Arithmetic properties of coefficients of L-functions of elliptic curves

Abstract

Let ∑ n ⩾ 1ann- s be the L-series of an elliptic curve E defined over the rationals without complex multiplication. In this paper, we present certain similarities between the arithmetic properties of the coefficients {an}n=1∞ and Euler’s totient function φ(n). Furthermore, we prove that both the set of n such that the regular polygon with | an| sides is ruler-and-compass constructible, and the set of n such that n- an+ 1 = φ(n) have asymptotic density zero. Finally, we improve a bound of Luca and Shparlinski on the counting function of elliptic pseudoprimes.