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The hyperbolic circle problem over Heegner points

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Pith's one-line read For Heegner points of different discriminants, the hyperbolic circle problem's error term is improved for the first time since 1977.

desk verdict First unconditional improvement of Selberg's 40-year-old bound for Heegner points of different discriminants; the proof is detailed, honest, and the apparent congruence issue evaporates on inspection. read the letter →

arxiv 2506.13883 v1 pith:RDQ6OPEX submitted 2025-06-16 math.NT

Dimitrios Chatzakos , Giacomo Cherubini , Stephen Lester , Morten S. Risager This is my paper · ORCID
classification math.NT MSC 11P2111N4511F6711E4511M32
keywords hyperboliccircleproblemHeegnerpointsmodulargroupRankin-SelbergL-functionsfractionalmomentstoric-periodformulabinaryquadraticformsSelberg'sbound
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a new error-term estimate for the counting function that records how many images of one Heegner point under the modular group fall within a given hyperbolic distance of another Heegner point, where the two points have different negative squarefree discriminants. It proves that this count equals the expected main term $(2\pi/\operatorname{vol}(\Gamma\backslash\mathbb{H}))X$ with error $O(X^{2/3}/(\log X)^{1/6})$, a logarithmic improvement over the 1977 bound $O(X^{2/3})$ that had never been beaten for any cofinite group or pair of points. The route goes through a new fractional moment estimate for central values of Rankin--Selberg $L$-functions attached to Hecke--Maass forms and $\theta$ series of imaginary quadratic fields, using a toric-period formula to convert products of eigenfunction values into products of $L$-values. If correct, this is the first unconditional evidence in a concrete arithmetic setting that the conjectured error $O(X^{1/2+\varepsilon})$ is approachable.

What carries the argument

The engine is the combination of a toric-period formula with fractional moments. The toric-period formula expresses $L(1/2,\phi_j\times f_\xi)/L(1,\operatorname{sym}^2\phi_j)$ as a constant multiple of $|\sum_{a\in\mathrm{Cl}_K}\xi(a)\phi_j(z_a)|^2$, so pointwise values of Hecke--Maass forms at Heegner points are controlled by central Rankin--Selberg $L$-values twisted by class group characters. The paper then estimates a fractional moment of the product of two such $L$-values using mollifiers $M_r(j,\xi,\xi')$ built from short Euler-product factors and the elementary inequality $2\sqrt{LL'}\le LM(M')^{-1}+L'M'M^{-1}$. Applying an iterated mollifier decomposition to split the spectral window into sets where the mollifiers expand into short Dirichlet polynomials, together with a new twisted first-moment asymptotic proved through the Kuznetsov formula, yields Theorem 1.2; the cancellation entering through Selberg-type orthogonality estimates fixes the exponent $\eta=1/4$ that becomes the $1/6$ in the error term via smoothing with a parameter $\delta=X^{-1/3}(\log X)^{-2\eta/3}$.

