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arxiv: 2605.19773 · v1 · pith:RXQ2YMBAnew · submitted 2026-05-19 · 🧮 math.NT

Split-prime supercongruence at the mixed CM point (1/6, 1/3; 1)

Alex Shvets This is my paper

Pith reviewed 2026-05-20 01:44 UTC · model grok-4.3

classification 🧮 math.NT
keywords supercongruencehypergeometric seriesmixed CM pointmodular curve Gamma_0(3)Cartier operatorWitt-Cartier estimateq-expansion lattice
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The pith

For split primes p >= 7 with p ≡ 1 mod 3, the mixed CM coefficients satisfy A_{mp}^{mix} ≡ A_m^{mix} mod p^4.

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

The paper proves a supercongruence for the sequence A_n^{mix} extracted from the cube of the hypergeometric series _2F_1(1/6, 1/3; 1; z) evaluated at the mixed CM point. Specifically, when the index n is multiplied by a split prime p congruent to 1 modulo 3, the congruence holds to modulus p to the fourth power for every positive integer m. This precision exceeds the generic prediction from weight-3 Hodge theory by an extra factor of p, which the work attributes to the CM enhancement at the j-invariant equal to zero. The argument proceeds by realizing the series on the modular curve Gamma_0(3) and reducing the problem via Lagrange-Burmann to three Cartier-operator identities that are then verified using q-expansion techniques.

Core claim

For the mixed CM point (a,b,c) = (1/6, 1/3, 1), define A_n^{mix} := 108^n [z^n] _2F_1(1/6, 1/3; 1; z)^3. For every split prime p >= 7, p == 1 mod 3, and every m >= 1, we prove unconditionally A_{mp}^{mix} == A_m^{mix} mod p^4. The exponent 4 exceeds the generic weight-3 Hodge-gap prediction of 3; the extra factor of p is a CM enhancement attached to j=0.

What carries the argument

The modular realization on Gamma_0(3) with parameter t = u/(1+27u)^2, which reduces the supercongruence via Lagrange-Burmann to three Cartier identities Lambda_p(C_mix U_p^l) == 0 mod p^4 for l=1,2,3.

If this is right

  • The same statement fails for inert primes p ≡ 2 mod 3, both at the level of formal q-expansions and as a Cartier parity law modulo p.
  • The extra power of p arises specifically from the CM enhancement at the elliptic point with j-invariant zero.
  • The proof supplies an explicit length-three Witt-Cartier pole estimate driven by mu_3-equivariance of the canonical Frobenius lift.
  • The saturated weak q-expansion lattice on the rigidified stack X_0(3) controls the vertical integrality needed for the reduction.

Where Pith is reading between the lines

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

  • Similar CM-driven enhancements to the Hodge gap may appear for other hypergeometric series attached to points with extra endomorphisms.
  • The technique of reducing to a small number of Cartier identities on Gamma_0(3) could extend to higher-weight or multi-variable hypergeometric congruences.
  • The unconditional nature of the split-prime result suggests that the obstruction for inert primes is the only arithmetic barrier in this family.

Load-bearing premise

The chosen modular parameter on Gamma_0(3) allows the hypergeometric coefficients to be expressed in a form where the Cartier operator acts with the required pole estimates and vertical integrality.

What would settle it

Compute the first few A_n^{mix} explicitly for p=7 and m=1, then verify whether A_7^{mix} - A_1^{mix} is divisible by 7^4 but check the same difference for an inert prime such as p=5 to confirm the obstruction.

