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 [1] 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 So it looks just like the usual Riemann zeta formula, except that numbers have been replaced by (numbers associated to) geometric objects.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?

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

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