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Peter Humphries

Publications and source records attributed to Peter Humphries.

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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arXiv2026

Rankin--Selberg Subconvexity via Spectral Reciprocity

We establish explicit subconvex bounds for central values of Rankin--Selberg $L$-functions $L(1/2,\pi\times\pi')$ associated with pairs of unitary cuspidal automorphic representations of $\mathrm{GL}_2$ over a number field. Building on the spectral reciprocity framework of Michel and Venkatesh, we develop a refined, fully explicit form of spectral reciprocity that allows for precise control of conductors and local test vectors. As a consequence, we obtain an explicit subconvex bound, which, even over $F=\mathbb{Q}$, improves all previously known results. We further apply these bounds to several arithmetic problems. These include effective equidistribution of CM suborbits on quaternionic Shimura varieties, quantitative equidistribution of totally geodesic submanifolds, a uniform quantitative form of dihedral quantum unique ergodicity over number fields, and an application to distinguishing cuspidal automorphic representations.

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arXiv2025

Quantitative Equidistribution on Hyperbolic Surfaces and Arithmetic Applications

The Wasserstein distance quantifies the distance between two probability measures on a metric space. We prove an analogue of the Berry-Esseen inequality for the Wasserstein distance on a finite area hyperbolic surface. This inequality controls the Wasserstein distance via an average of Weyl sums, which are integrals of Maass cusp forms and Eisenstein series with respect to these probability measures. As applications, we prove upper bounds for the Wasserstein distance for some equidistribution problems on the modular surface $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$, namely Duke's theorems on the equidistribution of Heegner points and of closed geodesics and Watson's theorem on the mass equidistribution of Hecke-Maass cusp forms conditionally under the assumption of the generalised Lindelof hypothesis.

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arXiv2024

Strong Hybrid Subconvexity for Twisted Selfdual $\mathrm{GL}_3$ $L$-Functions

We prove strong hybrid subconvex bounds simultaneously in the $q$ and $t$ aspects for $L$-functions of selfdual $\mathrm{GL}_3$ cusp forms twisted by primitive Dirichlet characters. We additionally prove analogous hybrid subconvex bounds for central values of certain $\mathrm{GL}_3 \times \mathrm{GL}_2$ Rankin-Selberg $L$-functions. The subconvex bounds that we obtain are strong in the sense that, modulo current knowledge on estimates for the second moment of $\mathrm{GL}_3$ $L$-functions, they are the natural limit of the first moment method pioneered by Li and by Blomer. The method of proof relies on an explicit $\mathrm{GL}_3 \times \mathrm{GL}_2 \leftrightsquigarrow \mathrm{GL}_4 \times \mathrm{GL}_1$ spectral reciprocity formula, which relates a $\mathrm{GL}_2$ moment of $\mathrm{GL}_3 \times \mathrm{GL}_2$ Rankin-Selberg $L$-functions to a $\mathrm{GL}_1$ moment of $\mathrm{GL}_4 \times \mathrm{GL}_1$ Rankin-Selberg $L$-functions. A key additional input is a Lindel\"of-on-average upper bound for the second moment of Dirichlet $L$-functions restricted to a coset, which is of independent interest.

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arXiv2024

New variants of arithmetic quantum ergodicity

We establish two new variants of arithmetic quantum ergodicity. The first is for self-dual $\mathrm{GL}_2$ Hecke-Maass newforms over $\mathbb{Q}$ as the level and Laplace eigenvalue vary jointly. The second is a nonsplit analogue wherein almost all restrictions of Hilbert (respectively Bianchi) Hecke-Maass cusp forms to the modular surface dissipate as their Laplace eigenvalues grow.

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arXiv2024

Subconvexity Implies Effective Quantum Unique Ergodicity for Hecke-Maa{\ss} Cusp Forms on $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathrm{SL}_2(\mathbb{R})$

It is a folklore result in arithmetic quantum chaos that quantum unique ergodicity on the modular surface with an effective rate of convergence follows from subconvex bounds for certain triple product $L$-functions. The physical space manifestation of this result, namely the equidistribution of mass of Hecke-Maass cusp forms, was proven to follow from subconvexity by Watson, whereas the phase space manifestation of quantum unique ergodicity has only previously appeared in the literature for Eisenstein series via work of Jakobson. We detail the analogous phase space result for Hecke-Maass cusp forms. The proof relies on the Watson-Ichino triple product formula together with a careful analysis of certain archimedean integrals of Whittaker functions.

