Abstract
We obtain new results concerning the Lang-Trotter conjectures on Frobenius traces and Frobenius fields over single and double parametric families of elliptic curves. We also obtain similar results with respect to the Sato-Tate conjecture. In particular, we improve a result of A. C. Cojocaru and the second author (2008) towards the Lang-Trotter conjecture on average for polynomially parameterized families of elliptic curves when the parameter runs through a set of rational numbers of bounded height. Some of the families we consider are much thinner than the ones previously studied.
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