Paul D. Nelson

activity

20102022

most citedEquidistribution of cusp forms in the level aspect

28 citations · 140 across the 20 of their papers we have counts for

collaborators

23 papers

math.NT2022★ 6 cited

Theta functions, fourth moments of eigenforms, and the sup-norm problem II

Ilya Khayutin, Paul D. Nelson, Raphael S. Steiner

For an $L^{2}$ -normalized holomorphic newform $f$ of weight $k$ on a hyperbolic surface of volume $V$ attached to an Eichler order of squarefree level in an indefinite quaternion alg…

math.NT2022

The subconvexity problem for Rankin-Selberg and triple product L-functions

Yueke Hu, Philippe Michel, Paul Nelson

In this paper we study the subconvexity problem for the Rankin-Selberg L-function and triple product L-function, allowing joint ramifications and conductor dropping range. We first…

math.NT2021★ 1 cited

Bounds for standard $L$ -functions

Paul D. Nelson

Let $π$ be a cuspidal automorphic representation of a general linear group over the rational numbers. We establish a subconvex bound for the standard $L$ -function of $π$ in the $t$ …

math.NT2021★ 12 cited

The Weyl bound for triple product L-functions

Valentin Blomer, Subhajit Jana, Paul D. Nelson

Let $π_{1}, π_{2}, π_{3}$ be three cuspidal automorphic representations for the group $SL (2, Z)$ , where $π_{1}$ and $π_{2}$ are fixed and $π_{3}$ has large conductor. We prove a su…

math.NT2020★ 1 cited

Spectral aspect subconvex bounds for $U_{n + 1} \times U_{n}$

Paul D. Nelson

Let $(π, σ)$ traverse a sequence of pairs of cuspidal automorphic representations of a unitary Gan--Gross--Prasad pair $(U_{n + 1}, U_{n})$ over a number field, with ${\rm U}…

math.NT2020

Quadratic Hecke sums and mass equidistribution

Paul D. Nelson

We consider the analogue of the quantum unique ergodicity conjecture for holomorphic Hecke eigenforms on compact arithmetic hyperbolic surfaces. We show that this conjecture follow…

math.NT2019★ 6 cited

Bounds for twisted symmetric square $L$ -functions via half-integral weight periods

Paul D. Nelson

We establish the first moment bound $φ \sum L (φ \otimes φ \otimes Ψ, \frac{1}{2}) ≪_{ε} p^{5/4 + ε}$ for triple product $L$ -functions, where $Ψ$ is a fixed H…

math.NT2019★ 11 cited

Eisenstein series and the cubic moment for PGL(2)

Paul D. Nelson

Following a strategy suggested by Michel--Venkatesh, we study the cubic moment of automorphic $L$ -functions on $PGL_{2}$ using regularized diagonal periods of products…

math.NT2019★ 4 cited

Analytic newvectors for $GL_{n} (R)$

Subhajit Jana, Paul D. Nelson

We relate the analytic conductor of a generic irreducible representation of $GL_{n} (R)$ to the invariance properties of vectors in that representation. The relation…

math.NT2019

Quantum variance on quaternion algebras, III

Paul D. Nelson

We determine the asymptotic quantum variance of microlocal lifts of Hecke--Maass cusp forms on the arithmetic compact hyperbolic surfaces attached to maximal orders in quaternion a…

math.NT2018★ 24 cited

The orbit method and analysis of automorphic forms

Paul D. Nelson, Akshay Venkatesh

We develop the orbit method in a quantitative form, along the lines of microlocal analysis, and apply it to the analytic theory of automorphic forms. Our main global application is…

math.NT2018

Subconvex bounds on GL(3) via degeneration to frequency zero

Roman Holowinsky, Paul D. Nelson

For a fixed cusp form $π$ on $GL_{3} (Z)$ and a varying Dirichlet character $χ$ of prime conductor $q$ , we prove that the subconvex bound \[ L(π\otimes χ, \tfr…

math.NT2017★ 2 cited

Some analytic aspects of automorphic forms on GL(2) of minimal type

Yueke Hu, Paul D. Nelson, Abhishek Saha

Let $π$ be a cuspidal automorphic representation of $P G L_{2} (A_{Q})$ of arithmetic conductor $C$ and archimedean parameter $T$ , and let $ϕ$ be an $L^{2}$ -normalized aut…