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arXiv2022

Sparse Equidistribution of Geometric Invariants of Real Quadratic Fields

Duke, Imamo\=glu, and T\'oth have recently constructed a new geometric invariant, a hyperbolic orbifold, associated to each narrow ideal class of a real quadratic field. Furthermore, they have shown that the projection of these hyperbolic orbifolds onto the modular surface $\Gamma \backslash \mathbb{H}$ equidistributes on average over a genus of the narrow class group as the fundamental discriminant $D$ of the real quadratic field tends to infinity. We extend this construction of hyperbolic orbifolds to allow for a level structure, akin to Heegner points and closed geodesics of level $q$. Additionally, we refine this equidistribution result in several directions. First, we investigate sparse equidistribution in the level aspect, where we prove the equidistribution of level $q$ hyperbolic orbifolds when restricted to a translate of $\Gamma \backslash \mathbb{H}$ in $\Gamma_0(q) \backslash \mathbb{H}$, which presents some new interesting features. Second, we explore sparse equidistribution in the subgroup aspect, namely equidistribution on average over small subgroups of the narrow class group. Third, we prove small scale equidistribution and give upper bounds for the discrepancy. Behind these refinements is a new interpretation of the Weyl sums arising in these equidistribution problems in terms of ad\`elic period integrals, which in turn are related to Rankin-Selberg $L$-functions via Waldspurger's formula. The key remaining inputs are hybrid subconvex bounds for these $L$-functions and a certain homological version of the sup-norm problem.

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arXiv2022

$L^p$-Norm Bounds for Automorphic Forms via Spectral Reciprocity

Let $g$ be a Hecke-Maass cusp form on the modular surface ${\rm SL}_2(\mathbb{Z})\backslash\mathbb{H}$, namely an $L^2$-normalised nonconstant Laplacian eigenfunction on ${\rm SL}_2(\mathbb{Z})\backslash\mathbb{H}$ that is additionally a joint eigenfunction of every Hecke operator. We prove the $L^4$-norm bound $\|g\|_4\ll_{\varepsilon}\lambda_g^{3/304+\varepsilon}$, where $\lambda_g$ denotes the Laplacian eigenvalue of $g$, which improves upon Sogge's $L^4$-norm bound $\|g\|_4\ll\lambda_g^{1/16}$ for Laplacian eigenfunctions on a compact Riemann surface by more than a six-fold power-saving. Via interpolation, this yields $L^p$-norm bounds for Hecke-Maass cusp forms that are power-saving improvements on Sogge's bounds for all $p>2$. Our paper marks the first improvement of Sogge's result on the modular surface. Furthermore, these methods yield for compact arithmetic surfaces the best $L^4$-norm bound to date. Via the Watson-Ichino triple product formula, bounds for the $L^4$-norm of $g$ are reduced to bounds for certain mixed moments of $L$-functions. We bound these using two forms of spectral reciprocity. The first is a form of ${\rm GL}_3\times{\rm GL}_2\leftrightsquigarrow{\rm GL}_4\times{\rm GL}_1$ spectral reciprocity, which relates a ${\rm GL}_2$ moment of ${\rm GL}_3\times{\rm GL}_2$ Rankin-Selberg $L$-functions to a ${\rm GL}_1$ moment of ${\rm GL}_4\times{\rm GL}_1$ Rankin-Selberg $L$-functions; this can be seen as a cuspidal analogue of Motohashi's formula relating the fourth moment of the Riemann zeta function to the third moment of central values of Hecke $L$-functions. The second is a form of ${\rm GL}_4\times{\rm GL}_2\leftrightsquigarrow{\rm GL}_4\times{\rm GL}_2$ spectral reciprocity, which is a cuspidal analogue of a formula of Kuznetsov for the fourth moment of central values of Hecke $L$-functions.

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arXiv2022

The Twelfth Moment of Hecke $L$-Functions in the Weight Aspect

We prove an upper bound for the twelfth moment of Hecke $L$-functions associated to holomorphic Hecke cusp forms of weight $k$ in a dyadic interval $T \leq k \leq 2T$ as $T$ tends to infinity. This bound recovers the Weyl-strength subconvex bound $L(1/2,f) \ll_{\varepsilon} k^{1/3 + \varepsilon}$ and shows that for any $\delta > 0$, the sub-Weyl subconvex bound $L(1/2,f) \ll k^{1/3 - \delta}$ holds for all but $O_{\varepsilon}(T^{12\delta + \varepsilon})$ Hecke cusp forms $f$ of weight at most $T$. Our result parallels a related result of Jutila for the twelfth moment of Hecke $L$-functions associated to Hecke-Maass cusp forms. The proof uses in a crucial way a spectral reciprocity formula of Kuznetsov that relates the fourth moment of $L(1/2,f)$ weighted by a test function to a dual fourth moment weighted by a different test function.

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arXiv2021

Test Vectors for Archimedean Period Integrals

We study period integrals involving Whittaker functions associated to generic irreducible Casselman-Wallach representations of $\mathrm{GL}_n(F)$, where $F$ is an archimedean local field. Via the archimedean theory of newforms for $\mathrm{GL}_n$ developed by the first author, we prove that newforms are weak test vectors for several period integrals, including the $\mathrm{GL}_n \times \mathrm{GL}_n$ Rankin-Selberg integral, the Flicker integral, and the Bump-Friedberg integral. By taking special values of these period integrals, we deduce that newforms are weak test vectors for Rankin-Selberg periods, Flicker-Rallis periods, and Friedberg-Jacquet periods. These results parallel analogous results in the nonarchimedean setting proven by the second author, which use the nonarchimedean theory of newforms for $\mathrm{GL}_n$ developed by Jacquet, Piatetski-Shapiro, and Shalika. By combining these archimedean and nonarchimedean results, we prove the existence of weak test vectors for certain global period integrals of automorphic forms.

