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arxiv: 1912.02909 · v2 · pith:P4AULWRKnew · submitted 2019-12-05 · 🧮 math.NT

Newton Polygons of Hecke Operators

Liubomir Chiriac , Andrei Jorza This is my paper

Pith reviewed 2026-05-24 15:06 UTC · model grok-4.3

classification 🧮 math.NT
keywords Newton polygonHecke operatorBuzzard-Calegari conjecturep-adic valuationexponential sumcusp formtrace
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The pith

If the Buzzard-Calegari polynomial has a vertex at n≤15, its Newton polygon agrees with that of T2 up to n for all large weights.

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

This paper verifies a truncated version of the Buzzard-Calegari conjecture by showing that the Newton polygon of the Hecke operator T2 on cusp forms matches the Newton polygon of the Buzzard-Calegari polynomial whenever the latter has a vertex at or before n=15. The authors first derive a formula for the p-adic valuations of exponential sums, then implement it to obtain the 2-adic valuations of the traces of T2. These computations confirm the polygon agreement for every weight that is sufficiently large. The result supplies concrete computational evidence for how the slopes of T2 behave in high weights.

Core claim

Whenever the Newton polygon of the Buzzard-Calegari polynomial has a vertex at some n≤15, this polygon coincides exactly with the Newton polygon of the Hecke operator T2 up to that n, and the agreement holds for all weights large enough. The verification rests on explicit computation of the 2-adic valuations of the relevant traces, obtained via a new formula for p-adic valuations of exponential sums.

What carries the argument

Formula for p-adic valuations of exponential sums, used to compute 2-adic valuations of traces of Hecke operators on spaces of cusp forms.

If this is right

  • The Newton polygon of T2 is completely determined by the Buzzard-Calegari polynomial up to any vertex at or before 15, for every large weight.
  • The truncated form of the Buzzard-Calegari conjecture holds for all weights beyond a fixed bound.
  • The same computational pipeline determines the initial segments of the Newton polygon of T2 for arbitrarily large weights.

Where Pith is reading between the lines

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

  • If the observed agreement continues for vertices beyond 15, the full Buzzard-Calegari conjecture would hold for all weights.
  • The valuation formula could be applied to Hecke operators at primes other than 2 to test analogous statements.
  • Extending the range of verified n would require only more computing power, not new theoretical input.

Load-bearing premise

The formula for p-adic valuations of exponential sums is correct and its implementation accurately computes the 2-adic valuations of the traces of the Hecke operators for all sufficiently large weights.

What would settle it

Finding one sufficiently large weight at which the Newton polygons of T2 and the Buzzard-Calegari polynomial differ at some vertex n≤15 would disprove the verified statement.

read the original abstract

In this computational paper we verify a truncated version of the Buzzard-Calegari conjecture on the Newton polygon of the Hecke operator $T_2$ for all large enough weights. We first develop a formula for computing $p$-adic valuations of exponential sums, which we then implement to compute $2$-adic valuations of traces of Hecke operators acting on spaces of cusp forms. Finally, we verify that if Newton polygon of the Buzzard-Calegari polynomial has a vertex at $n\leq 15$, then it agrees with the Newton polygon of $T_2$ up to $n$.

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

2 major / 2 minor

Summary. The paper claims to verify a truncated version of the Buzzard-Calegari conjecture on the Newton polygon of the Hecke operator T_2 for all large enough weights. It develops a formula for p-adic valuations of exponential sums, implements the formula to compute 2-adic valuations of traces of Hecke operators on spaces of cusp forms, and checks that if the Newton polygon of the Buzzard-Calegari polynomial has a vertex at n≤15 then the two polygons agree up to that n.

Significance. If the result holds, the work supplies concrete computational evidence for the conjecture in the range n≤15 together with an explicit formula for p-adic valuations of exponential sums that can be reused for further calculations. The paper performs a direct, parameter-free computational check whose correctness rests entirely on the derivation and implementation of the valuation formula.

major comments (2)
  1. [Abstract and final verification paragraph] The central verification is conditional on the Buzzard-Calegari polynomial possessing a vertex at some n≤15; the manuscript does not report an independent computation or proof that such a vertex exists, so the practical content of the agreement statement remains untested.
  2. [Section developing the valuation formula] The formula for p-adic valuations of exponential sums is presented as the key technical tool, yet the manuscript provides no explicit statement of the range of weights for which the implementation is guaranteed to be accurate, leaving the scope of the reported 2-adic trace valuations unclear.
minor comments (2)
  1. [Abstract] The abstract states the verification is performed 'for all large enough weights' while the concrete check is limited to n≤15; a clarifying sentence on the precise truncation would improve readability.
  2. [Implementation paragraph] No reference is given to the source code or data files used for the exponential-sum computations; adding a pointer to a public repository would strengthen reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract and final verification paragraph] The central verification is conditional on the Buzzard-Calegari polynomial possessing a vertex at some n≤15; the manuscript does not report an independent computation or proof that such a vertex exists, so the practical content of the agreement statement remains untested.

    Authors: The verification we perform is explicitly conditional, as stated in the abstract: we show that if the Buzzard-Calegari polynomial has a vertex at n≤15, then the Newton polygons agree up to that point. This is the truncated version of the conjecture that we are able to verify with our methods. We did not independently compute the Newton polygon of the Buzzard-Calegari polynomial, as that would require a separate implementation not related to our exponential sum formula. The practical content is the verification of the agreement under the stated hypothesis, which provides evidence for the conjecture in the cases where such vertices occur. To make this clearer, we will revise the abstract and the final paragraph to emphasize the conditional nature and note that the existence of the vertex is not addressed in this work. revision: yes

  2. Referee: [Section developing the valuation formula] The formula for p-adic valuations of exponential sums is presented as the key technical tool, yet the manuscript provides no explicit statement of the range of weights for which the implementation is guaranteed to be accurate, leaving the scope of the reported 2-adic trace valuations unclear.

    Authors: We agree that specifying the range of weights is important for clarity. In the revised version of the manuscript, we will include an explicit statement in the section developing the valuation formula, detailing the range of weights for which the implementation of the formula guarantees accurate computation of the 2-adic valuations of the traces. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivation consists of developing an explicit formula for p-adic valuations of exponential sums, implementing that formula to compute 2-adic valuations of Hecke traces for large weights, and then performing a direct numerical comparison of Newton polygons under a conditional hypothesis on the Buzzard-Calegari polynomial. None of these steps reduces by construction to a fitted parameter, a self-referential definition, or a load-bearing self-citation; the central claim is a conditional computational verification whose support lies outside the target result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable from the given text.

pith-pipeline@v0.9.0 · 5617 in / 1108 out tokens · 25960 ms · 2026-05-24T15:06:32.119943+00:00 · methodology

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Reference graph

Works this paper leans on

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