Stefan Patrikis's papers and notes Here are some papers I have written; these are not final versions and may change without notice.

Lifting and modularity of reducible mod p Galois representations (arXiv) (Inventiones Mathematicae (2021).
Joint with Najmuddin Fakhruddin and Chandrashekhar Khare. We extend the lifting methods of our earlier paper to produce geometric p-adic lifts of reducible G-valued Galois representations over totally real fields (assuming the representations are odd) or over function fields. We deduce new appications to modularity of residually reducible Galois representations, for GL_2 over totally real fields or for GL_n over function fields.
G-rigid local systems are integral (arXiv) (to appear in Transactions of the AMS)
Joint with Christian Klevdal. Let G be a reductive group, and let X be a smooth quasiprojective 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-Groechenig when G=GL_n, and it answers in the affirmative a conjecture of Simpson, a much weaker form of his conjecture that rigid local systems are motivic. Along the way we show that the connected component of the Zariski-closure of the monodromy group of any such local system is semisimple.
Potential automorphy of GSpin(2n+1)-valued Galois representations (arXiv) (Mathematische Zeitschrift (2021).
Joint with Shiang Tang. We prove a potential automorphy theorem for certain mod p and p-adic GSpin(2n+1)-valued Galois representations of the absolute Galois group of a totally real field. We also prove results on (automorphic) solvable descent for GSp(2n) over totally real fields, and combine these with the potential automorphy theorem to put GSpin(2n+1)-valued p-adic representations into compatible systems.
Lifting G-irreducible but GL_n-reducible Galois representations (arXiv) (Mathematics Research Letters Volume 27 (2020) Number 6)
Joint with Najmuddin Fakhruddin and Chandrashekhar Khare. This is a follow-up to our ``Relative deformation theory" paper below, in which we construct families of examples, when G is a classical group with n-dimensional standard representation, of G-irreducible but GL_n-reducible Galois representations to which our lifting methods apply. In particular, we can construct geometric lifts in situations, even for classical groups, where potential automorphy theorems do not apply.
Relative deformation theory and lifting irreducible Galois representations (arXiv) (Duke Mathematical Journal 170 (16), 3505-3599, (2021))
Joint with Najmuddin Fakhruddin and Chandrashekhar Khare. We study the problem of when an irreducible mod p Galois representation, valued in a general reductive group, of the Galois group of a number field can be lifted to a p-adic representation. For odd irreducible representations over totally real fields, we prove that existence of local (de Rham) lifts implies existence of global geometric lifts. (This greatly improves the results of ``Lifting irreducible Galois representations" below.) To that end, we combine a new ``relative deformation theory" method, which bypasses the usual global image difficulties in Ramakrishna-style lifting theorems, with results on the generic fibers of characteristic zero local lifting rings, which properly applied allow us to bypass the usual requirements of having formally smooth irreducible components of local lifting rings.
Lifting irreducible Galois representations (arXiv)
Joint with Najmuddin Fakhruddin and Chandrashekhar Khare. We study the problem of when an irreducible mod p Galois representation, valued in a general reductive group, of the Galois group of a number field can be lifted to a p-adic representation. For odd representations over totally real fields, we prove the existence of geometric lifts assuming some local ramification hypotheses and a certain multiplicity-free condition on the decomposition of the adjoint representation. For general number fields, we prove the existence of (non-geometric) lifts assuming that the mod p representation admits lifts everywhere locally. [The results of this paper are superseded by ``Relative deformation theory..." above, but some of the techniques are different and may still be of interest.]
Compatible systems of Galois representations associated to the exceptional group E6 (arXiv) (Forum of Mathematics, Sigma)
Joint with George Boxer, Frank Calegari, Matthew Emerton, Brandon Levin, and Keerthi Madapusi Pera. We construct (over any quadratic imaginary field) compatible systems of l-adic Galois representations, appearing in the cohomology of algebraic varieties, whose l-adic algebraic monodromy groups are the exceptional group E6.
G-valued Galois deformation rings when l≠p (arXiv) (Mathematics Research Letters)
Joint with Jeremy Booher. We study the generic fibers of local Galois deformation rings of G-valued mod p representations, in the l≠p case. In particular we show the generic fibers admit regular open dense subschemes and are equidimensional of dimension dim(G).
Generalized Kuga-Satake theory and good reduction properties of Galois representations (arXiv) (Algebra & Number Theory 2017, Vol. 11 No. 10)
This paper proves that the various l-adic realizations of a motivic Galois representation can be simultaneously lifted through a central torus quotient to geometric representations with uniformly (independent of l) bounded ramification set. This sharpening of the main Galois-theoretic results of Variations (see below) also yields a new proof of an analogous theorem of Wintenberger (lifting compatible systems through isogenies).
