2, 3, 5, Legendre: ± Trace Ratios in Families of Elliptic Curves

is one of the most familiar, and most studied, families of elliptic curves, often used for testing conjectures and for illustrating theorems. When we became interested in the Lang-Trotter conjecture [L-T, page 33] in the function field case, cf. [Ka-LTR], we turned to this family to do some computer experiments. This paper reports on an empirical discovery made in the course of those experiments, and on the theory which explains it. The explanation owes a great deal to Deligne, as will become clear below. In the experiments, we took an odd prime p, and tabulated, for each λ0 ∈ Fp \ {0, 1}, the ”trace of Frobenius” A(λ0,Fp) ∈ Z for the elliptic curve Eλ0 over Fp. Concretely, we have #Eλ0(Fp) = p+ 1− A(λ0,Fp). We calculated the numbers A(λ0,Fp) brutally, as the character sums A(λ0,Fp) = − ∑