Galois Representations with Non-Surjective Traces


Let E be an elliptic curve over Q , and let r be an integer. According to the Lang-Trotter conjecture, the number of primes p such that ap(E) = r is either finite, or is asymptotic to CE,r √ x/log x where CE,r is a non-zero constant. A typical example of the former is the case of rational `-torsion, where ap(E) = r is impossible if r ≡ 1 (mod `). We prove… (More)