Remarks on the Fourier coefficients of modular forms

TL;DR

This paper investigates the distribution of prime factors in a specific sequence derived from modular forms, proving under GRH and Artin's conjecture that infinitely many such numbers have a bounded number of prime factors.

Contribution

It extends Koblitz's conjecture to modular forms of weight ≥ 4, showing infinitely many primes with controlled prime factorization under certain hypotheses.

Findings

Infinitely many primes p with N_p(f) having bounded prime factors.

Application to about a hundred specific modular forms.

Conditional proof assuming GRH and Artin's Holomorphy Conjecture.

Abstract

We consider a variant of a question of N. Koblitz. For an elliptic curve $E/\Q$ which is not $\Q$-isogenous to an elliptic curve with torsion, Koblitz has conjectured that there exists infinitely many primes $p$ such that $N_p(E)=#E(\F_p)=p+1-a_p(E)$ is also a prime. We consider a variant of this question. For a newform $f$, without CM, of weight $k\geq 4$, on $\Gamma_0(M)$ with trivial Nebentypus $\chi_0$ and with integer Fourier coefficients, let $N_p(f)=\chi_0(p)p^{k-1}+1-a_p(f)$ (here $a_p(f)$ is the $p^{th}$-Fourier coefficient of $f$). We show under GRH and Artin's Holomorphy Conjecture that there are infinitely many $p$ such that $N_p(f)$ has at most $[5k+1+\sqrt{\log(k)}]$ distinct prime factors. We give examples of about hundred forms to which our theorem applies.