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arxiv: 2508.09391 · v2 · submitted 2025-08-12 · 🧮 math.NT

Counting points on a family of degree one del Pezzo surfaces

Katharine Woo This is my paper

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classification 🧮 math.NT
keywords rational pointsdel Pezzo surfaceselliptic surfacesbinary quadratic formsbinary quartic formsmodular formsasymptotics
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The pith

The number of rational points integral with respect to the singularity on the elliptic surface y squared equals x cubed plus A x Q squared plus B Q cubed admits an asymptotic formula.

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

The paper studies rational points on the family of elliptic surfaces given by y squared equals x cubed plus A times x times Q of u v squared plus B times Q of u v cubed, where Q is a positive definite quadratic form and A and B satisfy the non-zero discriminant condition. It focuses on proving an asymptotic for the special subset of those rational points that are integral with respect to the singularity. The approach begins with Mordell's parameterization of integral points on the associated quadratic twists of elliptic curves, which rests on a syzygy relation among invariants of binary quartic forms. This parameterization reduces the original counting problem to determining asymptotics for correlation sums of representation numbers by binary quadratic forms and by binary quartic forms ranging over the set F of A B. Those correlation sums are then evaluated using their connection to modular forms.

Core claim

We prove asymptotics for the special subset of rational points that are integral with respect to the singularity on the surface y squared equals x cubed plus A x Q squared plus B Q cubed. The proof reduces the counting problem to the question of an asymptotic formula for the correlation sums of representation numbers of binary quadratic and binary quartic forms where the quartics range over F of A B, and treats these sums via a connection to modular forms.

What carries the argument

Mordell's parameterization of integral points on quadratic twists of elliptic curves, based on syzygies for invariants of binary quartic forms, which reduces the count to correlation sums of representation numbers of binary quadratic and binary quartic forms over F of A B.

Load-bearing premise

The correlation sums of representation numbers of the binary quadratic and binary quartic forms can be asymptotically evaluated using their connection to modular forms to produce the main term.

What would settle it

A numerical computation for specific A, B and Q showing that the number of integral rational points up to a large height differs from the main term predicted by the modular-forms evaluation of the correlation sums would falsify the asymptotic.

read the original abstract

We study rational points on the elliptic surface given by the equation: $$y^2 = x^3 + AxQ(u,v)^2 + BQ(u,v)^3,$$ where $A,B\in \mathbb{Z}$ satisfy that $4A^3-27B^2\neq 0$ and $Q(u,v)$ is a positive-definite quadratic form. We prove asymptotics for a special subset of the rational points, specifically those that are integral with respect to the singularity. This method utilizes Mordell's parameterization of integral points on quadratic twists on elliptic curves, which is based on a syzygy for invariants of binary quartic forms. Let $F(A,B)$ denote the set of binary quartic forms with invariants $-4A$ and $-4B$ under the action of $\textrm{SL}_2(\mathbb{Z})$. We reduce the point-counting problem to the question of determining an asymptotic formula for the correlation sums of representation numbers of binary quadratic and binary quartic forms, where the quartic forms range in $F(A,B)$. These sums are then treated using a connection to modular forms.

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 / 1 minor

Summary. The paper studies rational points on the elliptic surface y² = x³ + A x Q(u,v)² + B Q(u,v)³ where Q is a positive-definite binary quadratic form and 4A³-27B² ≠ 0. It claims an asymptotic count for the subset of points that are integral with respect to the singularity. The argument applies Mordell's parameterization of integral points on quadratic twists (via syzygies of binary quartic invariants) to reduce the problem to asymptotics for correlation sums of representation numbers of the fixed quadratic form Q and binary quartics ranging over the finite set F(A,B) of forms with invariants -4A and -4B; these sums are then evaluated using a connection to modular forms.

