Introduction. Let N 2 1, k 2 2 be integers and let f be a cusp form of weight k for ro(N), which is a normalized eigenform for all the Hecke operators Tp, p 4 N. Let us write f (z) == an 2winz n>l for its Fourier expansion at ioo, and for simplicity of discussion, let us suppose that the an are rational integers. For each integral value of a, set 7rf,a(X) = #{P s x :aap = a}. If a = 0 and f is of CM-type (in the sense of Ribet [11]), we know that lrf,a(X) r(x)/2. In the remaining cases, (i.e. f is not of CM type or a * 0), Lang and Trotter [5] conjecture that l/2/log x if k = 2 Wf,a(X)-Cf,a loglog x if k = 3 1 if k 2 4 where Cf a is a constant which is generally (though not always) nonzero (Cf,a can be zero for example if a = 1, k = 2 and f corresponds to an elliptic curve having a nontrivial Q-rational torsion point). Moreover, Atkin and Serre [15] conjecture that for k 2 4, Manuscript received 4 April 1986; revised 6 April 1987. '.2Research partially supported by NSERC grants.
Let E be a fixed elliptic curve defined over the rational numbers. We prove that the number of primes p ≤ x such that E has supersingular reduction mod p is greater than for any positive δ and x sufficiently large. Here logkx is defined recursively as log(logk-1 x) and log1x = logx. We also establish several results related to the Lang-Trotter conjecture.
A famous conjecture of E. Artin [t] states that for any integer a 4= +_ I or a perfect square, there are infinitely many primes p for which a is a primitive root (modp). This conjecture was shown to be true if one assumes the generalized Riemann hypothesis by Hooley [5]. The purpose of this note is to exhibit a finite set S such that for some a eS, a is a primitive root (modp) for an infinity ofprimesp.To this end, let q, r and s denote three distinct primes. Define the following set:
S = {qs 2, q3r2, qZr, r3s 2, rZs, qZs3, qr 3, q3 rs z, rs 3, qZr3s, q3s, qr2s 3, qrs}.Theorem, For some a ~S, there is a 6 > 0 such that for at least 6x/log2x primes p < x, a is aprimitive root (modp).Our theorem is proved in the following way. First we show that there are at least cx/log2x primes p < x such that all odd prime divisors of (p -1) exceed x ~+'. For such primes, we prove that IF* = (q, r, s) with at most o (x/log2x) exceptional primes p < x. Hence, for at least cx/logZx primes p < x, IF* has a generator of the form q" r v s w for some u, v, w. The final step is to show that we can find u, v, w bounded by three. In fact, we can take a generator as in the set S above.
Lemma 1. Fix a prime q, and O < ~ < 88 If c~ = 88 -~, there is a constant c > 0 such thatcard(p
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