Frobenius Distributions and Galois Representations
by
Kumar Murty
University of Toronto

Let E be an elliptic curve defined over a number field K. If p is a prime ideal of good reduction, denote by a_p the integer

N(p) + 1 - |E(Fp)|

where N(p) denotes the norm of p, and F_p denotes the residue field. About twenty five years ago, Lang and Trotter proposed some conjectures on the values taken by this integer. Let a be an integer and let us set

\piE, a(x)   =  #{p:  N(p) <= x and ap = a}.

Then, Lang and Trotter conjectured that

\piE, a(x)  =  (cE, a + o(1))\surdx/logx

for some constant c_{E, a} which they specified explicitly. They presented a limited amount of numerical evidence to support their conjectures.

In this talk we will consider these conjectures and their analogues for a wide class of problems related to the distribution of Frobenius elements in Galois representations. We present a large amount of computational work, some of which seems to raise questions about the original conjectures. This is joint work with Kevin James and Cyrus Mehta.

Date received: March 20, 2000