✦ LIBER ✦
Curves of Every Genus with Many Points, I: Abelian and Toric Families
✍ Scribed by Andrew Kresch; Joseph L. Wetherell; Michael E. Zieve
- Publisher
- Elsevier Science
- Year
- 2002
- Tongue
- English
- Weight
- 157 KB
- Volume
- 250
- Category
- Article
- ISSN
- 0021-8693
No coin nor oath required. For personal study only.
✦ Synopsis
Let N q g denote the maximal number of F q -rational points on any curve of genus g over F q . Ihara (for square q) and Serre (for general q) proved that lim sup g→∞ N q g /g > 0 for any fixed q. Here we prove lim g→∞ N q g = ∞. More precisely, we use abelian covers of P 1 to prove lim inf g→∞ N q g / g/ log g > 0, and we use curves on toric surfaces to prove lim inf g→∞ N q g /g 1/3 > 0; we also show that these results are the best possible that can be proved using these families of curves. 2002 Elsevier Science (USA)