Ramanujan-style congruences for prime level

TL;DR

This paper proves new Ramanujan-style congruences between Eisenstein series and cusp forms at prime levels, refining previous results and analyzing the divisibility properties of Fourier coefficients.

Contribution

It establishes novel congruences for prime level modular forms and explores the algebraic and divisibility properties of their Fourier coefficients.

Findings

Refined congruences between Eisenstein series and cusp forms at prime levels.

Quantified non-divisibility of Fourier coefficients by certain primes.

Analyzed the algebraic degree of fields generated by Fourier coefficients.

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 $\Gamma_0(p)$. 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.