About

Peter Humphries is an Assistant Professor in the Department of Mathematics at the University of Virginia. His research focuses on Analytic Number Theory, Automorphic Forms, and Representation Theory. He is involved in exploring various aspects of Number Theory and Representation Theory, contributing to the advancement of these fields through his academic work.

Research topics

• Computer Science
• Physics
• Ophthalmology
• Neuroscience
• Biology
• Medicine
• Bioinformatics
• Endocrinology

Selected publications

• The second moment of Rankin–Selberg <i>L</i> -functions in conductor-dropping regimes

  Forum Mathematicum · 2026-04-29

  preprintOpen access1st authorCorresponding

  Abstract We prove an asymptotic formula for the second moment of L -functions associated to the Rankin–Selberg convolution of two holomorphic Hecke cusp forms with equal weight.

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• New Variants of Arithmetic Quantum Ergodicity

  Communications in Mathematical Physics · 2025-02-17 · 1 citations

  articleOpen access1st authorCorresponding

  Abstract We establish two new variants of arithmetic quantum ergodicity. The first is for self-dual $$\textrm{GL}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>GL</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> </mml:math> Hecke–Maaß newforms over $$\mathbb {Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> 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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• Sparse equidistribution of geometric invariants of real quadratic fields

  Journal of the European Mathematical Society · 2025-08-25 · 1 citations

  articleOpen access1st authorCorresponding

  Duke, Imamoḡlu, and Tóth 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 adèlic 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.

  PublisherDOI

• Lp$L^p$‐norm bounds for automorphic forms via spectral reciprocity

  Proceedings of the London Mathematical Society · 2025-06-01 · 1 citations

  articleOpen access1st authorCorresponding

  Abstract Let be a Hecke–Maaß cusp form on the modular surface , namely an ‐normalised non‐constant Laplacian eigenfunction on that is additionally a joint eigenfunction of every Hecke operator. We prove the ‐norm bound , where denotes the Laplacian eigenvalue of , which improves upon Sogge's ‐norm bound for Laplacian eigenfunctions on a compact Riemann surface by more than a six‐fold power‐saving. Interpolating with the sup‐norm bound due to Iwaniec and Sarnak, this yields ‐norm bounds for Hecke–Maaß cusp forms that are power‐saving improvements on Sogge's bounds for all . Our paper marks the first improvement of Sogge's result on the modular surface. Furthermore, these methods yield for compact arithmetic surfaces the best ‐norm bound to date. Via the Watson–Ichino triple product formula, bounds for the ‐norm of are reduced to bounds for certain mixed moments of ‐functions. We bound these using two forms of spectral reciprocity: identities between two different moments of central values of ‐functions. The first is a form of spectral reciprocity, which relates a moment of Rankin–Selberg ‐functions to a moment of Rankin–Selberg ‐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 ‐functions. The second is a form of spectral reciprocity, which is a cuspidal analogue of a formula of Kuznetsov for the fourth moment of central values of Hecke ‐functions.

  PublisherDOI

• New variants of arithmetic quantum ergodicity

  arXiv (Cornell University) · 2024-03-21

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

• Subconvexity implies effective quantum uniqueergodicity for Hecke–Maaß cusp forms on SL2(ℤ)∖SL2(ℝ)

  Essential Number Theory · 2024-09-26 · 2 citations

  articleOpen access

  Subconvexity implies effective quantum unique ergodicity for

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• ARCHIMEDEAN NEWFORM THEORY FOR $\operatorname {\mathrm {GL}}_n$

  Journal of the Institute of Mathematics of Jussieu · 2024-05-17 · 2 citations

  articleOpen access1st authorCorresponding

  Abstract We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman–Wallach representation of $\operatorname {\mathrm {GL}}_n(F)$ , where F is an archimedean local field, 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 $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_n$ and $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_{n - 1}$ 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 $\operatorname {\mathrm {GL}}_n$ over number fields. By-products of the proofs include new proofs of Stade’s formulæ and a new resolution of the test vector problem for archimedean Godement–Jacquet zeta integrals.

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• Zeros of Rankin–Selberg <i>L</i> -functions in families

  Compositio Mathematica · 2024-04-03 · 2 citations

  article1st authorCorresponding

  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 $L(s,\pi \times \widetilde {\pi })$ , in the sense of Bombieri–Vinogradov, for each $\pi \in \mathfrak {F}_n$ .

  PublisherDOI

• Issue Information

  The Medical Journal of Australia · 2024-05-05

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

  arXiv (Cornell University) · 2024-02-21

  preprintOpen access

  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.

  PublisherOA PDFDOI

Frequent coauthors

• Paul F. Kenna

  Royal Victoria Eye and Ear Hospital

  189 shared

• N. J. O’Higgins

  University College Dublin

  156 shared

• Marian M. Humphries

  Trinity College Dublin

  144 shared

• G. Jane Farrar

  Trinity College Dublin

  129 shared

• E. W. M. McDermott

  St. Vincent's University Hospital

  120 shared

• Matthew Campbell

  University of British Columbia

  105 shared

• P. M. Mercer

  96 shared

• M. J. Duffy

  95 shared

Education

• Ph.D., Mathematics

  Princeton University

  2017

• M.Phil., Mathematics

  Australian National University

  2012

• Ph.B. (Hons) (Science), Mathematics

  Australian National University

  2010