The Square Sieve and the Lang--Trotter Conjecture

[摘要] Let $E$ be an elliptic curve defined over $Q$ and withoutcomplex multiplication. Let $K$ be a fixed imaginary quadratic field.We find nontrivial upper bounds for the number of ordinary primes $p leq x$for which $Q(pi_p) = K$, where $pi_p$ denotes the Frobenius endomorphismof $E$ at $p$. More precisely, under a generalized Riemann hypothesiswe show that this number is $O_{E}(x^{slfrac{17}{18}}log x)$, and unconditionallywe show that this number is $O_{E, K}igl(frac{x(log log x)^{slfrac{13}{12}}}{(log x)^{slfrac{25}{24}}}igr)$. We also prove that the number of imaginary quadraticfields $K$, with $-disc K leq x$ and of the form $K = Q(pi_p)$, is$gg_Elogloglog x$ for $xgeq x_0(E)$. These results represent progress towardsa 1976 Lang--Trotter conjecture.

[学科分类] 数学（综合）