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12 5 $begingroup$ If $p$ is a prime then the zeta function for an algebraic curve $V$ over $mathbb{F}_p$ is defined to be $$zeta_{V,p}(s) := expleft(sum_{mgeq 1} frac{N_m}{m}(p^{-s})^mright). $$ where $N_m$ is the number of points over $mathbb{F}_{p^m}$ . I was wondering what is the motivation for this definition. The sum in the exponent is vaguely logarithmic. So maybe that explains the exponential? What sort of information is the zeta function meant to encode and how does it do it? Also, how does this end up being a rational function? algebraic-curves zeta-functions share | cite | improve this question asked 16 hours ago