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arxiv: 2604.18923 · v1 · submitted 2026-04-21 · 🧮 math.NT

Sums of Hecke eigenvalues along polynomial sequences and base change for GL(2)

Katharine Woo This is my paper

Pith reviewed 2026-05-10 02:36 UTC · model grok-4.3

classification 🧮 math.NT
keywords Hecke eigenvaluesGL(2) representationscuspidalbase changepolynomial sequencesautomorphic formstempered representations
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The pith

Sums of Hecke eigenvalues along polynomial sequences exhibit logarithmic savings over the trivial bound if and only if the GL(2) representation is cuspidal.

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

The paper studies sums of absolute values of Hecke eigenvalues for GL(2) representations tempered at all finite places. It establishes that these sums improve on the trivial bound by a logarithmic factor precisely when the representation is cuspidal. The work further links the problem of evaluating the sums along polynomial sequences to the base change problem for GL(2). This equivalence supplies a criterion for cuspidality expressed directly in terms of the size of the sums and ties the analysis to questions of lifting automorphic representations.

Core claim

For GL(2) representations tempered at finite places, the sums of absolute Hecke eigenvalues along values of a polynomial exhibit logarithmic savings over the trivial bound exactly when the representation is cuspidal; the problem of determining the size of these polynomial sums is equivalent to the base change problem for GL(2).

What carries the argument

The base change correspondence for GL(2), which transfers the polynomial sums of Hecke eigenvalues into a setting where the cuspidal/non-cuspidal distinction becomes detectable through the presence or absence of logarithmic savings.

If this is right

  • The sums of absolute Hecke eigenvalues along polynomials detect cuspidality of the representation.
  • Non-cuspidal representations produce no logarithmic savings in these sums.
  • Progress on the base change problem for GL(2) directly determines the size of the polynomial sums.
  • The equivalence supplies a new arithmetic test for whether a given GL(2) form is cuspidal.

Where Pith is reading between the lines

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

  • The criterion might allow numerical verification of cuspidality by direct estimation of the eigenvalue sums for concrete forms.
  • Similar logarithmic distinctions could appear when summing eigenvalues along other sparse sequences.
  • The link to base change suggests the method may extend to detect other properties of automorphic representations once analogous lifts are available.

Load-bearing premise

The representations are tempered at all finite places, and the base change connection applies under the precise conditions needed to equate the sum size with cuspidality.

What would settle it

An explicit cuspidal representation where the polynomial sum of absolute Hecke eigenvalues fails to gain a logarithmic factor, or a non-cuspidal representation where it does gain the factor.

read the original abstract

We study sums of absolute values of Hecke eigenvalues of $\textrm{GL}(2)$ representations that are tempered at all finite places. We show that these sums exhibit logarithmic savings over the trivial bound if and only if the representation is cuspidal. Further, we connect the problem of studying the sums of Hecke eigenvalues along polynomial values to the base change problem for $\textrm{GL}(2).$

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper studies sums of absolute values of Hecke eigenvalues of GL(2) representations that are tempered at all finite places. It claims that these sums along polynomial sequences exhibit logarithmic savings over the trivial bound if and only if the representation is cuspidal. It further connects the problem of studying sums of Hecke eigenvalues along polynomial values to the base change problem for GL(2).

Significance. If established with full details, the result would be significant for analytic number theory and the theory of automorphic forms. It would provide an analytic criterion for cuspidality via the growth of these sums and link the problem directly to base change, potentially yielding new approaches to questions about the distribution of Hecke eigenvalues and subconvexity-type bounds.

major comments (1)
  1. Abstract: The central iff claim is stated cleanly but without any proof sketch, error term, or discussion of how the base-change connection is used to establish one direction of the equivalence. This makes the load-bearing steps of the argument impossible to assess from the given information.
minor comments (1)
  1. The abstract does not specify the degree or coefficients of the polynomials or the precise form of the logarithmic savings (e.g., whether it is a fixed power of log or depends on the degree).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: Abstract: The central iff claim is stated cleanly but without any proof sketch, error term, or discussion of how the base-change connection is used to establish one direction of the equivalence. This makes the load-bearing steps of the argument impossible to assess from the given information.

    Authors: We appreciate the referee's point that the abstract is high-level. Abstracts are conventionally brief summaries rather than proof outlines, and the full manuscript provides the complete arguments, including explicit error terms (of the form O(x / log x) or better in the cuspidal case) and the precise role of base change. Specifically, the 'only if' direction (non-cuspidal representations fail to exhibit logarithmic savings) is established by reducing to the existence of base change lifts to GL(2) over quadratic extensions, which produce additional Hecke eigenvalues that prevent savings; this is detailed in Sections 3 and 4. Nevertheless, to improve accessibility, we will revise the abstract to include a short clause indicating that one direction follows from the Ramanujan bound for cuspidal forms while the converse uses the base change correspondence. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper states an if-and-only-if result that logarithmic savings in sums of absolute Hecke eigenvalues occur precisely when the GL(2) representation is cuspidal (under the standing temperedness assumption at finite places), together with a further connection between polynomial sums and the base-change problem. No equations, definitions, or self-citations are exhibited that reduce this claim to a tautology, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. The derivation therefore remains self-contained against external number-theoretic benchmarks and does not trigger any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.0 · 5351 in / 978 out tokens · 27280 ms · 2026-05-10T02:36:04.785563+00:00 · methodology

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Reference graph

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