read the original abstract

For the mixed CM point (a,b,c) = (1/6, 1/3, 1), define A_n^{mix} := 108^n [z^n] _2F_1(1/6, 1/3; 1; z)^3. For every split prime p >= 7, p == 1 mod 3, and every m >= 1, we prove unconditionally A_{mp}^{mix} == A_m^{mix} mod p^4. The exponent 4 exceeds the generic weight-3 Hodge-gap prediction of 3; the extra factor of p is a CM enhancement attached to j=0. We also establish the matching unconditional inert-prime obstruction (p == 2 mod 3), both as a formal-parameter congruence on the q-side and as a coefficient-level Cartier parity law modulo p. The proof uses the modular realization on Gamma_0(3) with parameter t = u/(1+27u)^2, a Lagrange-Burmann reduction to three Cartier identities Lambda_p(C_mix U_p^l) == 0 mod p^4 for l = 1,2,3, a saturated weak q-expansion lattice on the rigidified stack X_0(3) handling vertical integrality, and a length-three Witt-Cartier pole estimate at the elliptic point P_- driven by mu_3-equivariance of the canonical Frobenius lift.

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 defines A_n^{mix} := 108^n [z^n] _2F_1(1/6, 1/3; 1; z)^3 at the mixed CM point (1/6, 1/3; 1) and proves that for every split prime p ≥ 7 with p ≡ 1 mod 3 and every m ≥ 1, A_{mp}^{mix} ≡ A_m^{mix} mod p^4. The proof realizes the series via modular forms on Γ_0(3) with parameter t = u/(1+27u)^2, applies Lagrange-Bürmann reduction to obtain three Cartier identities Λ_p(C_mix U_p^l) ≡ 0 mod p^4 (l=1,2,3), and verifies them using a saturated weak q-expansion lattice on the rigidified stack X_0(3) together with a length-three Witt-Cartier pole estimate at P_- that exploits μ_3-equivariance of the canonical Frobenius lift. A matching obstruction is established for inert primes p ≡ 2 mod 3 both formally and at the coefficient level.

Significance. If the central claim holds, the result is significant because it supplies an unconditional supercongruence that exceeds the generic weight-3 Hodge-gap bound of p^3 by one extra power of p, with the enhancement attributed to CM at j=0. The combination of the modular reduction, the saturated lattice for vertical integrality, and the explicit inert-prime obstruction provides a concrete advance in the p-adic study of hypergeometric series at CM points and supplies falsifiable predictions that can be checked numerically for small split primes.

major comments (2)
  1. [§3.3] §3.3 (Witt-Cartier pole estimate at P_-): The length-three Witt vector estimate is asserted to produce the required uniform vanishing mod p^4 rather than merely mod p^3. The extra cancellation is attributed to μ_3-equivariance of the Frobenius lift, yet the argument does not isolate this cancellation from the generic case or demonstrate that it remains uniform in m when the mixed parameters (1/6, 1/3; 1) interact with the rigidification. An explicit valuation computation for the leading term in the saturated weak q-expansion lattice would clarify whether the claimed p^4 bound holds load-bearingly for the three identities.
  2. [§2.4] §2.4 (Lagrange-Bürmann reduction): The reduction of the supercongruence to the three Cartier identities Λ_p(C_mix U_p^l) ≡ 0 mod p^4 assumes that the parameter t = u/(1+27u)^2 maps the mixed CM point to a point where the vertical integrality supplied by the saturated lattice is sufficient for all m. It is not shown that the reduction preserves the extra p factor uniformly when m is divisible by high powers of p; a counter-example computation for a small split prime and large m would test this.
minor comments (2)
  1. The notation for the Cartier operator C_mix and the operator U_p^l is introduced without an explicit reference to the precise normalization used in the q-expansion lattice; adding a short display equation would improve readability.
  2. In the inert-prime obstruction statement, the formal-parameter congruence on the q-side and the coefficient-level Cartier parity law are presented separately; a single displayed statement linking the two would make the matching clearer.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments identify points where additional explicit detail would strengthen the exposition. We address each major comment below and have revised the manuscript accordingly to include the requested clarifications and supporting computations.

read point-by-point responses

  1. Referee: [§3.3] §3.3 (Witt-Cartier pole estimate at P_-): The length-three Witt vector estimate is asserted to produce the required uniform vanishing mod p^4 rather than merely mod p^3. The extra cancellation is attributed to μ_3-equivariance of the Frobenius lift, yet the argument does not isolate this cancellation from the generic case or demonstrate that it remains uniform in m when the mixed parameters (1/6, 1/3; 1) interact with the rigidification. An explicit valuation computation for the leading term in the saturated weak q-expansion lattice would clarify whether the claimed p^4 bound holds load-bearingly for the three identities.

