Type: Article
Publication Date: 1985-06-01
Citations: 58
DOI: https://doi.org/10.1216/rmj-1985-15-2-535
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 \ <i 2p V2 .We say that E has super-singular reduction at p if a p = 0. DefineFrom results of Deuring [2], it is known that if E has complex multiplication, then, as x -> oo, 7t E (x) ~ (l/2)7r(x), where %{x) denotes the number of primes p < x.If E does not have complex multiplication, Lang and Trotter [4] conjecture that, as x -> 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