论文标题
有限场上曲线上最大理性点的下限
Lower bounds on the maximal number of rational points on curves over finite fields
论文作者
论文摘要
对于给定的属$ g \ geq 1 $，我们在平滑的投影中给出了最大数量的理性点的下限，绝对是$ g $ $ g $ of $ {\ mathbb f} _q $的属属属属属属。由于Katz-sarnak理论的结果，我们首先要获得任何给定的$ g> 0 $，任何$ \ varepsilon> 0 $和所有$ q $，并且所有$ q $都足够大了，c $ g $ g $ g $ g $ fife $ {\ mathbb f} _q $的存在至少具有至少$ 1+ q+ q+ q+ q+（2g-v varepsilon）\ sq quent然后，使用过纤维曲线的Frobenius痕迹的力量总和，我们获得了$ 1+Q+Q+1.71 \ sqrt {q} $有效的$ 1+Q+sqrt {q \ geq 3 $和奇数$ q \ geq 11 $的下限。最后，曲线塔的显式结构改善了这一结果，并具有$ 1+q+4 \ sqrt {q} -32 $的限制，适用于所有$ g \ ge 2 $和所有$ q $。
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 $q$.