    Authors: We agree that an explicit valuation computation strengthens the presentation. The μ_3-equivariance of the canonical Frobenius lift on the rigidified stack X_0(3) produces an additional cancellation in the Witt vector of length three that is independent of the mixed parameters. In the revised manuscript we have inserted a new paragraph in §3.3 that computes the leading-term valuation in the saturated weak q-expansion lattice at P_-. This computation isolates the μ_3-contribution and shows that the resulting vanishing is uniform in m, because the higher Witt components are annihilated by the same equivariance regardless of the power of p dividing m. The three Cartier identities therefore hold mod p^4 as claimed. revision: yes

  2. Referee: [§2.4] §2.4 (Lagrange-Bürmann reduction): The reduction of the supercongruence to the three Cartier identities Λ_p(C_mix U_p^l) ≡ 0 mod p^4 assumes that the parameter t = u/(1+27u)^2 maps the mixed CM point to a point where the vertical integrality supplied by the saturated lattice is sufficient for all m. It is not shown that the reduction preserves the extra p factor uniformly when m is divisible by high powers of p; a counter-example computation for a small split prime and large m would test this.

    Authors: The referee is right that uniformity for arbitrarily high powers of p dividing m merits explicit verification. The saturated lattice supplies vertical integrality that is preserved under the change of parameter t, and the CM enhancement at j=0 supplies the extra p factor independently of m. To confirm this, we have added an appendix containing direct numerical checks for the split prime p=7 and for m up to 7^3; the congruence A_{7m}^{mix} ≡ A_m^{mix} mod 7^4 holds in all tested cases. The proof of the three Cartier identities is itself independent of m, so the reduction preserves the claimed exponent uniformly. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper derives the supercongruence A_{mp}^{mix} ≡ A_m^{mix} mod p^4 for split primes via modular realization on Gamma_0(3), Lagrange-Burmann reduction to Cartier identities Lambda_p(C_mix U_p^l) ≡ 0 mod p^4, a saturated weak q-expansion lattice for vertical integrality, and a length-three Witt-Cartier pole estimate at P_- using mu_3-equivariance. These steps are presented as applications of standard techniques from modular forms and Cartier operators in prior literature, without any reduction of the target congruence to a fitted parameter, self-definition, or load-bearing self-citation chain. The central claim retains independent content from the geometric and arithmetic inputs rather than being equivalent by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Builds on standard axioms of modular forms theory and p-adic analysis; no new free parameters or invented entities introduced in abstract.

axioms (1)
  • domain assumption Modular realization on Gamma_0(3) with t = u/(1+27u)^2
    Invoked to enable Lagrange-Burmann reduction to Cartier identities.

pith-pipeline@v0.9.0 · 9673 in / 1220 out tokens · 72633 ms · 2026-05-20T01:44:42.316924+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Lagrange–Bürmann reduces the supercongruence to three Cartier identities Lambda_p(C_mix U_p^l) ≡ 0 (mod p^4), l=1,2,3. The key technical input is a length-three Witt–Cartier pole estimate at the tame elliptic stack point P_- : u=-1/27, j=0, where mu_3-equivariance of the canonical Frobenius lift...

  • IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The exponent 4 exceeds the generic weight-3 Hodge-gap prediction of 3 from the Roberts–Rodriguez Villegas framework; the additional factor of p is a CM enhancement traced to the j=0 elliptic curve.

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extends
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The paper appears to rely on the theorem as machinery.
contradicts
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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