arXiv:2608.14495v1

Shimura curves of discriminant 14 and 15 and associated Heun Functions

Harshavardhan Reddy, Devendra Tiwari

math.NT11F0333E3011J9111G18

Paper metadata

Original submission: August 14, 2026
Claimed MSC: 11F03, 33E30, 11J91, 11G18
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗

Abstract

The Shimura curve of discriminant $D$ for $D=14, 15$ is uniformized by a subgroup of an arithmetic quadrilateral Fuchsian group $(2, 2, 2, q)$ , where $q=4, 6$ . We relate the generator of the ring of quaternionic modular forms on this Shimura curve to explicit Heun functions for the quadrilateral group. We also discuss how the Picard-Fuchs equation of the associated family of abelian surfaces has solutions that are modular forms on $X^{D}(1) / W_{D}$ , where $W_D$ is the full group of Atkin-Lehner involutions. This leads us to completely describe the rational exceptional sets of the associated Heun functions, and the algebraic values attained by the Heun function on these points, for example ${\rm He}\left( 81, \frac{1}{2}; \frac{1}{3}, \frac{1}{6}, \frac{1}{2}, \frac{1}{2};-\frac{729}{112}\right)= \left( \frac{2^2 \cdot 3^3 \cdot 5^3}{7^5} \right)^{\frac{1}{6}}$ .

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsContains unsupported statements

The modular-form presentations, Heun equations after correcting two uniquely determined symbols, and rational exceptional-set classifications are supported. The three asserted exact nontrivial Heun values are not formally verified because the paper does not certify the required root choices and omits the computations needed for two of them.

Theorems 3.1–3.2 and Proposition 3.5Correct

The modular-form generators and Heun realizations are supported

Pages 8–11 · Theorems 3.1–3.2, Lemmas 3.3–3.4, and Proposition 3.5 · arXiv:2608.14495v1

The valence formula gives the displayed dimensions, and the prescribed zeros of $D\tau$ produce generators of weights $4,8,14$ for discriminant $14$ and weights $4,8,10,12$ for discriminant $15$ . Their listed relations have the required weights and Hilbert-series counts. Logarithmic differentiation of the weight-four generator gives the first-derivative coefficient of the appropriate Heun equation, while the local normalization at $z_8$ and $z_{12}$ yields the identities in Proposition 3.5. The two literal differential-equation symbol defects are reported separately as typos and have unique corrections.

Theorems 4.1–4.2Correct

The rational exceptional sets agree with the CM tables

Pages 12–13 · Theorems 4.1–4.2 · arXiv:2608.14495v1

The modular-form expression reduces algebraicity to equality of the fundamental CM field with that of the chosen base point. For discriminant $14$ , Elkies's Table 5 contains $\tau=75/16$ with CM discriminant $-72$ , whose fundamental field discriminant is $-8$ , together with the base value $\tau=0$ ; these give $\{0,75/16\}$ . For discriminant $15$ , Table 6 gives the four projective rational values $0,\infty,243,-729/112$ attached to the fundamental field of discriminant $-3$ . This verifies the two classifications independently of the contradictory paragraph following Theorem 4.1.

Elkies, Shimura curve computations, Tables 5–6 ↗

Theorem 5.1 · three nontrivial valuesNot able to verify

The exact Heun values are not established by a verifiable calculation

Pages 13–14 and 17 · both occurrences of Theorem 5.1 · arXiv:2608.14495v1

The paper asserts $H_{15}(243)=\frac{\sqrt3}{2}+\frac{i}{4},\qquad F_{14}(75/16)=\frac{\sqrt7}{3},$ and $H_{15}(-729/112)=\left(\frac{2^2 3^3 5^3}{7^5}\right)^{1/6}.$ For the first value, the displayed algebra determines only a fourth power and the branch is chosen by an unspecified numerical calculation. For the second, the proof says only that the calculation is the same and displays a covering map, without carrying out the derivative and automorphy-factor calculation. For the third, no Yang-polynomial roots, limiting values, or error bounds are supplied, and the final branch is again selected by an unspecified numerical approximation. These omissions are concrete nontrivial obligations; no fully verified independent exact derivation was obtained, and no contradiction to the displayed values was found.

02Proofs5 reported findingsContains incorrect or incomplete proofs

The proof following Theorem 4.1 contradicts both the theorem and its cited table. The exact-value arguments do not rigorously select algebraic branches, and two of the three calculations are only asserted rather than displayed. The differential-equation notation also contains two harmless, uniquely repairable typos.