Residual irreducibility of compatible systems (arXiv) (IMRN rnw241)
Joint with Andrew Snowden and Andrew Wiles. We show that a compatible system of absolutely irreducible l-adic representations of the Galois group of a number field remains irreducible after reduction modulo l for a density one set of primes l. This generalizes work of Barnet-Lamb, Gee, Geraghty, and Taylor in the case of Hodge-Tate regular compatible systems.
Deformations of Galois representations and exceptional monodromy, II: raising the level (arXiv) (Mathematische Annalen 2017, vol. 368)
This paper enhances the Galois deformation techniques of Deformations of Galois representations and exceptional monodromy (below) to incorporate a ``level-raising'' result that was originally established in type A1 by Khare and Ramakrishna. The paper also gives more flexible applications to the construction of geometric Galois representations with exceptional monodromy groups, producing, for instance, examples for almost all l rather than a density one set of l.
Anabelian geometry and descent obstructions on moduli spaces (arXiv) (Algebra & Number Theory 2016, Vol. 10 no. 6)
Joint with Felipe Voloch and Yuri Zarhin. This paper studies the section conjecture of anabelian geometry and the sufficiency of the finite descent obstruction for moduli spaces of principally polarized abelian varieties, and to a lesser extent for moduli of curves.
Deformations of Galois representations and exceptional monodromy (arXiv) (Inventiones Mathematicae 205 (2))
For any simple algebraic group G of exceptional type, this paper constructs geometric l-adic representations with algebraic monodromy group G. Along the way it establishes a generalization to any reductive group of Ravi Ramakrishna's lifting methods (see `Lifting symplectic Galois representations,' below).
Generalized Kuga-Satake theory and rigid local systems, II: rigid Hecke eigensheaves (arXiv) (Algebra & Number Theory 2016, Vol. 10 no. 7 )
This paper uses rigid Hecke eigensheaves, building on Yun's work on the construction of motives with exceptional Galois groups, to produce more robust examples of `generalized Kuga-Satake theory' outside the Tannakian category of motives generated by abelian varieties. It also strengthens the sense in which Yun's original motives are `motivic,' by generalizing a theorem of de Cataldo-Migliorini that roughly says `the factors of the decomposition theorem are motivated.'
Generalized Kuga-Satake theory and rigid local systems, I: the middle convolution (arXiv), in Recent Advances in Hodge Theory, eds M. Kerr and G. Pearlstein, Cambridge University Press
This paper produces the first non-trivial examples of generalized Kuga-Satake theory for motives not generated by abelian varieties. The essential tools are Katz's `middle convolution' and a description, due to Bogner and Reiter, of rank 4, symplectically rigid local systems on the puncture line.
On the sign of regular algebraic polarizable automorphic representations (arXiv), Math. Ann. 362 (2015)
This paper shows that a parity condition on automorphic representations considered in the Paris Book Project is superfluous. It then generalizes the construction of automorphic Galois representations to the `mixed-parity' case, finding associated geometric projective representations. Finally, some of the Galois lifting results in `Variations on a theorem of Tate' are optimized.
Mumford-Tate groups of polarizable Hodge structures (arXiv), Proceedings of the AMS 144 (2016)
This paper classifies the possible Mumford-Tate groups of polarizable rational Hodge structures.
Automorphy and irreducibility of some l-adic representations (arXiv), Comp. Math. vol. 151 #2 (2015)
Joint with Richard Taylor. This paper removes the irreducibility hypotheses from potential automorphy theorems for (regular, polarizable, pure) compatible systems, with applications to meromorphic continuation of motivic L-functions, and `cuspidality implies irreducibility.' We have included many examples of regular motives, constructed via Katz's theory of rigid local systems, to which the potential automorphy theorem applies.
Variations on a theorem of Tate (pdf) (Memoirs of the AMS, Vol. 258, No. 1238, 2019)
More or less my thesis. The `theorem of Tate' is the result that all projective representations of the absolute Galois group of a number field lift to actual representations. This paper discusses refinements, automorphic and motivic analogues, and a number of related questions.
Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves (pdf)
Joint with G. Grigorov, A. Jorza, C. Tarnita, and W. Stein. Appeared as Math. Comp. 78 (2009), 2397-2425.
Lifting symplectic Galois representations
My undergraduate thesis, which proves a generalization to symplectic groups of any rank of Ravi Ramakrishna's lifting theorem from Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur, Ann. of Math. (2) 156 no. 1 (2002), 115-154. These results have mostly been majorized by applications of potential automorphy theorems; one thing that may still be of interest is the calculation of the symplectic crystalline local deformation ring (in the Fontaine-Laffaille range). Available upon request.