Lang-Trotter and Sato-Tate conjectures on average
We discuss recent results towards the
Lang-Trotter and Sato-Tate conjectures.
While the Sato-Tate conjecture has recently been proved,
in the case of the Lang-Trotter conjecture even the Extended Riemann
Hypothesis is not powerful enough to establish the expected result.
We show that various techniques from analytic number theory help to
establish new results for these conjectures on average over natural families
of elliptic curve.
These include the curves
Y2 = X3 + aX + b where a and b
the intervals of the form |a| &le A, b &le B and curves
Y2 = X3 +
A(t)X + B(t), where A and B are fixed polynomials and t runs through
the set of Farey fractions of order T.
Some of the results are obtained jointly with Bill Banks (Missouri) and
Alina Cojocaru (Univ. of Illinois at Chicago).