Character sums to prime power moduli evaluated at binary quadratic forms
Pith reviewed 2026-05-21 22:37 UTC · model grok-4.3
The pith
Short character sums to prime power moduli at binary quadratic forms admit non-trivial estimates via p-adic methods.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We establish estimates for short character sums to prime power moduli evaluated at binary quadratic forms. This complements estimates established by Heath-Brown for such character sums to squarefree moduli. Our approach uses p-adic analysis. More precisely, we use tools from the p-adic theory of exponential sums, as initiated by Milićević.
What carries the argument
Tools from the p-adic theory of exponential sums initiated by Milićević, applied directly to character sums whose arguments are binary quadratic forms.
If this is right
- The same p-adic method yields bounds that are uniform in the prime-power exponent.
- Applications that previously required the modulus to be squarefree can now include prime-power moduli.
- The estimates remain effective when the quadratic form has small discriminant.
Where Pith is reading between the lines
- The approach may extend to moduli that are products of a fixed number of prime powers.
- Similar p-adic analysis could apply to higher-degree forms or to sums weighted by additive characters.
- The bounds might be combined with existing squarefree estimates to handle arbitrary moduli via the Chinese remainder theorem.
Load-bearing premise
The p-adic exponential-sum machinery developed for other settings transfers without essential change to character sums evaluated at binary quadratic forms when the modulus is a prime power.
What would settle it
An explicit computation of the sum for a small prime power, say modulus 8 or 9, and a fixed binary quadratic form that produces a value larger than the claimed bound.
read the original abstract
We establish estimates for short character sums to prime power moduli evaluated at binary quadratic forms. This complements estimates established by Heath-Brown for such character sums to squarefree moduli. Our approach uses $p$-adic analysis. More precisely, we use tools from the $p$-adic theory of exponential sums, as initiated by Mili\'cevi\'c.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper establishes estimates for short character sums to prime power moduli evaluated at binary quadratic forms, complementing Heath-Brown's results for squarefree moduli. The approach relies on p-adic analysis, specifically tools from the p-adic theory of exponential sums initiated by Milićević.
Significance. If the central estimates hold with the claimed error terms, the work would provide a meaningful extension of character sum bounds to the prime-power setting for quadratic forms, building directly on independent prior results by Heath-Brown and Milićević. This could have applications in analytic number theory involving short sums and quadratic forms.
major comments (2)
- [§2 (Approach and reduction)] The central claim requires an explicit reduction of the multiplicative character sum ∑ χ(Q(x,y)) to a p-adic exponential sum while preserving shortness. The manuscript must detail this step (likely in §2 or §3) to show how the quadratic form's discriminant interacts with the p-adic valuation without introducing factors that invalidate the shortness or the application of Milićević's stationary phase or Weyl differencing techniques.
- [Theorem 1.1] The error terms and range of summation for the short sums are not specified in the abstract; the main theorem (presumably Theorem 1.1 or 1.2) should state the precise dependence on the prime power p^k, the length of the sum, and the discriminant of Q to allow verification of the claimed bounds.
minor comments (2)
- [Introduction] Notation for the binary quadratic form Q and the character χ should be introduced consistently in the introduction before being used in the statements of results.
- [§1] A brief comparison table or paragraph contrasting the new bounds with Heath-Brown's squarefree case would improve readability.
Simulated Author's Rebuttal
We thank the referee for their careful reading of the manuscript and for the constructive major comments. We address each point below and will incorporate revisions to improve clarity and explicitness.
read point-by-point responses
-
Referee: [§2 (Approach and reduction)] The central claim requires an explicit reduction of the multiplicative character sum ∑ χ(Q(x,y)) to a p-adic exponential sum while preserving shortness. The manuscript must detail this step (likely in §2 or §3) to show how the quadratic form's discriminant interacts with the p-adic valuation without introducing factors that invalidate the shortness or the application of Milićević's stationary phase or Weyl differencing techniques.
Authors: We agree that the reduction step merits a more explicit and self-contained treatment. Section 2 of the manuscript already sketches the passage from the character sum ∑ χ(Q(x,y)) to a p-adic exponential sum via the p-adic theory initiated by Milićević, and the interaction between the discriminant of Q and the p-adic valuation is used to control the phase. To address the referee’s concern directly, we will expand this section with a detailed derivation that isolates each step, verifies that the length of the resulting exponential sum remains short relative to p^k, and confirms that no extraneous factors arise that would obstruct the subsequent application of stationary-phase or Weyl-differencing arguments. The revised text will make these verifications explicit. revision: yes
-
Referee: [Theorem 1.1] The error terms and range of summation for the short sums are not specified in the abstract; the main theorem (presumably Theorem 1.1 or 1.2) should state the precise dependence on the prime power p^k, the length of the sum, and the discriminant of Q to allow verification of the claimed bounds.
Authors: We accept that the abstract would benefit from a concise indication of the main ranges and error terms. The precise dependence on p^k, the summation length, and the discriminant of Q is already stated in the body of Theorem 1.1. In the revised version we will augment the abstract with a short sentence summarizing these dependencies and will review the wording of Theorem 1.1 to ensure every parameter appears explicitly. These changes will make the claimed bounds immediately verifiable without altering the mathematical content. revision: yes
Circularity Check
No circularity: derivation applies external p-adic tools to new setting
full rationale
The paper states that it establishes estimates by using tools from the p-adic theory of exponential sums initiated by Milićević, an independent prior result, and complements Heath-Brown's work on squarefree moduli. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided abstract or description. The central approach is presented as a direct application of external machinery to character sums at binary quadratic forms modulo prime powers, without reducing the claimed estimates to the inputs by construction.
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.