arXiv subjects
Publications and source records attributed to 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 $t$-aspect. More generally, we address the spectral aspect in the case of uniform parameter growth.
To assess the ability of current AI systems to correctly answer research-level mathematics questions, we share a set of ten math questions which have arisen naturally in the research process of the authors. The questions had not been shared publicly until now; the answers are known to the authors of the questions but will remain encrypted for a short time.
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 algebras. Our result extends those of Luo--Sarnak--Zhao concerning the non-compact modular surface. The results of this article's prequel (which involved the theta correspondence, Rallis inner product formula and equidistribution of translates of elementary theta functions) reduce the present task to some local problems over the reals involving the construction and analysis of microlocal lifts via integral operators on the group. We address these here using an analytic incarnation of the method of coadjoint orbits.
We give a soft proof of a uniform upper bound for the local factors in the triple product formula, sufficient for deducing effective and general forms of quantum unique ergodicity (QUE) from subconvexity.
We establish an explicit global spectral decomposition of shifted convolution sums and the second moment of automorphic $L$-functions for Maass forms with explicit integral transforms as well as explicit inversion formulae over every local field.
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.
For $L$-functions attached to automorphic representations of unitary groups $U_{n+1}\times U_n$, we establish a subconvex bound valid in certain horizontal aspects, where the set of ramified places is allowed to vary.
Let $(π,σ)$ traverse a sequence of pairs of cuspidal automorphic representations of a unitary Gan--Gross--Prasad pair $({\rm U}_{n+1},{\rm U}_n)$ over a number field, with ${\rm U}_n$ anisotropic. We assume that at some distinguished archimedean place, the pair stays away from the conductor dropping locus, while at every other place, the pair has bounded ramification and satisfies certain local conditions (in particular, temperedness). We prove that the subconvex bound \[ L(π\times σ,1/2) \ll C(π\times σ)^{1/4 - δ} \] holds for any fixed \[ δ< \frac{1}{8 n^5 + 28 n^4 + 42 n^3 + 36 n^2 + 14 n}. \] Among other ingredients, the proof employs a refinement of the microlocal calculus for Lie group representations developed with A. Venkatesh and an observation of S. Marshall concerning the geometric side of the relative trace formula.
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 algebra over $\mathbb{Q}$, we prove the sup-norm estimate \[ \| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_ε (k V)^{\frac{1}{4}+ε} \] with absolute implied constant. For a cuspidal Maaß newform $φ$ of eigenvalue $λ$ on such a surface, we prove that \[ \|φ\|_{\infty} \ll_{λ,ε} V^{\frac{1}{4}+ε}. \] We establish analogous estimates in the setting of definite quaternion algebras.
Let $π_1, π_2, π_3$ be three cuspidal automorphic representations for the group ${\rm SL}(2, \Bbb{Z})$, where $π_1$ and $π_2$ are fixed and $π_3$ has large conductor. We prove a subconvex bound for $L(1/2, π_1 \otimes π_2 \otimes π_3)$ of Weyl-type quality. Allowing $π_3$ to be an Eisenstein series we also obtain a Weyl-type subconvex bound for $L(1/2 + it, π_1 \otimes π_2)$.
We consider the analogue of the quantum unique ergodicity conjecture for holomorphic Hecke eigenforms on compact arithmetic hyperbolic surfaces. We show that this conjecture follows from nontrivial bounds for Hecke eigenvalues summed over quadratic progressions. Our reduction provides an analogue for the compact case of a criterion established by Luo--Sarnak for the case of the non-compact modular surface. The novelty is that known proofs of such criteria have depended crucially upon Fourier expansions, which are not available in the compact case. Unconditionally, we establish a twisted variant of the Holowinsky--Soundararajan theorem involving restrictions of normalized Hilbert modular forms arising via base change.
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 an asymptotic formula for averages of Gan--Gross--Prasad periods in arbitrary rank. The automorphic form on the larger group is held fixed, while that on the smaller group varies over a family of size roughly the fourth root of the conductors of the corresponding $L$-functions. Ratner's results on measure classification provide an important input to the proof. Our local results include asymptotic expansions for certain special functions arising from representations of higher rank Lie groups, such as the relative characters defined by matrix coefficient integrals as in the Ichino--Ikeda conjecture.
