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THE FIXED TRACE CASE
Fixed traces from the quadratic field
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4-division abelian extension associated vector space assume asymptotic behavior characteristic polynomial class field theory closed subgroup compute concludes the proof congruence conjecture conjugation contained converges corresponding cosets cube cyclic cyclotomic field define denote density function determine discriminant disjoint division points elliptic curve equal fiber product follows at once formula Frobenius elements Galois extension Galois group Galois representation given GL2-extension Haar measure Harish transform Hence homomorphism imaginary quadratic field integral inverse image isomorphism kernel mod 9 mod q ND ED Pr notation number of primes obtain odd prime positive integer predicted value probabilistic model ramified rationals reduction mod roots of unity Sato-Tate scalar matrices sequence Serre curves Serre's subgroup set of primes splits and stabilizes splits completely stable at level subgroup of index subset supersingular primes Tchebotarev Theorem 7.1 trace of Frobenius unramified write