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Refined spectral reciprocity yields explicit subconvex bounds for Rankin-Selberg L-functions over number fields

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2026-06-27 11:16 UTC pith:CRJDCYBG

load-bearing objection The paper refines the Michel-Venkatesh spectral reciprocity to produce an explicit subconvex saving for Rankin-Selberg L-functions that beats prior explicit bounds even over Q, with several arithmetic applications.

arxiv 2606.11451 v1 pith:CRJDCYBG submitted 2026-06-09 math.NT

Rankin--Selberg Subconvexity via Spectral Reciprocity

Peter Humphries , Liyang Yang This is my paper
classification math.NT
keywords Rankin-Selberg L-functionssubconvexityspectral reciprocityautomorphic representationsGL(2)equidistributionquantum unique ergodicityShimura varieties
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a fully explicit version of spectral reciprocity for pairs of cuspidal automorphic representations of GL(2) over a number field. The refinement gives precise control over conductors and local test vectors, which is used to derive subconvex bounds on the central values of the associated Rankin-Selberg L-functions. These bounds improve all previously known explicit results, including those when the base field is the rationals. The estimates are applied to obtain effective equidistribution results for CM suborbits on quaternionic Shimura varieties, quantitative equidistribution of totally geodesic submanifolds, and a uniform form of dihedral quantum unique ergodicity over number fields.

Core claim

By developing a fully explicit form of spectral reciprocity that allows precise control of conductors and local test vectors, we obtain an explicit subconvex bound for L(1/2, π × π'), which improves all previously known results even over the rationals, and apply these bounds to effective equidistribution of CM suborbits on quaternionic Shimura varieties, quantitative equidistribution of totally geodesic submanifolds, uniform quantitative dihedral quantum unique ergodicity over number fields, and distinguishing cuspidal automorphic representations.

What carries the argument

The refined, fully explicit spectral reciprocity identity for pairs of GL(2) automorphic forms, which supplies precise control over conductors and local test vectors.

Load-bearing premise

The refined spectral reciprocity identity supplies precise control of conductors and local test vectors sufficient to optimize the resulting subconvex exponent.

What would settle it

A numerical check for an explicit pair of Maass forms over Q showing that L(1/2, π × π') exceeds the claimed subconvex bound for sufficiently large conductor, or a direct counterexample to the explicit error terms in the reciprocity identity.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Effective equidistribution of CM suborbits on quaternionic Shimura varieties follows from the bounds.
  • Quantitative equidistribution of totally geodesic submanifolds holds.
  • A uniform quantitative form of dihedral quantum unique ergodicity over number fields is obtained.
  • The bounds allow distinguishing certain cuspidal automorphic representations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit reciprocity approach supplies a template that could be applied in other settings where similar identities can be made fully explicit.
  • The resulting bounds may combine with existing methods to treat additional families of L-functions attached to GL(2) forms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The paper establishes explicit subconvex bounds for central values of Rankin-Selberg L-functions L(1/2, π imes π') for pairs of unitary cuspidal automorphic representations of GL_{2} over a number field. Building on the Michel-Venkatesh spectral reciprocity framework, it develops a refined, fully explicit form of the reciprocity identity that provides precise control of conductors and local test vectors. This yields an explicit subconvex bound improving all prior results even over F=Q. The bounds are applied to 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 distinguishing cuspidal automorphic representations.

Significance. If the explicit subconvex bounds hold with the claimed improvements, the work would advance subconvexity results for Rankin-Selberg L-functions by supplying stronger, explicit exponents and constants usable in effective arithmetic applications. The refined explicit reciprocity identity and its applications to equidistribution and QUE problems constitute the main strengths; the explicitness of the bounds is a clear asset for downstream arithmetic consequences.

minor comments (3)
  1. [Abstract] Abstract: the claimed improvement over prior results is stated without naming the precise subconvex exponent or the previous best exponent; adding the explicit numerical comparison would strengthen the claim.
  2. The applications section should include a brief table or explicit statement of the numerical saving obtained in each arithmetic consequence to make the impact of the main bound transparent.
  3. Notation for the refined spectral reciprocity identity should be introduced with a displayed equation early in the text so that subsequent conductor estimates can be cross-referenced directly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work on explicit subconvexity for Rankin-Selberg L-functions and the associated arithmetic applications. The recommendation for minor revision is noted with appreciation. No specific major comments appear in the report, so we have no points requiring detailed rebuttal at this stage.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation builds on the Michel-Venkatesh spectral reciprocity framework by introducing a refined, fully explicit version that supplies improved conductor and local test vector control. This refinement is presented as original work enabling new explicit subconvex bounds, rather than reducing by construction to prior fitted parameters or self-citations. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citation chains appear in the provided abstract and description. The central claim remains an explicit improvement over prior results and is stated as falsifiable via constants, consistent with standard analytic number theory practice.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new entities; ledger left empty pending full text.

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read the original abstract

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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