric lifts. For general number fields, we show without any oddness or multiplicity condition that the representation admits a p-adic lift if it does everywhere locally.

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arXiv2016

Residual irreducibility of compatible systems

We show that if $\{ρ_{\ell}\}$ is a compatible system of absolutely irreducible Galois representations of a number field then the residual representation $\overlineρ_{\ell}$ is absolutely irreducible for $\ell$ in a density 1 set of primes. The key technical result is the following theorem: the image of $ρ_{\ell}$ is an open subgroup of a hyperspecial maximal compact subgroup of its Zariski closure with bounded index (as $\ell$ varies). This result combines a theorem of Larsen on the semi-simple part of the image with an analogous result for the central torus that was recently proved by Barnet-Lamb, Gee, Geraghty, and Taylor, and for which we give a new proof.

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arXiv2016

Anabelian geometry and descent obstructions on moduli spaces

We study the section conjecture of anabelian geometry and the sufficiency of the finite descent obstruction to the Hasse principle for the moduli spaces of principally polarized abelian varieties and of curves over number fields. For the former we show that the section conjecture fails and the finite descent obstruction holds for a general class of adelic points, assuming several well-known conjectures. This is done by relating the problem to a local-global principle for Galois representations. For the latter, we prove some partial results that indicate that the finite descent obstruction suffices. We also show how this sufficiency implies the same for all hyperbolic curves.

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arXiv2016

Deformations of Galois representations and exceptional monodromy, II: raising the level

Building on lifting results of Ramakrishna, Khare and Ramakrishna proved a purely Galois-theoretic level-raising theorem for two-dimensional odd representations of the Galois group of Q. In this paper, we generalize these techniques from type A1 to general (semi-)simple groups. We then strengthen our previous results on constructing geometric Galois representations with exceptional monodromy groups, achieving such constructions for almost all l, rather than a density-one set, and achieving greater flexibility in the Hodge numbers of the lifts; the latter improvement requires the new level-raising result.

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arXiv2016

Generalized Kuga-Satake theory and good reduction properties of Galois representations

In previous work we described when a single geometric representation, valued in a linear algebraic group, of the Galois group of a number field lifts through a central torus quotient to a geometric representation. In this paper we prove a much sharper result for systems of l-adic representations, such as the l-adic realizations of a motive, having common "good reduction" properties. Namely, such systems admit geometric lifts with good reduction outside a common finite set of primes. The method yields new proofs of theorems of Tate (the original result on lifting projective representations over number fields) and Wintenberger (an analogue of our main result in the setting of lifting through a central isogeny).

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arXiv2015

Deformations of Galois representations and exceptional monodromy

For any simple algebraic group $G$ of exceptional type, we construct geometric $\ell$-adic Galois representations with algebraic monodromy group equal to $G$, in particular producing the first such examples in types $\mathrm{F}_4$ and $\mathrm{E}_6$. To do this, we extend to general reductive groups Ravi Ramakrishna's techniques for lifting odd two-dimensional Galois representations to geometric $\ell$-adic representations.

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arXiv2014

Generalized Kuga-Satake theory and rigid local systems, II: rigid Hecke eigensheaves

This paper uses rigid Hecke eigensheaves, building on Yun's work on the construction of motives with exceptional Galois groups, to produce the first robust examples of `generalized Kuga-Satake theory' outside the Tannakian category of motives generated by abelian varieties. To strengthen our description of the `motivic' nature of Kuga-Satake lifts, we digress to establish a result that should be of independent interest: for any quasi-projective variety over a (finitely-generated) characteristic zero field, the associated weight-graded of its intersection cohomology arises from a motivated motive in the sense of André, and in particular from a classical homological motive if one assumes the Standard Conjectures. This extends work of de Cataldo and Migliorini.

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