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Edgar Assing

Publications and source records attributed to Edgar Assing.

At least 19 recordsLinked to original sources
arXiv2026

Spectral fourth moments of Hecke--Maa{\ss} cusp forms

In this note, we establish essentially optimal bounds for certain spectral moments of automorphic forms for $\mathrm{GL}(2)$. More precisely, we consider the family of Hecke--Maa{\ss} cusp forms with spectral parameter in a dyadic interval and study the fourth moment of these forms evaluated at a Heegner point. We additionally present applications of our main result to the shifted convolution problem involving the sum of two squares function $r(n)$ as well as to pointwise Diophantine exponents.

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arXiv2025

On the spectral aspect density hypothesis and application

We prove that the density of non-tempered (at any $p$-adic place) cuspidal representations for $\mathrm{GL}_n(\mathbb{Z})$, varying over a family of representations ordered by their infinitesimal characters, is small -- confirming Sarnak's density hypothesis in this set-up. Among other ingredients, the proof uses tools from microlocal analysis for Lie group representations as developed by Nelson and Venkatesh. As an application, we prove that the Diophantine exponent of the $\mathrm{SL}_n(\mathbb{Z}[1/p])$-action on $\mathrm{SL}_n(\mathbb{R})/\mathrm{SO}_n(\mathbb{R})$ is \emph{optimal} -- resolving a conjecture of Ghosh, Gorodnik, and Nevo.

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arXiv2025

The orbit method in number theory through the sup-norm problem for $\operatorname{GL}(2)$

The orbit method in its quantitative form due to Nelson and Venkatesh has played a central role in recent advances in the analytic theory of higher rank $L$-functions. The goal of this note is to explain how the method can be applied to the sup-norm problem for automorphic forms on $\operatorname{PGL}(2)$. Doing so, we prove a new hybrid bound for newforms $\varphi$ of large prime-power level $N = p^{4n}$ and large eigenvalue $\lambda$. It states that $\| \varphi \|_\infty \ll_p (\lambda N)^{5/24 + \varepsilon}$, recovering the result of Iwaniec and Sarnak spectrally and improving the local bound in the depth aspect for the first time in this non-compact setting. We also provide an exposition of the microlocal tools used, illustrating and motivating the theory through the classical case of $\operatorname{PGL}(2)$, following notes and lectures of Nelson and Venkatesh.

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arXiv2025

Isogenies of CM Elliptic Curves

Given two CM elliptic curves over a number field and a natural number $m$, we establish a polynomial lower bound (in terms of $m$) for the number of rational primes $p$ such that the reductions of these elliptic curves modulo a prime above $p$ are $m$-isogenous. The proof relies on higher Green functions and theorems of Gross-Zagier and Gross-Kohnen-Zagier. A crucial observation is that the Fourier coefficients of incoherent Eisenstein series can be approximated by those of coherent Eisenstein series of increasing level. Another key ingredient is an explicit upper bound for the Petersson norm of an arbitrary elliptic modular form in terms of finitely many of its Fourier coefficients at the cusp infinity, which is a result of independent interest.

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arXiv2024

Unipotent mixing for general moduli

In this note we prove a joint equidistribution result for discrete low lying horocycles. This generalizes previous work of Blomer and Michel, where it was crucially assumed that the number of discrete points is prime.

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arXiv2024

Local analysis of the Kuznetsov formula and the density conjecture

We prove Sarnak's spherical density conjecture for the principal congruence subgroup of SL(n, Z) of arbitrary level. Applications include a complete version of Sarnak's optimal lifting conjecture for principal congruence subgroups of SL(n, Z), as well as a transfer of the density theorem to certain co-compact situations. The main ingredients are new lower bounds for Whittaker functions and strong estimates for the cardinality of ramified Kloosterman sets.

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arXiv2024

An observation concerning highly ramified $\epsilon$-factors

In this note we prove a quantitative stability result for the $\epsilon$-factors associated to generic irreducible representations of $\textrm{GL}_n(F)$ under twists by highly ramified characters, where $F$ is a non-archimedean local field.

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arXiv2023

A note on Sarnak's density hypothesis for $\textrm{Sp}_4$

In this note we prove Sarnak's (spherical) density hypothesis for the full discrete spectrum of the quotients $\Gamma_{\textrm{pa}}(q)\backslash \textrm{Sp}_4(\mathbb{R})$, where $\Gamma_{\textrm{pa}}(q)$ are paramodular groups with square-free level $q$. To derive this estimate we upgrade a density estimate established via the (pre)-Kuznetsov formula, which only accounts for the generic part, using Arthur's parametrization of the discrete spectrum.

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arXiv2022

The density conjecture for principal congruence subgroups

We prove Sarnak's density conjecture for the principal congruence subgroup of SL_n(Z) of squarefree level and discuss various arithmetic applications. The ingredients include new bounds for local Whittaker functions and Kloosterman sums.

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arXiv2021

The sup-norm problem beyond the newform

In this note we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation $\pi$ we consider a special small $GL_2(\mathbb{Z}_p)$-type $V$ in $\pi$ and proof global sup-norm bounds for an average over an orthonormal basis of $V$. We achieve a non-trivial saving when the dimension of $V$ grows.

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arXiv2020

Vorono\"i summation for half-integral weight automorphic forms

A general Vorono\"i summation formula for the (metaplectic) double cover of $\text{GL}_2$ is derived via the representation theoretic framework \`a la Ichino--Templier. The identity is also formulated classically and used to establish Vorono\"i summation formulae for half-integral weight modular forms and Maa{\ss} forms.

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arXiv2020

Uniform Titchmarsh divisor problems

Asymptotic formulae for Titchmarsh-type divisor sums are obtained with strong error terms that are uniform in the shift parameter. This applies to more general arithmetic functions such as sums of two squares, improving the error term in the representation of the number as a sum of a prime and two squares, and to Fourier coefficients of cusp forms, generalizing a result of Pitt.

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arXiv2019

Vorono\"{i} summation via switching cusps

We consider the Fourier expansion of a Hecke (resp.\ Hecke--Maa\ss) cusp form of general level $N$ at the various cusps of $\Gamma_{0}(N)\bs\Hb$. We explain how to compute these coefficients via the local theory of $p$-adic Whittaker functions and establish a classical Vorono\"i summation formula allowing an arbitrary additive twist. Our discussion has applications to bounding sums of Fourier coefficients and understanding the (generalised) Atkin--Lehner relations.

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arXiv2018

Adelic Voronoi Summation and Subconvexity for GL(2) L-Functions in the depth Aspect

In this paper we establish a very flexible and explicit Voronoi summation formula. This is then used to prove an almost Weyl strength subconvexity result for automorphic $L$-functions of degree two in the depth aspect. That is, looking at twists by characters of prime power conductor. This is the natural $p$-adic analogue to the well studied $t$-aspect.

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arXiv2017

On sup-norm bounds part I: ramified Maa{\ss} newforms over number fields

We prove new upper bounds for the sup-norm of Hecke Maa{\ss} newforms on $GL(2)$ over a number field. Our newforms are more general than those considered in a recent paper by Blomer, Harcos, Maga, and Mili\`cevi\`c: we do not require square free level. Furthermore, we allow for non-trivial central character. Over the rationals we cover the best bounds as obtained by Saha.

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arXiv2017

On the size of p-adic Whittaker functions

In this paper we tackle a question raised by N. Templier and A. Saha concerning the size of Whittaker new vectors appearing in infinite dimensional representations of GL(2) over non-archimedean fields. We derive precise bounds for such functions in all possible situations. Our main tool is the p-adic method of stationary phase.

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