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arXiv · 2111.15287

Ramanujan-style congruences for prime level

Arvind Kumar·Moni Kumari·Pieter Moree·Sujeet Kumar Singh
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Abstract

We establish Ramanujan-style congruences modulo certain primes $\ell$ between an Eisenstein series of weight $k$, prime level $p$ and a cuspidal newform in the $\varepsilon$-eigenspace of the Atkin-Lehner operator inside the space of cusp forms of weight $k$ for $Γ_0(p)$. Under a mild assumption, this refines a result of Gaba-Popa. We use these congruences and recent work of Ciolan, Languasco and the third author on Euler-Kronecker constants, to quantify the non-divisibility of the Fourier coefficients involved by $\ell.$ The degree of the number field generated by these coefficients we investigate using recent results on prime factors of shifted prime numbers.

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Arvind Kumar, Moni Kumari, Pieter Moree, Sujeet Kumar Singh. 2022-10-14. Ramanujan-style congruences for prime level. https://arxiv.org/abs/2111.15287

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