arXiv subjects

Anna Szumowicz

Publications and source records attributed to Anna Szumowicz.

5 recordsLinked to original sources

arXiv2023

Uniform bounds on the Harish-Chandra characters

Let $\mathbf{G}$ be a connected reductive algebraic group over a $p$-adic local field $F$. In this paper we study the asymptotic behaviour of the trace characters $θ_π$ evaluated at a regular element $γ$ of $\mathbf{G}(F)$ as $π$ varies among supercuspidal representations of $\mathbf{G}(F)$. Kim, Shin and Templier conjectured that $\frac{θ_π(γ)}{{\rm deg}(π)}$ tends to $0$ when $π$ runs over irreducible supercuspidal representations of $\textbf{G}(F)$ with unitary central character and the formal degree of $π$ tends to infinity. For $\textbf{G}$ semisimple we prove that the trace character is uniformly bounded on $γ$ under the assumption, which is expected to hold true for every $\textbf{G} (F)$, that all irreducible supercuspidal representations of $\textbf{G}(F)$ are compactly induced from an open compact modulo center subgroup. Moreover, we give an explicit upper bound in the case of $γ$ ellitpic.

math.RT↗

arXiv2022

Orbits of cuspidal types on $\textrm{GL}_{p}(\mathcal{O}_{F})$

Let $F$ be a non-Archimedean local field and let $\mathcal{O}_{F}$ be its ring of integers. The orbit of an irreducible representation $ρ$ of $\mathrm{GL}_n(\mathcal{O}_F)$ is a conjugacy class in $\mathfrak{gl}_n(\mathcal{O}_F)$ attached to $ρ$ by means of Clifford's theory. We give a description of orbits of cuspidal types on $\mathrm{GL}_{p}( \mathcal{O}_{F})$, with $p$ prime. We determine which of them are regular and we provide an example which shows that the orbit of a representation does not always determine whether it is a cuspidal type or not.

math.RT↗

arXiv2022

Simultaneous $\mathfrak{p}$-orderings and equidistribution

Let $D$ be a Dedekind domain. Roughly speaking, a simultaneous $\mathfrak{p}$-ordering is a sequence of elements from $D$ which is equidistributed modulo every power of every prime ideal in $D$ as well as possible. Bhargava asked which subsets of the Dedekind domains admit simultaneous $\mathfrak{p}$-orderings. We give an overview on the progress in this problem. We also explain how it relates to the theory of integer valued polynomials and list some open problems.

math.NT↗

arXiv2022

On the optimal rate of equidistribution in number fields

Let $k$ be a number field. We study how well can finite sets of $\mathcal O_k$ equidistribute modulo powers of prime ideals, for all prime ideals at the same time. Our main result states that the optimal rate of equidistribution in $\mathcal O_k$ predicted by the local contstraints cannot be achieved unless $k=\mathcal Q$. We deduce that $\mathcal Q$ is the only number field where the ring of integers $\mathcal O_k$ admits a simultaneous $\frak p$-ordering, answering a question of Bhargava. Along the way we establish a non-trivial upper bound on the number of solutions $x\in \mathcal O_k$ of the inequality $|N_{k/\mathcal Q}(x(a-x))|\leq X^2$ where $X$ is a positive real parameter and $a\in\mathcal O_k$ is of norm at least $e^{-B}X$ for a fixed real number $B$. The latter can be translated as an upper bound on the average number of solutions of certain unit equations in $\mathcal O_k$.

math.NT↗

arXiv2016

Simultaneous $p$-orderings and minimising volumes in number fields

In the paper "On the interpolation of integer-valued polynomials" (Journal of Number Theory 133 (2013), pp. 4224--4232.) V. Volkov and F. Petrov consider the problem of existence of the so-called $n$-universal sets (related to simultaneous $p$-orderings of Bhargava) in the ring of Gaussian integers. We extend their results to arbitrary imaginary quadratic number fields and prove an existence theorem that provides a strong counterexample to a conjecture of Volkov-Petrov on minimal cardinality of $n$-universal sets. Along the way, we discover a link with Euler-Kronecker constants and prove a lower bound on Euler-Kronecker constants which is of the same order of magnitude as the one obtained by Ihara.

math.NT↗