arXiv subjects

Dylan Pentland

Publications and source records attributed to Dylan Pentland.

5 recordsLinked to original sources

arXiv2025

Syntomification and crystalline local systems

Let $p$ be a prime, and let $\mathrm{X}$ be a smooth $p$-adic formal scheme over $\mathrm{Spf} \mathcal{O}_K$ where $K/\mathbf{Q}_p$ is a finite extension. We show that reflexive sheaves on the stack $\mathrm{X}^{\mathrm{Syn}}$ are equivalent to $\mathbf{Z}_p$-lattices in crystalline local systems on the rigid generic fiber $\mathrm{X}_\eta$, and then use this to study the essential image of the \'{e}tale realization functor on the isogeny category of perfect complexes on $\mathrm{X}^{\mathrm{Syn}}$. We also show that when $\mathrm{X}/\mathrm{Spf} \mathcal{O}_K$ is smooth and proper that $\mathsf{Perf}(\mathrm{X}^{\mathrm{Syn}})[1/p]$ is equivalent to a category of admissible filtered $F$-isocrystals in perfect complexes.

math.NT↗

arXiv2021

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

Let $F$ be a local field over $\mathbf{Q}_p$ or $\mathbf{F}_p((t))$, and let $D$ be a central simple division algebra over $F$ of degree $d$. In the $p$-adic case, we assume $p>de+1$ where $e$ is the ramification degree over $\mathbf{Q}_p$; otherwise, we need only assume $p$ and $d$ are coprime. For the subgroup $I_1=1+\varpi_D \mathcal{O}_D$ of $D^\times$ we determine the structure of $\mathrm{H}^1(I_1, π)$ as a representation of $D^\times/I_1$ for an arbitrary smooth irreducible representation $π$ of $D^\times$. We use this to compute the group $\mathrm{Ext}^1_{D^\times}(π,π')$ for arbitrary smooth irreducible representations $π$ and $π'$ of $D^\times$. In the $p$-adic case, via Poincaré duality we can compute the top cohomology groups and compute the highest degree extensions.

math.NT↗

arXiv2021

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

We study the sequence of zeta functions $Z(C_p,T)$ of a generic Picard curve $C:y^3=f(x)$ defined over $\mathbb{Q}$ at primes $p$ of good reduction for $C$. We define a degree 9 polynomial $ψ_f\in \mathbb{Q}[x]$ such that the splitting field of $ψ_f(x^3/2)$ is the $2$-torsion field of the Jacobian of $C$. We prove that, for all but a density zero subset of primes, the zeta function $Z(C_p,T)$ is uniquely determined by the Cartier-Manin matrix $A_p$ of $C$ modulo $p$ and the splitting behavior modulo $p$ of $f$ and $ψ_f$; we also show that for primes $\equiv 1 \pmod{3}$ the matrix $A_p$ suffices and that for primes $\equiv 2 \pmod{3}$ the genericity assumption on $C$ is unnecessary. An element of the proof, which may be of independent interest, is the determination of the density of the set of primes of ordinary reduction for a generic Picard curve. By combining this with recent work of Sutherland, we obtain a practical deterministic algorithm that computes $Z(C_p,T)$ for almost all primes $p \le N$ using $N\log(N)^{3+o(1)}$ bit operations. This is the first practical result of this type for curves of genus greater than 2.

math.NT↗

arXiv2021

Filtrations on block subalgebras of reduced universal enveloping algebras

We study the interaction between the block decompositions of reduced universal enveloping algebras in positive characteristic, the PBW filtration, and the nilpotent cone. We provide two natural versions of the PBW filtration on the block subalgebra $A_λ$ of the restricted universal enveloping algebra $\mathcal{U}_χ(\mathfrak{g})$ and show these are dual to each other. We also consider a shifted PBW filtration for which we relate the associated graded algebra to the algebra of functions on the Frobenius neighbourhood of $0$ in the nilpotent cone and the coinvariants algebra corresponding to $λ$. In the case of $\mathfrak{g}=\mathfrak{sl}_2(k)$ in characteristic $p>2$ we determine the associated graded algebras of these filtrations on block subalgebras of $\mathcal{U}_0(\mathfrak{sl}_2)$. We also apply this to determine the structure of the adjoint representation of $\mathcal{U}_0(\mathfrak{sl}_2)$.

math.RT↗

arXiv2020

Coefficients of Gaussian Polynomials Modulo $N$

The $q$-analogue of the binomial coefficient, known as a $q$-binomial coefficient, is typically denoted $\left[{n \atop k}\right]_q$. These polynomials are important combinatorial objects, often appearing in generating functions related to permutations and in representation theory. Stanley conjectured that the function $f_{k,R}(n) = \#\left\{i : [q^{i}] \left[{n \atop k}\right]_q \equiv R \pmod{N}\right\}$ is quasipolynomial for $N=2$. We generalize, showing that this is in fact true for any integer $N\in \mathbb{N}$ and determine a quasi-period $π'_N(k)$ derived from the minimal period $π_N(k)$ of partitions with at most $k$ parts modulo $N$.

math.CO↗