Proof of Theorem 4.1Incorrect as written

The proof omits $75/16$ and concludes the wrong exceptional set

Page 12 · paragraph after Theorem 4.1 · arXiv:2608.14495v1

The theorem states that the rational exceptional set is $\{0,75/16\}$ , but the proof says that Table 5 has only the point $\tau=0$ and concludes $\{0\}$ . It also identifies $\tau=0$ with $z_4$ and $\mathbb Q(\sqrt{-7})$ , although the table on page 4 identifies $\tau(z_8)=0$ and the relevant base field is $\mathbb Q(\sqrt{-2})$ . Elkies's Table 5 explicitly lists $75/16$ with discriminant $-72$ , hence the same fundamental field discriminant $-8$ . Repair classification: Verified repair. Replace the contradictory paragraph by the two-row CM-field comparison for $0$ and $75/16$ and exclude the remaining rational CM rows because their fundamental fields differ.

Elkies, Shimura curve computations, Table 5 ↗

Theorem 5.1(i)Incomplete as written

The fourth-root branch at $243$ is selected without a rigorous bound

Pages 13–14 · final paragraph of the proof of Theorem 5.1(i) · arXiv:2608.14495v1

The preceding calculation determines $H_{15}(243)^4$ , leaving four possible fourth roots. The proof then says that a numerical calculation of the first few digits rules out three roots, but gives neither those digits nor a certified approximation and error bound tied to the analytically continued normalized Heun branch. Branch selection is necessary for the exact displayed value. Repair classification: Plausible repair only. A certified continuation with an explicit error smaller than half the separation of the four candidates, or an exact argument controlling the branch along a specified path, would complete this step.

Theorem 5.1(ii) · discriminant $14$Incomplete as written

The claimed analogous computation at $75/16$ is not carried out

Page 14 · part (ii) of the proof of Theorem 5.1 · arXiv:2608.14495v1

After displaying the level- $3$ covering and involution, the paper says only that the proof is the same as part (i). It does not identify the required preimages, compute the two derivative limits and automorphy factor, derive a power of $F_{14}(75/16)$ , or select its branch. These are the substantive ingredients that produced the first value and do not follow merely by substituting the new covering formula. Repair classification: No repair supplied; the corresponding exact calculation and branch determination are required.

Section 5.4Incomplete as written

The Yang-polynomial evaluation at $-729/112$ is not reproducible

Page 17 · proof of the second Theorem 5.1 · arXiv:2608.14495v1

The factorization uses $w_5=5-e+f$ , but $f$ is not defined in the manuscript's presentation of the quaternion algebra. More importantly, the proof supplies none of the roots obtained from $\Psi_2$ and $\Psi_7$ , none of the limiting calculations for the factors described as $0$ and $\infty$ , and no numerical value or certified error bound for selecting the final algebraic branch. Thus the displayed product cannot be checked from the paper. Repair classification: No repair supplied; the quaternion element, exact root data, limiting ratios, and certified branch selection must be given.

Equations (4) and Lemma 3.4Typo

Two Heun-equation symbols are copied from the wrong context

Pages 6 and 11 · Equation (4) and Lemma 3.4 · arXiv:2608.14495v1

Equation (4) ends with the dependent variable $f$ , while the same equation denotes it by $w$ in the two preceding terms; replace $f$ by $w$ . Lemma 3.4 prints the discriminant- $14$ numerator $(1/8)(3/8)\tau-q$ inside the discriminant- $15$ equation. Section 2.4 and Proposition 3.5 uniquely determine the replacement $(1/3)(1/6)\tau-1/2,$ together with $a=81$ . Both corrections are mechanical notation repairs and do not alter the argument or overall statement status.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.14495v1
Authors listed: Harshavardhan Reddy, Devendra Tiwari
Audit date: August 18, 2026

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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. 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P(delta)

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Shimura curves of discriminant 14 and 15 and associated Heun Functions

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The modular-form generators and Heun realizations are supported

The rational exceptional sets agree with the CM tables

The exact Heun values are not established by a verifiable calculation

The proof omits 75/1675/1675/16 and concludes the wrong exceptional set

The fourth-root branch at 243243243 is selected without a rigorous bound

The claimed analogous computation at 75/1675/1675/16 is not carried out

The Yang-polynomial evaluation at −729/112-729/112−729/112 is not reproducible

Two Heun-equation symbols are copied from the wrong context

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Authenticate with ORCID, then require an exact arXiv match

The proof omits $75/16$ and concludes the wrong exceptional set

The fourth-root branch at $243$ is selected without a rigorous bound

The claimed analogous computation at $75/16$ is not carried out

The Yang-polynomial evaluation at $-729/112$ is not reproducible