How good are the Weil bounds for curves?

Abstract:

A compact surface with $g$ holes can be endowed with the structure of complex manifold (which for $g > 0$ is not unique). When this is done it turns out that it can be embedded into projective space as the zero locus of polynomial equations. If the equations have integer coefficients it makes sense to reduce them modulo a prime $p$. \\ For most primes $p$ the resulting projective algebraic curve $C$ over the field with $p$ elements will be smooth. Weil proved that the number $N_C$ of points (with coordinates in $\Z/p$) always satisfies $|N_C- p - 1| \leq 2 g \sqrt{p}$. \\ A natural question is therefore: for fixed $g$ if $t$ is an integer satisfying $|t | \leq 2 g \sqrt{p}$ is there a smooth projective curve $C$ over $\Z/p$ such that $N_C= p+ 1 -t$. \\ In this talk I will explain how the theory of Abelian varieties over finite fields can be used to give partial answers to this question when $g \leq 3$. The ideas used mostly date back to Serre’s lectures on this topic in the 1980s. The theory used largely pre-dates the 1980s with some improvements from the thesis of Everett Howe.