What would settle it

For a non-genus class group character $\xi$ of $\mathbb{Q}(\sqrt d)$ and any $\xi'$ of $\mathbb{Q}(\sqrt{d'})$, numerically compute $S(x)=\sum_{p\le x}\lambda_\xi(p)\lambda_{\xi'}(p)/p$ for $x$ up to, say, $10^9$; if $|S(x)|$ is not eventually bounded (for instance, if it grows like a positive power of $\log x$), Lemma 4.5(1) fails and the fractional-moment saving (1.5) is false, so the error term in Theorem 1.1 would not follow from this argument. Conversely, one could test Theorem 1.2 directly by evaluating the $L$-value sum for $T$ around $10^4$ and checking whether its ratio to $T^2/(\log T)^{1/4}$ stays bounded.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is Theorem 1.1: for $\Gamma=\mathrm{SL}_2(\mathbb{Z})$ and two Heegner points $z_d,z_{d'}$ with squarefree discriminants $d\neq d'$, $d,d'<0$, the hyperbolic lattice-point count has main term $(2\pi/\operatorname{vol}(\Gamma\backslash\mathbb{H}))X$ and error $O(X^{2/3}/(\log X)^{1/6})$. The logarithmic saving is produced by Theorem 1.2, a fractional moment estimate: over a spectral window $T<t_j\le 2T$, the sum of $\sqrt{L(1/2,\phi_j\times f_\xi)L(1/2,\phi_j\times f_{\xi'})}/L(1,\operatorname{sym}^2\phi_j)$ is $O(T^2/(\log T)^{1/4})$ for all class group characters $\xi,\xi'$ of the two fields. Because a toric-period formula isolates $|\phi_j(z_d)|$ up to constants as such an $L$-value ratio, and because summing these ratios over characters dominates $|\phi_j(z_d)|$, the theorem yields the spectral input $\sum_{T<t_j\le 2T}|\phi_j(z_d)\phi_j(z_{d'})|\ll T^2/(\log T)^{1/4}$; a spectral pre-trace formula then converts this into the improved error term. The paper also derives matching statements for counting binary quadratic forms and $\mathrm{SL}_2(\mathbb{Z})$-equivalence classes of pairs of forms, and an averaged second-moment estimate $O(X(\log X)^{3/4})$ for the error term. The proof is written under the standing assumption $d\equiv 1\pmod 4$, a restriction not stated in Theorem 1.1.

Load-bearing premise

The proof's logarithmic saving rests on unproved-in-this-paper cancellation in the prime sums $\sum_{p\le x}\lambda_\xi(p)\lambda_{\xi'}(p)/p$: these must be $O(1)$ when at least one character is not a genus character, and exactly $B\log\log x+O(1)$ in the genus case; the theorem also assumes $d\equiv 1\pmod 4$ throughout, a restriction not stated in Theorem 1.1.

Editorial extensions

If this is right

  • For any Heegner pair $z_d,z_{d'}$ of distinct squarefree discriminants, the counting function's error term is $O(X^{2/3}/(\log X)^{1/6})$, beating the universal 1977 bound by a logarithmic factor.
  • The spectral sum $\sum_{T<t_j\le 2T}|\phi_j(z_d)\phi_j(z_{d'})|$ is $O(T^2/(\log T)^{1/4})$, stronger than the bound suggested by the random-wave model when the two points are treated as independent.
  • The quadratic-form counting function $n_d(x)$ and the summed class-number function $\sum_{0<-\Delta\le x}h(d,d',\Delta)$ both inherit error terms $O(x^{2/3}/(\log x)^{1/6})$.
  • Averaged over $X\le x\le 2X$, the variance of the error term is $O(X(\log X)^{3/4})$, improving previously known logarithmic powers for Heegner points.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same machinery should in principle give logarithmic improvements for pairs of Heegner points with one point fixed and the other varying over shrinking balls, provided the fractional moment estimate can be localized spectrally.
  • Because the fractional-moment bound has the exponent $1/4$ predicted by Keating--Snaith heuristics, the logarithmic saving in Theorem 1.1 is likely the optimal output of this method; further progress toward $X^{1/2+\varepsilon}$ would need a genuinely new idea.
  • The restriction to $d\equiv 1\pmod 4$ looks removable by the same proof after adjusting the theta-series setup for even discriminants; if so, Theorem 1.1's statement as written is conservative.
  • The logarithmic saving may extend to the variance of the error term when Heegner points are averaged over discriminants, since the spectral bound is sufficiently uniform in $d,d'$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper proves that for the full modular group and for Heegner points z_d, z_d' attached to distinct negative squarefree fundamental discriminants d, d', the hyperbolic circle-counting function satisfies N(X;z_d,z_d') = (2π/vol(Γ\H)) X + O(X^{2/3}/(log X)^{1/6}). This is the first unconditional improvement over Selberg's classical O(X^{2/3}) bound in this setting. The strategy is to combine Waldspurger's formula, which expresses |φ_j(z_d)|^2 through central values of Rankin-Selberg L-functions L(1/2, φ_j × f_ξ), with a new twisted first-moment estimate proved by Kuznetsov's trace formula (Theorem 3.1) and a fractional-moment estimate for products of such central values (Theorem 1.2). A pre-trace formula argument (Proposition 5.1) converts the resulting spectral bound into the counting-function error term. The paper also derives applications to counting binary quadratic forms and a second-moment bound for the error term.

