Order drop, Hecke descent, and a mod $p^4$ supercongruence for symmetric-cube hypergeometric coefficients
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| Format: | Preprint |
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2026
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| _version_ | 1866911651495673856 |
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| author | Shvets, Alex |
| author_facet | Shvets, Alex |
| contents | We prove that the symmetric-cube coefficients $A_n=(-27)^n[z^n]\,_2F_1(1/3,1/3;1;z)^3$ satisfy the supercongruence $A(mp)\equiv A(m) \bmod p^4$ for every prime $p\geq 5$ and every $m\geq 1$. The proof rests on three ingredients: (i) the modular identification $F(t(τ))=η(τ)^9/η(3τ)^3$ with $t(τ)=η(3τ)^{12}/η(τ)^{12}$, whose logarithmic derivative is the weight-5 Eisenstein series $C(q)=3E_5(χ_0,χ_3)$ on $Γ_0(3)$; (ii) exact congruences $c_{mp^r}\equiv c_{mp^{r-1}} \bmod p^{4r}$ for the coefficients of $C$, combined with a Lagrange-Burmann extraction; and (iii) a Hecke descent on weakly holomorphic forms, where the defect is expanded in the two-dimensional space of weight-5 forms on $Γ_0(3)$ with character $χ_3$, spanned by $C$ and $tC$, via a cusp-adapted basis, with the second cusp handled by the Fricke involution $W_3$. As an independent result, we show that the Mao-Tian cubic recurrence drops from order 3 to order 2 at the specialization $(1/3,1/3,1)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_06238 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Order drop, Hecke descent, and a mod $p^4$ supercongruence for symmetric-cube hypergeometric coefficients Shvets, Alex Number Theory 11A07, 11F11, 33C20 We prove that the symmetric-cube coefficients $A_n=(-27)^n[z^n]\,_2F_1(1/3,1/3;1;z)^3$ satisfy the supercongruence $A(mp)\equiv A(m) \bmod p^4$ for every prime $p\geq 5$ and every $m\geq 1$. The proof rests on three ingredients: (i) the modular identification $F(t(τ))=η(τ)^9/η(3τ)^3$ with $t(τ)=η(3τ)^{12}/η(τ)^{12}$, whose logarithmic derivative is the weight-5 Eisenstein series $C(q)=3E_5(χ_0,χ_3)$ on $Γ_0(3)$; (ii) exact congruences $c_{mp^r}\equiv c_{mp^{r-1}} \bmod p^{4r}$ for the coefficients of $C$, combined with a Lagrange-Burmann extraction; and (iii) a Hecke descent on weakly holomorphic forms, where the defect is expanded in the two-dimensional space of weight-5 forms on $Γ_0(3)$ with character $χ_3$, spanned by $C$ and $tC$, via a cusp-adapted basis, with the second cusp handled by the Fricke involution $W_3$. As an independent result, we show that the Mao-Tian cubic recurrence drops from order 3 to order 2 at the specialization $(1/3,1/3,1)$. |
| title | Order drop, Hecke descent, and a mod $p^4$ supercongruence for symmetric-cube hypergeometric coefficients |
| topic | Number Theory 11A07, 11F11, 33C20 |
| url | https://arxiv.org/abs/2604.06238 |