Average Frobenius distributions for elliptic curves with 3-torsion (Q1763999)

Let \(E/\mathbb Q\) be a non-CM elliptic curve and, for each prime \(p\), let \(a_p(E)=p+1-\#E(\mathbb F_p)\) be the trace of Frobenius. The Lang-Trotter conjecture says that \[ \pi_E^r(X):=\#\{p\leq X:\;a_p(E)=r\}\sim C(E,r)\sqrt{X}/\log X, \] where the constant \(C(E,r)\) is given explicitly in terms of the Galois representation attached to \(E\). \textit{E. Fouvry} and \textit{M. R. Murty} [Can. J. Math. 48, No. 1, 81--104 (1996; Zbl 0864.11030)] for \(r=0\) and \textit{C. David} and \textit{F. Pappalardi} [Int. Math. Res. Not. 1999, No. 4, 165--183 (1999; Zbl 0934.11033)] for general \(r\) proved that a version of the Lang-Trotter conjecture holds if one averages over all elliptic curves ordered by the size of their Weierstrass coefficients. Here the author examines the Lang-Trotter conjecture for elliptic curves which possess rational 3-torsion points. He proves that if one averages over all such elliptic curves then one obtains an asymptotic similar to the one predicted by Lang and Trotter.