Split-prime supercongruence at the mixed CM point (1/6, 1/3; 1)
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911698419449856 |
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| author | Shvets, Alex |
| author_facet | Shvets, Alex |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_19773 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Split-prime supercongruence at the mixed CM point (1/6, 1/3; 1) Shvets, Alex Number Theory 11A07, 11F33, 11G07, 11G15, 11S80, 14G45 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. |
| title | Split-prime supercongruence at the mixed CM point (1/6, 1/3; 1) |
| topic | Number Theory 11A07, 11F33, 11G07, 11G15, 11S80, 14G45 |
| url | https://arxiv.org/abs/2605.19773 |