Arizona Winter School 2000
Katz's Project Description

Mysterious formulae involving the numbers of points in some families of elliptic curves

The project is to find a (good) explanation of the following experimental fact. for each prime p at least five, we look at the weierstrass equations y² = 4x³ - g₂x - g₃ with coefficients g₂ and g₃ in the prime field F_p and for which the discriminant (g₂)³ - 27(g₃)² is equal to ONE. For each such elliptic curve, its number of F_p points (including the point at infinity) we write as 1 + p - A(p, g₂, g₃). Then we form the sum of A(p, g₂, g₃) over all the g₂ and g₃ in the prime field F_p with discrimint ONE, call this sum B(p). okay, it is not hard to show that for prime p which are 3 mod 4, we have B(p) = 0. And one finds experimentally that for a prime p which is 1 mod 4, if we write p as the sum of two squares a² + b² with a and b both positive and with a ODD, then we have the miraculous formula

How come?

Here is a similar fact, in this case provably true, but even here I don't know a really satisfying explanation. One takes, for each prime p at least three, all the modified legendre equations y² = a(a-1)x(x-1)(x-a) over F_p, now with a in F_p but not 0 or 1. one writes the number of F_p points on this curve as 1 + p - A(p,a), then one forms the sum of A(p,a) over all the a not 0 or 1 in F_p, call this sum B(p). The notation is not as bad as it seems, because this B(p) is 0 for primes p which are 3 mod 4, and if p is 1 mod 4, if we write p as the sum of two squares a² + b² with a and b both positive and with a ODD, then we have the SAME formula

How come? And why is "this" B(p) equal to the one above? In other words, can one see that the two B(p)'s are equal to each other without knowing what their common numerical value is? How can one predict such phenomena?

good luck, nick katz