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Peter Vang Uttenthal

Publications and source records attributed to Peter Vang Uttenthal.

11 recordsLinked to original sources
arXiv2026

Positive kernels in Lie group representations and Fourier interpolation

For a given function $f$ on a discrete subspace $Γ$ of a noncompact Riemannian symmetric space $G/K$, we construct analytic, noncompactly supported functions $W$ on $G/K$ with $W|_Γ=f$ that, in addition, satisfy an invariant differential equation on $G/K$. The key step is to regard $W$ as an orthogonal projection onto a space of coherent states in a representation of the Lie group $G$ in a Hilbert space that admits a strictly positive definite reproducing kernel. The techniques are implemented for the heat kernel on $G/K$, and $W$ is expressed in terms of certain closed formulas for linearly independent coherent states.

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arXiv2026

Convergent series of coherent states

For a discrete subspace $Γ$ of a homogeneous space $X$ and a positive definite reproducing kernel Hilbert space $\mathscr{H}(X)$, a convergence proof is given for series of partial Whitney functions over $Γ$ that arise in the span of coherent states in $\mathscr{H}(X)$.

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arXiv2026

Covers of Bruhat-Tits trees

Let $G$ be a locally compact group and let $\widetilde{G}$ be a central extension that splits over a maximal compact subgroup $K$ of $G$. We derive an explicit cocycle that lifts the natural action of $G$ on the homogeneous space $G/K$ to an action of $\widetilde{G}$. As an application, for a non-Archimedean local field $F$, we construct a connected locally finite tree on which the metaplectic covers of $\operatorname{GL}_2(F)$ act by automorphisms, providing a geometric analog of the Bruhat--Tits tree of $\operatorname{GL}_2(F)$. Furthermore, under suitable transitivity assumptions, we prove that $(\widetilde{G},\widetilde{K})$ is a Gelfand pair. Finally, we describe the associated parabolic and contraction subgroups with respect to $\widetilde{G}$ from the perspective of the geometry of the constructed tree.

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arXiv2025

Parameters of solvable automorphic forms

In a letter from Tate to Serre dated March 26, 1974, Tate suggested a classification of weight one modular forms of prime level in terms of their associated odd Artin representations. This paper carries out an analogous classification of Maass wave forms of prime power level in terms of complex even representations. The parameters are identified with techniques from class field theory and Galois representations. The classification reveals that there exist distinct Maass cusp forms of tetrahedral type on $Γ_1(\ell)$ that remain inequivalent modulo $3$ for $\ell = 7687, 16363$ and $20887$, and that these $\ell$ are the three smallest such primes.

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arXiv2025

A classification of even representations onto 3-adic SL(2)

This paper gives a classification of even representations onto $\operatorname{SL}(2,\mathbb{Z}_3)$ of prime conductor. In addition, an explicit algorithm based on global class field theory is exhibited, computing an exhaustive series of such even representations.

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arXiv2025

Level-raising of even representations of tetrahedral type and equidistribution of lines in the projective plane

The distribution of primes raising the level of even Galois representations of tetrahedral type is studied. Data are presented on primes $v\leq 10^8$ raising the level of $3$-adic even representations of various conductors. Based on the data, a conjecture is formulated concerning the distribution of certain lines in the plane. By an application of Wiles' formula, the conjecture is shown to imply that the density of primes raising the level of a $p$-adic even representation is $$\frac{p-1}{p},$$ in agreement with the density of $2/3$ for $p=3$ observed in the data.

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arXiv2025

On flat even deformation rings

In the presence of a nontrivial dual Selmer group, certain global even deformation rings are shown to be finite and flat over $\mathbb{Z}_p$. Previously, flatness was only known in established cases of Langlands reciprocity in the odd parity. By techniques from global class field theory, explicit examples of even representations are computed to which the results apply. For even representations $\overlineρ$ in an explicit family, it is observed that if Leopoldt's conjecture is true for a certain number field attached to $\overlineρ$, then the global even deformation ring is flat at the minimal level.

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arXiv2025

Whitney extensions on symmetric spaces

In 1934, H. Whitney introduced the problem of extending a function on a set of points in $\mathbb{R}^n$ to an analytic function on the ambient space. In this article we prove Whitney type extension theorems for data on some homogeneous spaces. We use harmonic analysis on the homogeneous spaces and representation theory of compact as well as noncompact reductive groups.

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arXiv2024

Elliptic curves and spin

In the early 2000s, Ramakrishna asked the question: For the elliptic curve $$ E: y^2 = x^3 - x, $$ what is the density of primes $p$ for which the Fourier coefficient $a_p(E)$ is a cube modulo $p$? As a generalization of this question, Weston--Zaurova formulated conjectures concerning the distribution of power residues of degree $m$ of the Fourier coefficients of elliptic curves $E/\mathbb{Q}$ with complex multiplication. In this paper, we prove their conjecture for cubic residues using the analytic theory of spin. Our proof works for all elliptic curves $E$ with complex multiplication.

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arXiv2023

Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity

This paper concerns the distribution of Selmer ranks in a family of even Galois representations in even residual characteristic obtained by allowing ramification at auxiliary primes. The main result is a Galois cohomological analogue of a theorem of Friedlander, Iwaniec, Mazur and Rubin on the distribution of Selmer ranks in a family of twists of elliptic curves. The Selmer groups are constructed as prescribed by the Galois cohomological method for GL(2): At each ramified place, the local Selmer condition is the tangent space of a smooth quotient of the local deformation ring. By methods of global class field theory, the Selmer group at the minimal level is computed explicitly. The infinitude of primes for which the Selmer rank increases by one is proved, and the density of such primes is shown to be 1/192. The proof combines Wiles' formula and the global reciprocity law. The result has implications for the algebraic structure of even deformation rings and the distribution of their presentations in families.

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