Remarks on the Fourier coefficients of modular forms

الأصل · EN

We consider a variant of a question of N. Koblitz. For an elliptic curve E/ which is not -isogenous to an elliptic curve with torsion, Koblitz has conjectured that there exists infinitely many primes p such that Nₚ(E)=#E(ₚ)=p+1-aₚ(E) is also a prime. We consider a variant of this question. For a newform f, without CM, of weight k≥ 4, on Γ₀(M) with trivial Nebentypus χ₀ and with integer Fourier coefficients, let Nₚ(f)=χ₀(p)pᵏ⁻¹+1-aₚ(f) (here aₚ(f) is the pth-Fourier coefficient of f). We show under GRH and Artin's Holomorphy Conjecture that there are infinitely many p such that Nₚ(f) has at most [5k+1+√(k)] distinct prime factors. We give examples of about hundred forms to which our theorem applies.