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Average Frobenius distributions for elliptic curves with nontrivial rational torsion. (English) Zbl 1151.11022

Summary: Let \(E/{\mathbb Q}\) denote an elliptic curve, let \(a_p(E) = p+1-\#E({\mathbb F}_p)\) and for \(r\in {\mathbb Z}\), let \(\pi_E^r(X) := \#\{p \leq X : a_p(E) = r\}\). Lang and Trotter have conjectured that \(\pi_E^r(X) \sim C_{E,r} \frac{\sqrt{X}}{\log{X}}\), where \(C_{E,r}\) is an explicit constant depending only on \(E\) and \(r\). In this paper we consider the Lang-Trotter conjecture for various families of elliptic curves with prescribed torsion structure. We prove that the Lang-Trotter conjecture holds in an average sense for these families of curves.

MSC:

11G05 Elliptic curves over global fields
11F80 Galois representations
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