Patrick Allen

I am interested in algebraic number theory; in particular, the connection between Galois representations and automorphic forms. I am especially interested in the conjectures of Fontaine–Mazur and Langlands, and their interaction with the deformation theory of Galois representations, p-adic automorphic forms, and Selmer groups.

Publications

The papers posted here or on the arXiv may differ from the published versions.

Modularity of GL₂(F_p) representations over CM fields.
(with Chandrashekhar Khare and Jack Thorne) To appear in Camb. J. Math.
Potential automorphy over CM fields. (Video, Another video)
(with Ana Caraiani, Frank Calegari, David Helm, Toby Gee, Bao V. Le Hung, James Newton, Peter Scholze, Richard Taylor, and Jack A. Thorne) Ann. of Math. (2) 197 (2023), no. 3, 897–1113. arXiv
Congruences like Atkin's for the partition function.
(with Scott Ahlgren and Shiang Tang) Trans. Amer. Math. Soc. Ser. B 9 (2022), 1044–1064. arXiv
Modularity of PGL₂(F_p) representations over totally real fields.
(with Chandrashekhar Khare and Jack Thorne) Proc. Natl. Acad. Sci. USA 118 (2021), no. 33, e2108064118. arXiv
Automorphy lifting for residually reducible l-adic Galois representations, II.
(with James Newton and Jack A. Thorne) Compos. Math. 156 (2020), no. 11, 2399–2422. arXiv
Monodromy for some rank two Galois representations over CM fields. (Slides)
(with James Newton) Doc. Math. 25 (2020), 2487–2506. arXiv
On automorphic points in polarized deformation rings.
Amer. J. Math. 141 (2019), no. 1, 119–167. arXiv
Deformations of polarized automorphic Galois representations and adjoint Selmer groups. (Video)
Duke Math. J. 165 (2016), no. 13, 2407–2460. arXiv
Finiteness of unramified deformation rings.
(with Frank Calegari) Algebra Number Theory 8 (2014), no. 9, 2263–2272. arXiv
Modularity of nearly ordinary 2-adic residually dihedral Galois representations.
Compositio Math. 150 (2014), no. 8, 1235–1346. arXiv
On the multiplicity of linear recurrence sequences.
J. Number Theory 126 (2007), no. 2, 212–216.

Other writing

The following are some things I've written that are not (currently) intended for publication.

Deformations of Hilbert modular Galois representations and adjoint Selmer groups (Many, but not all, of the results here have generalized in the above preprint, Deformations of polarized automorphic Galois representations and adjoint Selmer groups).
Nonarchimedean Local Fields. Course notes written for a graduate course on local fields and local class field theory via Lubin–Tate theory. The presentation is a mix of Serre, Iwasawa, and Yoshida with a little more detail.