Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Lower bounds on the maximal number of rational points on curves over
  finite fields   For a given genus $g \geq 1$, we give lower bounds for the maximal number of
rational points on a smooth projective absolutely irreducible curve of genus
$g$ over ${\mathbb F}_q$. As a consequence of Katz-Sarnak theory, we first get
for any given $g>0$, any $\varepsilon>0$ and all $q$ large enough, the
existence of a curve of genus $g$ over ${\mathbb F}_q$ with at least $1+q+
(2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces
of Frobenius of hyperelliptic curves, we get a lower bound of the form
$1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$. Finally, explicit
constructions of towers of curves improve this result, with a bound of the form
$1+q+4 \sqrt{q} -32$ valid for all $g\ge 2$ and for all $q$.

21
The categories of this paper are: adap-org
15
The categories of this paper are: nucl-ex
22
The categories of this paper are: chao-dyn
31
The categories of this paper are: ao-sci
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
35
The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics
29
The categories of this paper are: funct-an
23
The categories of this paper are: solv-int
8
The categories of this paper are: hep-ex