Publications

Extensions of mod p representations of division algebras over non-Archimedean local fields

Published in Journal de Théorie des Nombres de Bordeaux, 2023

(With Andrew Keisling) This paper computes the Ext^1 groups of arbitrary smooth irreducible mod p representations of division algebras over a non-Archimedean local field of residue characteristic p (with mild assumptions on p). We also give partial results for higher extensions.

Recommended citation: Andrew Keisling and Dylan Pentland. "Extensions of mod p representations of division algebras over non-Archimedean local fields." arXiv preprint arXiv:2110.00705 (2023). https://arxiv.org/pdf/2110.00705.pdf

Computing L-Polynomials of Picard curves from Cartier-Manin matrices

Published in Mathematics of Computation, 2022

(With Sualeh Asif and Francesc Fité) This paper gives a fast and practical algorithm to compute L-polynomials of generic Picard curves over Q for all primes p of good reduction up to some integer N. We prove an interesting theoretical result to achieve this: for all but a density zero subset of primes, we can either compute the L-polynomial from the Cartier-Manin matrix or from this matrix and the splitting behavior of f and a polynomial psi_f in Q[x] related to the 2-torsion of the Jacobian. These show that surprisingly little work is needed to compute the L-polynomial at p from the L polynomial modulo p (which is found by the Cartier-Manin matrix).

Recommended citation: Asif, Sualeh, Francesc Fité, and Dylan Pentland. "Computing L-polynomials of Picard curves from Cartier–Manin matrices." Mathematics of Computation 91.334 (2022): 943-971. https://arxiv.org/pdf/2010.07247.pdf

Filtrations on block subalgebras of reduced enveloping algebras

Published in Journal of Algebra and its Applications, 2022

(With Andrei Ionov) We study several natural filtrations related to the PBW filtration of block subalgebra of restricted universal enveloping algebras. We show how to study the associated graded using the nilpotent cone, and compute it in the case of SL2.

Recommended citation: Ionov, Andrei, and Dylan Pentland. "Filtrations on block subalgebras of reduced enveloping algebras." Journal of Algebra and Its Applications 21.12 (2022): 2350001. https://arxiv.org/pdf/2009.07477.pdf

Coefficients of Gaussian Polynomials Modulo N

Published in Electronic Journal of Combinatorics, 2020

This paper proves a conjecture of Richard Stanley about reductions modulo N of the coefficients of q-binomial coefficients (hence the change to the less used “Gaussian Polynomials”). Namely, it shows that the residue counting function for n choose k is quasipolynomial as a corollary of a more interesting result about the decomposition of the coefficients modulo N into various periodic classes.

Recommended citation: Pentland, Dylan. "Coefficients of Gaussian Polynomials Modulo N." The Electronic Journal of Combinatorics (2020): P2-58. https://www.combinatorics.org/ojs/index.php/eljc/article/view/V27i2p58/pdf