Significance. If the proof is correct, this is a genuine breakthrough: it removes the forty-year-old barrier at O(X^{2/3}) for the hyperbolic circle problem in a natural arithmetic family, and it introduces a fractional-moment technique for Rankin-Selberg L-functions attached to class-group theta series that is likely to have further applications. The proof is unusually detailed: the Kuznetsov transform estimates, the mollifier iteration, and the spectral-to-counting conversion are all written out, and the external inputs are explicitly identified. The main potential risk identified by the stress test is the reliance of Lemma 4.5 on Selberg orthogonality results of Liu-Ye and Avdispahić-Smajlović; on reading the paper, I do not find a defect here, since the needed sums are standard Rankin-Selberg consequences and the citations are to the appropriate general theorems. I also see no circularity: the target error bound is never assumed, and no parameters are fitted to the desired output.

minor comments (4)
  1. [Theorem 1.1 and Section 2.1] Theorem 1.1 states that d,d' are 'different squarefree discriminants' but does not state the standing assumption d≡1 mod 4 introduced at the start of Section 2.1. For a negative squarefree fundamental discriminant this congruence is automatic, but the theorem should either say 'fundamental discriminants' or explicitly include d,d'≡1 mod 4 so that its hypotheses visibly match those of Theorem 1.2 and Proposition 5.1.
  2. [Section 4, Lemma 4.5] Lemma 4.5 is the load-bearing input that sets the exponent θ≥1/4 in (4.40), and its proof is only a citation to [39,40,41,1]. Because the theta series f_ξ of Section 2.3 are weight-one forms with nebentypus, may be non-self-contragredient, and include genus characters for which the form is Eisenstein rather than cuspidal, the manuscript should state precisely which theorem in the cited papers applies to each of the four assertions, or give a short Rankin-Selberg verification. I believe the standard results do cover the situation, but the current presentation leaves the verification to the reader at the single most delicate point of the argument.
  3. [Equations (4.9) and (4.41)] The text defines C1>100 and also says it is 'sufficiently large', but the inequality in (4.41) requires C1 to exceed a much larger absolute constant, depending on the numerical constants in (4.34). Please make the quantifier uniform by stating once that C1 is chosen larger than all absolute constants appearing in the subsequent estimates, so that (4.41) is visibly valid.
  4. [Section 3.2, first line] There is a duplicated word in 'To prove Theorem 3.1 we we will use the same-sign Kuznetsov trace formula'; this should be corrected to 'we will use'.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity found: the logarithmic saving is derived from independently cited Selberg orthogonality estimates, Waldspurger's formula, and first-moment asymptotics; self-citations are not load-bearing.

full rationale

The paper's claimed derivation chain is not circular. Theorem 1.1 is deduced from Proposition 5.1, whose hypothesis (a bound on the spectral sum over Heegner points) is verified through (2.18) and Theorem 1.2. Theorem 1.2 is proved from the twisted first-moment asymptotics of Theorem 3.1, which is established independently via the Kuznetsov trace formula, an approximate functional equation, and bounds on the Kuznetsov transform; no parameter is fitted to the target error term. The one genuinely imported analytic input is Lemma 4.5, which uses Selberg orthogonality estimates from [39,40,41,1]; these are external published theorems, and their conclusions (O(1) or B log log x + O(1) for the relevant sums) are exactly what sets the exponent theta >= 1/4. The apparent d = 1 mod 4 restriction in Theorem 1.1 is automatic for negative squarefree discriminants under the definition in Section 2.1 and is therefore not an overclaim. The self-citations [13,14,52] appear as background and as a technical estimate in the pre-trace formula step; they do not assume the target theorem, and the cited estimates are independently published. Thus no equation of the paper reduces to its own output by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters and no new entities. The central claim rests on external theorems: Waldspurger's formula, Selberg orthogonality estimates for class group characters, spectral decomposition via Kuznetsov and pre-trace formulas, and Nordentoft's Eisenstein bound. These are background results, not assumptions that hide the target conclusion.

