arXiv subjects

Misja F. A. Steinmetz

Publications and source records attributed to Misja F. A. Steinmetz.

4 recordsLinked to original sources

arXiv2023

Packets of Serre weights for generic locally reducible two-dimensional Galois representations

Suppose $K/\mathbb{Q}_p$ is finite and $\overline{r}\colon G_K\to \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ is a reducible Galois representation. In this paper we prove that we can use the results by the author in [Ste22] to obtain a decomposition of the set of Serre weights $W(\overline{r})$ into a disjoint union of at most $(e+1)^f$ 'packets' of weights (where $f$ is the residue degree and $e$ the ramification degree of $K$) under the assumption that $\overline{r}$ is weakly generic. Thereby, we improve on results of Diamond--Savitt in [DS15] which give a similar decomposition, by rather different methods, under the assumption that $\overline{r}$ is strongly generic. We show that our definition of weak genericity is optimal for the results of this paper to hold when $e=1$. However, we expect that for $e=2$ one of the main results of this paper still holds under weaker hypotheses than the ones used in this paper.

math.NT↗

arXiv2022

Explicit Serre weights for two-dimensional Galois representations over a ramified base

Given a totally real number field $F$ and a mod $p$ Galois representation $ρ\colon G_F\to \mathrm{GL}_2(\bar{\mathbf{F}}_p)$, we propose an explicit definition of the set of Serre weights $W(ρ)$ attached to $ρ$. We prove that our explicit definition is equivalent to previous definitions available in the literature. As a consequence we obtain an explicit Serre's modularity conjecture for Hilbert modular forms over totally real number fields. Our work generalises previous work of Dembélé--Diamond--Roberts and Calegari--Emerton--Gee--Mavrides which together give explicit and equivalent sets of weights when $p$ is unramified in $F$.

math.NT↗

arXiv2022

Explicit Serre weights for GL_2 via Kummer theory

We give an explicit formulation of the weight part of Serre's conjecture for GL_2 using Kummer theory. This avoids any reference to p-adic Hodge theory. The key inputs are a description of the reduction modulo p of crystalline extensions in terms of certain "G_K-Artin-Scheier cocycles" and a result of Abrashkin which describes these cocycles in terms of Kummer theory. An alternative explicit formulation in terms of local class field theory was previously given by Dembele-Diamond-Roberts in the unramified case and by the second author in general. We show that the description of Dembele-Diamond-Roberts can be recovered directly from ours using the explicit reciprocity laws of Brueckner-Shaferevich-Vostokov. These calculations illustrate how our use of Kummer theory eliminates certain combinatorial complications appearing in these two papers.

math.NT↗

arXiv2015

Asypmtotic Behaviour of the Conjugacy Probability of the Alternating Group

For $G$ a finite group, let $κ(G)$ be the probability that $σ,τ\in G$ are conjugate, when $σ$ and $τ$ are chosen independently and uniformly at random. In this paper, we extend the methods of Blackburn et al [2012] to show that $n^2κ(A_n) \to A$ as $n\to\infty$ for a constant $A$, which we will give explicitly.

math.GR↗