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Lower bounds on the maximal number of rational points on curves over finite fields.
- Source :
-
Mathematical Proceedings of the Cambridge Philosophical Society . Jan2024, Vol. 176 Issue 1, p213-238. 26p. - Publication Year :
- 2024
-
Abstract
- 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: We show that the bound $1+q+4 \sqrt{q} -32$ is valid for all $g\ge 2$ and for all q. [ABSTRACT FROM AUTHOR]
- Subjects :
- *RATIONAL numbers
*FINITE fields
Subjects
Details
- Language :
- English
- ISSN :
- 03050041
- Volume :
- 176
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Mathematical Proceedings of the Cambridge Philosophical Society
- Publication Type :
- Academic Journal
- Accession number :
- 174324277
- Full Text :
- https://doi.org/10.1017/S0305004123000476