On the parity of coefficients of eta powers
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916496785014784 |
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| author | Charlton, Steven Mauth, Lukas Medvedovsky, Anna |
| author_facet | Charlton, Steven Mauth, Lukas Medvedovsky, Anna |
| contents | We consider a special subsequence of the Fourier coefficients of powers of the Dedekind $η$-function, analogous to the sequence $δ_\ell := 24^{-1} \pmod{\ell}$ on which exceptional congruences of the partition function are supported. Therefrom we define a notion of density $D(r)$ for a normalized eta-power $η^r$ measuring the proportion of primes $\ell$ for which the order at infinity of $U_\ell (η^r)$ modulo 2 is maximal. We relate $D(r)$ to a notion of density measuring nonzero prime Fourier coefficients introduced by Bellaïche, and use this to completely classify the vanishing of and establish upper bounds for $D(r)$. Furthermore, for several infinite families of $η$ powers corresponding to dihedral/CM mod-2 modular forms in the sense of Nicholas-Serre and Bellaïche, we explicitly compute the densities $D$. We rely on Galois-theoretic techniques developed by Bellaïche in level 1 and extend these to level 9. En passant we take the opportunity to communicate proofs of two of Bellaïche's unpublished results on densities of mod-$2$ modular forms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_17638 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the parity of coefficients of eta powers Charlton, Steven Mauth, Lukas Medvedovsky, Anna Number Theory Primary: 11F33, Secondary: 11F03, 11F20, 11F80, 11P83, 11R45 We consider a special subsequence of the Fourier coefficients of powers of the Dedekind $η$-function, analogous to the sequence $δ_\ell := 24^{-1} \pmod{\ell}$ on which exceptional congruences of the partition function are supported. Therefrom we define a notion of density $D(r)$ for a normalized eta-power $η^r$ measuring the proportion of primes $\ell$ for which the order at infinity of $U_\ell (η^r)$ modulo 2 is maximal. We relate $D(r)$ to a notion of density measuring nonzero prime Fourier coefficients introduced by Bellaïche, and use this to completely classify the vanishing of and establish upper bounds for $D(r)$. Furthermore, for several infinite families of $η$ powers corresponding to dihedral/CM mod-2 modular forms in the sense of Nicholas-Serre and Bellaïche, we explicitly compute the densities $D$. We rely on Galois-theoretic techniques developed by Bellaïche in level 1 and extend these to level 9. En passant we take the opportunity to communicate proofs of two of Bellaïche's unpublished results on densities of mod-$2$ modular forms. |
| title | On the parity of coefficients of eta powers |
| topic | Number Theory Primary: 11F33, Secondary: 11F03, 11F20, 11F80, 11P83, 11R45 |
| url | https://arxiv.org/abs/2411.17638 |