## Conference "Arithmetic, Geometry and Coding Theory"## March 14 - 18, 2011## CIRM, Marseille, France |

**Organisers:** Yves Aubry *(Université de la Méditerranée, IML)*, Christophe Ritzenthaler *(Université de la Méditerranée, IML)*, Alexey Zykin *(Mathematical Department of HSE, Laboratoire Poncelet, IITP)*

For a curve C over a finite field Fq (projective, nonsingular, absolutely irreducible) we denote by g(C) (resp. N(C)) the genus (resp. the number of F_q-rational points) of C. The classical Hasse-Weil Theorem says that for given q and g = g(C),
q + 1 - 2g \sqrt{q} \leq N(C) \leq q + 1 + 2g \sqrt{q};
i.e. N(C) lies in a finite interval. A lot of effort has been put into improving the upper bound of this interval,
partly motivated by applications of curves with `many' points in coding theory and cryptography, but also
since `the question represents an attractive mathematical challenge' (van der Geer).
In this talk, we change the point of view slightly: we fix the finite field F_q and a non-negative integer N
and ask for all possible values of g such that there exists a curve C over F_q of genus g,
having exactly N rational points. As follows immediately from the Hasse-Weil Theorem, g must satisfy the
condition
g \geq (N - q - 1)/ (2\sqrt{q})
Our main result is

**Theorem 1. **Given a finite field F_q and an integer N \geq 0, there is an integer g0 \geq 0 such that for every
g \geq g0, there exists a curve C over F_q with g(C) = g and N(C) = N.

A generalization of this result is

**Theorem 2.** Given a finite field F)q and non-negative integers b_1,..., b_m. Then there exists an integer g_1
with the following property: for every g \geq g_1 there exists a curve C over F_q with g(C) = g such that C has
exactly b_r points of degree r, for r = 1,..., m.

Theorem 2 has a nice interpretation in terms of the L-polynomial (the numerator of the zeta function) of
the curve C. One can - under certain conditions - prescribe the first coefficients of the L-polynomial of C
arbitrarily.