arXiv subjects

Thomas Rüd

Publications and source records attributed to Thomas Rüd.

6 recordsLinked to original sources

arXiv2026

Measures on Cameron's treelike classes and applications to tensor categories

Measures on Fra\"iss\'e classes are a key input in the Harman--Snowden (2022) construction of tensor categories. Treelike Fra\"iss\'e classes provide a particularly tractable source of examples. In this paper, we complete the classification of measures on Cameron's elementary treelike classes. In particular, for the class $\partial \mathfrak{T}_3(n)$ of node-colored rooted binary tree structures with $n$ colors, we classify measures by an explicit bijection with directed rooted trees edge-labeled by $\{1, \dots, n\}$ with a distinguished vertex, yielding $(2n+2)^n$ distinct $\mathbb{Z}\left[\frac{1}{2}\right]$-valued measures. For each $n \geq 1$, we use a family of measures $\mu_n^I$ and their supports $\partial \mathfrak{T}_3(n)^{\mathrm{ord}}_I$ (where $I \subseteq \{1, \dots, n\}$) to construct the Karoubi envelopes $\mathbf{Rep}(\partial \mathfrak{T}_3(n)^{\mathrm{ord}}_I;\mu^I_n)$, producing infinite families of semisimple tensor categories with superexponential growth that cannot be obtained via Deligne's interpolation of representation categories. We also prove the nonexistence of measures on the $n$-colored tree class $C_n\mathfrak{T}$ for $n \geq 2$ and the labeled tree class $L \mathfrak{T}$, extending Snowden's results for uncolored trees.

math.CO↗

arXiv2025

A comparison problem for abelian surfaces and descent for symplectic orbital integrals

To answer a question about the distribution of products of elliptic curves in isogeny classes of abelian surfaces defined over finite fields, we compute specific orbital integrals in the group $\mathrm{GSp}_4$. More precisely, we compute integrals over the orbits of elements in the subgroup $\mathrm{GL}_2\times_{\det} \mathrm{GL}_2$. As a first step towards a complete solution of the problem, this article contains explicit computations for arbitrary orbital integrals of spherical functions over this subgroup, and also compute orbital integrals over $\mathrm{GSp}_4$ in a large number of cases.

math.NT↗

arXiv2024

A Projective Twist on the Hasse Norm Theorem

A finite extension of global fields $L/K$ satisfies the Hasse norm principle if any nonzero element of $K$ has the property that it is a norm locally if and only if it is a norm globally. In 1931, Hasse proved that any cyclic extension satisfies the Hasse norm principle, providing a novel approach to extending the local-global principle to equations with degree greater than $2$. In this paper, we introduce the projective Hasse norm principle, generalizing the Hasse norm principle to multiple fields and asking whether a projective line that contains a norm locally in every field must also contain a norm globally in every field. We show that the projective Hasse norm principle is independent from the conjunction of Hasse norm principles in all of the constituent fields in the general case, but that the latter implies the former when the fields are all Galois and independent. We also prove an analogue of the Hasse norm theorem for the projective Hasse norm theorem, namely that the projective Hasse norm principle holds in all cyclic extensions.

math.NT↗

arXiv2022

Counting abelian varieties over finite fields via Frobenius densities

Let $[X,λ]$ be a principally polarized abelian variety over a finite field with commutative endomorphism ring; further suppose that either $X$ is ordinary or the field is prime. Motivated by an equidistribution heuristic, we introduce a factor $ν_v([X,λ])$ for each place $v$ of $\mathbb Q$, and show that the product of these factors essentially computes the size of the isogeny class of $[X,λ]$. The derivation of this mass formula depends on a formula of Kottwitz and on analysis of measures on the group of symplectic similitudes, and in particular does not rely on a calculation of class numbers.

math.NT↗

arXiv2020

Explicit Tamagawa numbers for certain algebraic tori over number fields

Given a number field extension $K/k$ with an intermediate field $K^+$ fixed by a central element of the corresponding Galois group of prime order $p$, we build an algebraic torus over $k$ whose rational points are elements of $K^\times$ sent to $k^\times$ via the norm map $N_{K/K^+}$. The goal is to compute the Tamagawa number of that torus explicitly via Ono's formula that expresses it as a ratio of cohomological invariants. A fairly complete and detailed description of the cohomology of the character lattice of such a torus is given when $K/k$ is Galois. Partial results including the numerator are given when the extension is not Galois, or more generally when the torus is defined by an étale algebra. We also present tools developed in SAGE for this purpose, allowing us to build and compute the cohomology and explore the local-global principles for such an algebraic torus. Particular attention is given to the case when $[K:K^+]=2$ and $K$ is a CM-field. This case corresponds to tori in $\mathrm{GSp}_{2n}$, and most examples will be in that setting. This is motivated by the application to abelian varieties over finite fields and the Hasse principle for bilinear forms.

math.NT↗

arXiv2018

On Uniform Admissibility of Unitary and Smooth Representations

Let $G$ be a locally compact totally disconnected topological group. Under a necessary mild assumption, we show that the irreducible unitary representations of $G$ are uniformly admissible if and only if the irreducible smooth representations of $G$ are uniformly admissible. We also show that the latter property is inherited by finite-index subgroups and overgroups of $G$.

math.RT↗