Explicit formulae and the Lang-Trotter conjecture
V. Kumar Murty · Rocky Mountain Journal of Mathematics · 1985
Introduction Let E be an elliptic curve defined over Q, of conductor N.For a prime p X N, the reduction E p of E (mod p) is an elliptic curve defined over the field F^, of/? elements.Denote by N p the number of points of Ep which are rational over F^, and write a p = p 4-1 -N p .Then it is known that \a p \ oo, 7t E (x) ~ (l/2)7r(x), where %{x) denotes the number of primes p oo, Serre [6] has shown that, for any e > 0, 7C E (x) < £ x/(logx) 5/4 " £ and on the assumption of the Riemann Hypothesis for all Artin L-functions, 7t E (x) < * 3/4 -For each p, write a p = 2p 1/2 cos d p with d p e [0, %].Then it is conjectured by Sato and Tate that for any interval / in (0,7r),for a certain (specified) measure ju E (cf.[5]).Attached to E, there is a family of /-adic representations p/.Gal(Q/Q) -Gl^Z,) such that if p X fN, and o p is a Frobenius element at p, then pX 0 ^) has