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The Square Sieve and the Lang--Trotter Conjecture

  Published:2005-12-01
 Printed: Dec 2005
  • Alina Carmen Cojocaru
  • Etienne Fouvry
  • M. Ram Murty
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Abstract

Let $E$ be an elliptic curve defined over $\Q$ and without complex 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 endomorphism of $E$ at $p$. More precisely, under a generalized Riemann hypothesis we show that this number is $O_{E}(x^{\slfrac{17}{18}}\log x)$, and unconditionally we show that this number is $O_{E, K}\bigl(\frac{x(\log \log x)^{\slfrac{13}{12}}} {(\log x)^{\slfrac{25}{24}}}\bigr)$. We also prove that the number of imaginary quadratic fields $K$, with $-\disc K \leq x$ and of the form $K = \Q(\pi_p)$, is $\gg_E\log\log\log x$ for $x\geq x_0(E)$. These results represent progress towards a 1976 Lang--Trotter conjecture.
Keywords: Elliptic curves modulo $p$, Lang--Trotter conjecture, applications of sieve methods Elliptic curves modulo $p$, Lang--Trotter conjecture, applications of sieve methods
MSC Classifications: 11G05, 11N36, 11R45 show english descriptions Elliptic curves over global fields [See also 14H52]
Applications of sieve methods
Density theorems
11G05 - Elliptic curves over global fields [See also 14H52]
11N36 - Applications of sieve methods
11R45 - Density theorems
 

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