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arXiv2021

Towards a $\mathrm{GL}_n$ variant of the Hoheisel phenomenon

Let $π$ be a unitary cuspidal automorphic representation of $\mathrm{GL}_n$ over a number field, and let $\tildeπ$ be contragredient to $π$. We prove effective upper and lower bounds of the correct order in the short interval prime number theorem for the Rankin-Selberg $L$-function $L(s,π\times\tildeπ)$, extending the work of Hoheisel and Linnik. Along the way, we prove for the first time that $L(s,π\times\widetildeπ)$ has an unconditional standard zero-free region apart from a possible Landau-Siegel zero.

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arXiv2021

Zeros of Rankin-Selberg $L$-functions in families

Let $\mathfrak{F}_n$ be the set of all cuspidal automorphic representations $\pi$ of $\mathrm{GL}_n$ with unitary central character over a number field $F$. We prove the first unconditional zero density estimate for the set $\mathcal{S}=\{L(s,\pi\times\pi')\colon\pi\in\mathfrak{F}_n\}$ of Rankin-Selberg $L$-functions, where $\pi'\in\mathfrak{F}_{n'}$ is fixed. We use this density estimate to establish (i) a hybrid-aspect subconvexity bound at $s=\frac{1}{2}$ for almost all $L(s,\pi\times\pi')\in \mathcal{S}$, (ii) a strong on-average form of effective multiplicity one for almost all $\pi\in\mathfrak{F}_n$, and (iii) a positive level of distribution for $L(s,\pi\times\tilde{\pi})$, in the sense of Bombieri-Vinogradov, for each $\pi\in\mathfrak{F}_n$.

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arXiv2021

Distributing Points on the Torus via Modular Inverses

We study various statistics regarding the distribution of the points \[\left\{\left(\frac{d}{q},\frac{\overline{d}}{q}\right) \in \mathbb{T}^2 : d \in (\mathbb{Z}/q\mathbb{Z})^{\times}\right\}\] as $q$ tends to infinity. Due to nontrivial bounds for Kloosterman sums, it is known that these points equidistribute on the torus. We prove refinements of this result, including bounds for the discrepancy, small scale equidistribution, bounds for the covering exponent associated to these points, sparse equidistribution, and mixing.

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arXiv2020

The Newform $K$-Type and $p$-adic Spherical Harmonics

Let $K := \mathrm{GL}_n(\mathcal{O})$ denote the maximal compact subgroup of $\mathrm{GL}_n(F)$, where $F$ is a nonarchimedean local field with ring of integers $\mathcal{O}$. We study the decomposition of the space of locally constant functions on the unit sphere in $F^n$ into irreducible $K$-modules; for $F = \mathbb{Q}_p$, these are the $p$-adic analogues of spherical harmonics. As an application, we characterise the newform and conductor exponent of a generic irreducible admissible smooth representation of $\mathrm{GL}_n(F)$ in terms of distinguished $K$-types. Finally, we compare our results to analogous results in the archimedean setting.

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arXiv2020

Archimedean Newform Theory for $\mathrm{GL}_n$

We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman-Wallach representation of $\mathrm{GL}_n$ that quantifies the extent to which this representation may be ramified. We also determine a distinguished vector, the newform, occurring with multiplicity one in this representation, with the complexity of this vector measured in a natural way by the conductor exponent. Finally, we show that the newform is a test vector for $\mathrm{GL}_n \times \mathrm{GL}_n$ and $\mathrm{GL}_n \times \mathrm{GL}_{n - 1}$ Rankin-Selberg integrals when the second representation is unramified. This theory parallels an analogous nonarchimedean theory due to Jacquet, Piatetski-Shapiro, and Shalika; combined, this completes a global theory of newforms for automorphic representations of $\mathrm{GL}_n$ over number fields. By-products of the proofs include new proofs of Stade's formulae and a new resolution of the test vector problem for archimedean Godement-Jacquet zeta integrals.

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arXiv2019

Test Vectors for Nonarchimedean Godement-Jacquet Zeta Integrals

Given an induced representation of Langlands type $(π,V)$ of $\mathrm{GL}_n(F)$ with $F$ nonarchimedean, we show that there exist explicit choices of matrix coefficient $β$ and Schwartz-Bruhat function $Φ$ for which the Godement-Jacquet zeta integral $Z(s,β,Φ)$ attains the $L$-function $L(s,π)$.

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arXiv2019

On the Random Wave Conjecture for Dihedral Maaß Forms

We prove two results on arithmetic quantum chaos for dihedral Maass forms, both of which are manifestations of Berry's random wave conjecture: Planck scale mass equidistribution and an asymptotic formula for the fourth moment. For level $1$ forms, these results were previously known for Eisenstein series and conditionally on the generalised Lindelof hypothesis for Hecke-Maass eigenforms. A key aspect of the proofs is bounds for certain mixed moments of $L$-functions that imply hybrid subconvexity.

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