occasional meanderings in physics' brave new world

Monday, July 23, 2007

Riemann Rocks

If we don't worry about complex numbers, various forms of the Riemann Hypothesis have known to be true for some time now. Bombieri came up with a simple proof for algebraic curves over finite fields  in the early 1970s.

Let $k$ be the finite field with $q$ elements and $C$ a curve over this field. The zeta function is about divisors on $C$, which we can think of as formal collections of points on the curve. Prime divisors will be labelled $P$. Now let $d(P) = [k_{P} : k]$, which is the degree of a field extension $k_{P}$, the so-called residue field. The variables $t$ and $s$ will be used, where $t = q^{-s}$. The Euler product formula for the zeta function takes the form

$\sum_{D} N(D)^{-s} = \prod_{P} (1 - N(P)^{-s})^{-1}$

where the sum is over divisors and the product over prime divisors, and $N(P) = q^{d(P)}$. So it looks just like the usual Riemann zeta formula, except that numbers have been replaced by (numbers associated to) geometric objects. If we let

$L(t) = \zeta (t) (1 - t) (1 - qt)$

and let $a_{i}$ be the inverses of the roots of $L(t)$, then the Riemann hypothesis has the following simple form: the $a_{i}$ all satisfy $| a_{i} | = \sqrt{q}$. There are $2g$ such $a_{i}$ and the set can be ordered into two parts so that

$a_{i} a_{g + i} = q$

The functional equation is

$\zeta (1 - s) = (q^{2g - 2})^{s - 0.5} \zeta (s)$

and in all this $g$ is basically the genus of the curve. Do these kind of primes $P$ look more like respectable categorical objects?

 C. J. Moreno, Algebraic Curves over Finite Fields, Cambridge (1991)