Divisors of Fourier coefficients of two newforms

Contact: Pieter Moree (moree@mpim-bonn.mpg.de)

For a non-CM elliptic curve $E/ \mathbb Q $ with conductor $N$ and and for a prime $p\nmid N$, $N_p(E):=p+1-a(p)$ denotes the number of points of the reduction of $E$ modulo $p$. Assume that $E$ is not $\mathbb Q$-isogenous to an elliptic curve with torsion. In 1988, N. Koblitz conjectured that $$\{p\le x: p\nmid N,~ N_p(E) \text{ is prime}\}\sim C_E \frac{x}{(\log x)^2},$$ where $C_E$ is a positive constant depending on $E$. In particular, $N_p(E)$ is prime infinitely often when $p$ runs over the set of primes. This conjecture is still open but there are many results towards this in the literature.

In general, the obvious variant of Koblitz's conjecture is not true for modular forms of higher weights. However, Kirti Joshi in 2012 gave an estimate for the primes $p$ for which $p^{k-1}+1-a_f(p)$ is an almost prime, i.e., has few prime divisors, where $a_f(p)$ is the (integer) $p$th Fourier coefficient of a newform $f$.

In this talk, we will obtain an estimate for the set of primes $p$ such that $a_1(p)-a_2(p)$ is an almost prime, where $a_{1}(n)$ and $a_{2}(n)$ are rational integral Fourier coefficients of two distinct non-CM newforms of weight $k$. More precisely, under GRH, we will give a non-trivial lower bound for the number of primes $p$ such that $$\omega(a_1(p)-a_2(p)) \le [ 7k+{1}/{2}+k^{1/5} ],$$ where $\omega(n)$ denotes the number of distinct prime divisors of an integer $n$. As an application, we obtain a variant of multiplicity one result for newforms via congruences.