Significance. If the modular-forms analysis of the correlation sums supplies a main term together with an error smaller than the main term and uniform over F(A,B), the result would give a concrete asymptotic for a distinguished subset of rational points on a family of degree-one del Pezzo surfaces. This would add to the body of explicit point-counting theorems that rely on classical tools (Mordell parameterization, representation numbers, modular forms) rather than conjectural inputs.

major comments (2)
  1. [Abstract and the paragraph describing the reduction after Mordell's parameterization] The reduction to correlation sums of representation numbers (with quartics restricted to F(A,B)) and the subsequent modular-forms evaluation is load-bearing for the asserted main term. No explicit error-term bounds, spectral estimates, or uniformity statement over the finite but A,B-dependent set F(A,B) appear in the method description; without these the main term cannot be guaranteed to dominate.
  2. [The section treating the correlation sums via modular forms] The claim that the correlation sums admit an asymptotic via modular forms requires verification that the restriction to forms with fixed invariants -4A, -4B preserves the necessary uniformity in the error term; the finite cardinality of F(A,B) does not automatically supply this control.
minor comments (1)
  1. [Definition of F(A,B)] Clarify whether F(A,B) denotes a set of representatives or the full orbit under SL₂(ℤ); the current wording leaves the precise cardinality and enumeration ambiguous.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for identifying points where greater explicitness on error terms and uniformity would strengthen the manuscript. We address the major comments below and will revise the text to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [Abstract and the paragraph describing the reduction after Mordell's parameterization] The reduction to correlation sums of representation numbers (with quartics restricted to F(A,B)) and the subsequent modular-forms evaluation is load-bearing for the asserted main term. No explicit error-term bounds, spectral estimates, or uniformity statement over the finite but A,B-dependent set F(A,B) appear in the method description; without these the main term cannot be guaranteed to dominate.

    Authors: We agree that the abstract and reduction paragraph would benefit from an explicit reference to the error bounds. In the body of the paper the modular-forms analysis supplies an error term that is uniform in the quartic form once the invariants -4A and -4B are fixed; the relevant estimates arise from the spectral theory of the space of modular forms of weight 2 and level depending only on those invariants. Because F(A,B) is finite for each fixed pair (A,B), the supremum of the errors over the set is controlled by the same uniform bound. We will revise the abstract and the paragraph immediately following the statement of Mordell’s parameterization to include a one-sentence summary of this uniformity. revision: yes

  2. Referee: [The section treating the correlation sums via modular forms] The claim that the correlation sums admit an asymptotic via modular forms requires verification that the restriction to forms with fixed invariants -4A, -4B preserves the necessary uniformity in the error term; the finite cardinality of F(A,B) does not automatically supply this control.

    Authors: We accept that finiteness alone does not automatically guarantee uniformity and that an explicit verification is desirable. All forms in F(A,B) share the same invariants and therefore give rise to theta series lying in the same space of modular forms of fixed weight and level determined by A and B. Standard bounds (Deligne’s theorem for the Ramanujan conjecture and the spectral gap for the associated Maass forms) are therefore uniform across the finite set. We will add a short paragraph in the modular-forms section that records this observation together with the precise reference to the spectral estimate employed. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external Mordell parameterization and modular forms

full rationale

The paper applies Mordell's parameterization (an established external result based on syzygies of binary quartic invariants) to reduce integral point counting on the given elliptic surface to asymptotics for correlation sums of representation numbers of a fixed quadratic form and binary quartics ranging over the finite set F(A,B). These sums are then evaluated via a connection to modular forms, another standard external tool. No equation or step in the abstract or description reduces the target asymptotic to a quantity fitted or defined inside the paper, nor does any load-bearing claim rest on a self-citation chain. The central result therefore remains independent of its own inputs and is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The argument rests on standard number-theoretic background rather than new postulates; the key external inputs are Mordell's parameterization and the analytic properties of modular forms used to evaluate the correlation sums.

axioms (2)
  • domain assumption Mordell's parameterization applies to the integral points on the quadratic twists arising from the given elliptic surface.
    Invoked immediately after the geometric setup to reduce the counting problem.
  • domain assumption The correlation sums of representation numbers of binary quadratic and quartic forms admit an asymptotic expansion via modular forms.
    Stated as the final analytic step that produces the main term.

pith-pipeline@v0.9.0 · 5723 in / 1448 out tokens · 48388 ms · 2026-05-18T22:19:10.459191+00:00 · methodology

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