assumptions (5)
  • standard math Waldspurger-Zhang formula, equation (2.17): L(1/2, phi_j x f_xi)/L(1, sym^2 phi_j) = c_d^{-1} |sum_{a in Cl_K} xi(a) phi_j(z_a)|^2.
    Imported theorem; it is the bridge that replaces Heegner point values by central L-values and supplies non-negativity in (2.18).
  • standard math Selberg orthogonality estimates for class group characters: sum_{p <= x} lambda_xi(p) lambda_xi'(p)/p is O(1) in the non-genus case and B_{xi,xi'} log log x + O(1) in the genus case (Lemma 4.5).
    Invoked in Lemma 4.5 and around (4.15), citing [39,40,41,1]. These cancellations determine the exponent theta in Theorem 1.2 and hence the 1/6 saving in Theorem 1.1.
  • standard math Non-negativity of the central values L(1/2, phi_j x f_xi) >= 0, from (2.17).
    Used in Section 4 to drop absolute values, apply inequality (1.7), and justify the mollifier arguments.
  • standard math Kuznetsov trace formula, approximate functional equation, and Weil bound for Kloosterman sums yield the twisted first moment Theorem 3.1 for the test function h(t) in (3.1).
    The main analytic engine; the proof in Section 3 depends on standard spectral theory, Stirling bounds, and the Weil bound.
  • standard math Sup-norm bound for the Eisenstein series E(z, 1/2 + it) << y^{1/2}(|t| + 1)^{1/3 + epsilon}, cited to Nordentoft [47].
    Controls the continuous spectrum contribution in Proposition 5.1 and Corollary 1.5.

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Cite this review

Pith. "Pith review of The hyperbolic circle problem over Heegner points." pith.science (2026). https://pith.science/paper/RDQ6OPEX

@misc{pith2026250613883,
  author       = {Pith},
  title        = {Pith review of: The hyperbolic circle problem over Heegner points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RDQ6OPEX}},
  note         = {Machine review of arXiv:2506.13883}
}
read the original abstract

For the full modular group, we obtain a logarithmic improvement on Selberg's long-standing bound for the error term of the counting function in the hyperbolic circle problem over Heegner points of different discriminants. The main ingredients in our method are Waldspurger's formula, twisted first moments of certain Rankin-Selberg convolutions, and a new fractional moment estimate.

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Spectral fourth moments of Hecke--Maa{\ss} cusp forms

    math.NT 2026-07 conditional novelty 7.0 of 10

    For Heegner points, the spectral fourth moment of Hecke–Maass cusp forms in a dyadic interval is O(T^{2+ε}), matching the Lindelöf-on-average prediction.

  2. The hyperbolic lattice counting problem in large dimensions

    math.NT 2025-06 conditional novelty 6.0 of 10

    In hyperbolic space H^n (n≥3) with a cocompact lattice, the averaged error in lattice counting diverges, and under two conjectures the local average over the quotient is O(X^{n-2+ε}).

  3. Quadratic forms of modular forms

    math.NT 2025-07 conditional novelty 5.0 of 10

    Under GRH and analytic continuation hypotheses, the paper completes a decorrelation conjecture for products of Hecke eigenforms and proves new ℓ^p-norm bounds for quadratic forms in the Hecke basis.

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