Lang–Trotter and Sato–Tate distributions in single and double parametric families of elliptic curves

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 parameterised 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.27 page(s