Let $θ$ be an elementary theta function, such as the classical Jacobi theta function. We establish a spectral decomposition and surprisingly strong asymptotic formulas for $\langle |θ|^2, φ\rangle$ as $φ$ traverses a sequence of Hecke-translates of a nice enough fixed function. The subtlety is that typically $|θ|^2 \notin L^2$. Applications to the subconvexity, quantum variance and $4$-norm problems are indicated.
We establish the first moment bound $$ \sum_φ L(φ\otimes φ\otimes Ψ, \tfrac{1}{2}) \ll_{\varepsilon} p^{5/4+\varepsilon} $$ for triple product $L$-functions, where $Ψ$ is a fixed Hecke-Maass form on $\operatorname{SL}_2(\mathbb{Z})$ and $φ$ runs over the Hecke-Maass newforms on $Γ_0(p)$ of bounded eigenvalue. The proof is via the theta correspondence and analysis of periods of half-integral weight modular forms. This estimate is not expected to be optimal, but the exponent $5/4$ is the strongest obtained to date for a moment problem of this shape. We show that the expected upper bound follows if one assumes the Ramanujan conjecture in both the integral and half-integral weight cases. Under the triple product formula, our result may be understood as a strong level aspect form of quantum ergodicity: for a large prime $p$, all but very few Hecke-Maass newforms on $Γ_0(p) \backslash \mathbb{H}$ of bounded eigenvalue have very uniformly distributed mass after pushforward to $\operatorname{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$. Our main result turns out to be closely related to estimates such as $$ \sum_{|n| < p} L(Ψ\otimes χ_{n p},\tfrac{1}{2}) \ll p, $$ where the sum is over $n$ for which $n p$ is a fundamental discriminant and $χ_{n p}$ denotes the corresponding quadratic character. Such estimates improve upon bounds of Duke-Iwaniec.
We relate the analytic conductor of a generic irreducible representation of $\mathrm{GL}_n(\mathbb{R})$ to the invariance properties of vectors in that representation. The relationship is an analytic archimedean analogue of some aspects of the classical non-archimedean newvector theory of Casselman and Jacquet--Piatetski-Shapiro--Shalika. We illustrate how this relationship may be applied in trace formulas to majorize sums over automorphic forms on $\mathrm{PGL}_n(\mathbb{Z}) \backslash \mathrm{PGL}_n(\mathbb{R})$ ordered by analytic conductor.
In this paper we give quantitative local test vectors for Waldspurger's period integral (i.e., a toric period on $\text{GL}_2$) in new cases with joint ramifications. The construction involves minimal vectors, rather than newforms and their variants. This paper gives a uniform treatment for the matrix algebra and division algebra cases under mild assumptions, and establishes an explicit relation between the size of the local integral and the finite conductor $C(π\timesπ_{χ^{-1}})$. As an application, we combine the test vector results with the relative trace formula, and prove a hybrid type subconvexity bound which can be as strong as the Weyl bound in proper range.
Following a strategy suggested by Michel--Venkatesh, we study the cubic moment of automorphic $L$-functions on $\operatorname{PGL}_2$ using regularized diagonal periods of products of Eisenstein series. Our main innovation is to produce vectors whose integral transforms achieve arbitrarily weighted moments. Applications include general Motohashi-type identities and Weyl-type subconvex bounds for some families of $L$-functions, extending some results of Conrey--Iwaniec and Petrow--Young to the number field setting. We deduce improved estimates for representation numbers of ternary quadratic forms over number fields and for the prime geodesic theorem on arithmetic hyperbolic $3$-folds.
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 groups GL(1) or PGL(2), let H(G) denote the assertion that subconvexity holds for G-twists of the adjoint $L$-function of $π$, with polynomial dependence upon the conductor of the twist. We show that H(GL(1)) implies H(PGL(2)). In geometric terms, H(PGL(2)) corresponds roughly to an instance of arithmetic quantum unique ergodicity with a power savings in the error term, H(GL(1)) to the special case in which the relevant sequence of measures is tested against an Eisenstein series.