math.NT2017★ 4 cited

Quantum variance on quaternion algebras, II

Paul D. Nelson

A method for determining quantum variance asymptotics on compact quotients attached to non-split quaternion algebras is developed in general and applied to "microlocal lifts" in th…

math.NT2017★ 16 cited

Subconvex equidistribution of cusp forms: reduction to Eisenstein observables

Paul D. Nelson

Let $π$ traverse a sequence of cuspidal automorphic representations of GL(2) with large prime level, unramified central character and bounded infinity type. For G either of the gro…

math.NT2016

Analytic isolation of newforms of given level

Paul D. Nelson

We describe a method for understanding averages over newforms on $Γ_{0} (q)$ in terms of averages over all forms of some level. The method is simplest when $q$ is divisible by the cub…

math.NT2016★ 6 cited

Microlocal lifts and quantum unique ergodicity on $G L (2, Q_{p})$

Paul D. Nelson

We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of $G L (2, Q_{p})$ for automorphic forms belonging to the principal series. We interpr…

math.NT2016★ 3 cited

Quantum variance on quaternion algebras, I

Paul D. Nelson

We determine the quantum variance of a sequence of families of automorphic forms on a compact quotient arising from a non-split quaternion algebra. Our results compare to those obt…

math.NT2012★ 3 cited

Evaluating modular forms on Shimura curves

Paul D. Nelson

Let f be a newform, as specified by its Hecke eigenvalues, on a Shimura curve X. We describe a method for evaluating f. The most interesting case is when X arises as a compact quot…

math.NT2012★ 1 cited

Bounds for Rankin--Selberg integrals and quantum unique ergodicity for powerful levels

Paul D. Nelson, Ameya Pitale, Abhishek Saha

Let f be a classical holomorphic newform of level q and even weight k. We show that the pushforward to the full level modular curve of the mass of f equidistributes as qk -> infini…

math.NT2012

Stable averages of central values of Rankin-Selberg L-functions: some new variants

Paul D. Nelson

As shown by Michel-Ramakrishan (2007) and later generalized by Feigon-Whitehouse (2008), there are "stable" formulas for the average central L-value of the Rankin-Selberg convoluti…

math.NT2010★ 28 cited

Equidistribution of cusp forms in the level aspect

Paul D. Nelson

Let f traverse a sequence of classical holomorphic newforms of fixed weight and increasing squarefree level q tending to infinity. We prove that the pushforward of the mass of f to…

math.NT2010★ 12 cited

Mass equidistribution of Hilbert modular eigenforms

Paul D. Nelson

Let F be a totally real number field, and let f traverse a sequence of non-dihedral holomorphic eigencuspforms on GL(2)/F of weight (k_1,...,k_n), trivial central character and ful…

Theta functions, fourth moments of eigenforms, and the sup-norm problem II

The subconvexity problem for Rankin-Selberg and triple product L-functions

Bounds for standard L-functions

The Weyl bound for triple product L-functions

Spectral aspect subconvex bounds for Un+1​×Un​

Quadratic Hecke sums and mass equidistribution

Bounds for twisted symmetric square L-functions via half-integral weight periods

Eisenstein series and the cubic moment for PGL(2)

Analytic newvectors for GLn​(R)

Quantum variance on quaternion algebras, III

The orbit method and analysis of automorphic forms

Subconvex bounds on GL(3) via degeneration to frequency zero

Some analytic aspects of automorphic forms on GL(2) of minimal type

Quantum variance on quaternion algebras, II

Subconvex equidistribution of cusp forms: reduction to Eisenstein observables

Analytic isolation of newforms of given level

Microlocal lifts and quantum unique ergodicity on GL(2,Qp​)

Quantum variance on quaternion algebras, I

Evaluating modular forms on Shimura curves

Bounds for Rankin--Selberg integrals and quantum unique ergodicity for powerful levels

Stable averages of central values of Rankin-Selberg L-functions: some new variants

Equidistribution of cusp forms in the level aspect

Mass equidistribution of Hilbert modular eigenforms

Bounds for standard $L$ -functions

Spectral aspect subconvex bounds for $U_{n + 1} \times U_{n}$

Bounds for twisted symmetric square $L$ -functions via half-integral weight periods

Analytic newvectors for $GL_{n} (R)$

Microlocal lifts and quantum unique ergodicity on $G L (2, Q_{p})$