Ramanujan-style congruences for prime level

2021/11/30 by Arvind Kumar, Moni Kumari, Kumar, Arvind +5
Mathematics · #11F11 #11F33 #11F80 #11N37 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2111.15287

openalex publication_date 2021/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish Ramanujan-style congruences modulo certain primes ℓ between an Eisenstein series of weight k, prime level p and a cuspidal newform in the ε-eigenspace of the Atkin-Lehner operator inside the space of cusp forms of weight k for Γ₀(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 ℓ. 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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