38 citations · 72 across the 10 of their papers we have counts for
11 papers
Congruences like Atkin's for the partition function
Scott Ahlgren, Patrick B. Allen, Shiang Tang
Let be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form where and are…
Modularity of -representations over totally real fields
Patrick B. Allen, Chandrashekhar B. Khare, Jack A. Thorne
We study an analogue of Serre's modularity conjecture for projective representations , where $…
Automorphy lifting for residually reducible -adic Galois representations, II
Patrick B. Allen, James Newton, Jack A. Thorne
We prove new automorphy lifting theorems for residually reducible Galois representations of unitary type in which the residual representation is permitted to have an arbitrary numb…
Modularity of -representations over CM fields
Patrick B. Allen, Chandrashekhar Khare, Jack A. Thorne
We prove that many representations , where is a CM field, arise from modular elliptic c…
Monodromy for some rank two Galois representations over CM fields
Patrick B. Allen, James Newton
We investigate local-global compatibility for cuspidal automorphic representations for GL(2) over CM fields that are regular algebraic of weight . We prove that for a Dirich…
Potential automorphy over CM fields
Patrick B. Allen, Frank Calegari, Ana Caraiani +7
Let be a CM number field. We prove modularity lifting theorems for regular -dimensional Galois representations over without any self-duality condition. We deduce that al…
On automorphic points in polarized deformation rings
Patrick B. Allen
For a fixed mod automorphic Galois representation, -adic automorphic Galois representations lifting it determine points in universal deformation space. In the case of modula…
Finiteness of unramified deformation rings
Patrick B. Allen, Frank Calegari
We prove that the universal unramified deformation ring of a continuous Galois representation (for a totally…
Deformations of polarized automorphic Galois representations and adjoint Selmer groups
Patrick B. Allen
We prove the vanishing of the geometric Bloch-Kato Selmer group for the adjoint representation of a Galois representation associated to regular algebraic polarized cuspidal automor…
Modularity of nearly ordinary 2-adic residually dihedral Galois representations
Patrick B. Allen
We prove modularity of some two dimensional, 2-adic Galois representations over totally real fields that are nearly ordinary and that are residually dihedral. We do this by employi…
New Bounds on van der Waerden-type Numbers for Generalized 3-term Arithmetic Progressions
Patrick Allen, Bruce M. Landman, Holly Meeks
Let a and b be positive integers with a \leq b. An (a,b)-triple is a set {x,ax+d,bx+ 2d}, where x,d \geq 1. Define T(a,b;r) to be the least positive integer n such that any r-color…