The Maximum or Minimum Number of Rational Points on Genus Three Curves over Finite Fields

Kristin Lauter; Jean-Pierre Serre

doi:10.1023/A:1020246226326

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that for all finite fields Fq, there exists a curve C over Fq of genus 3 such that the number of rational points on C is within 3 of the Serre–Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over Fq.

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Top, Jaap 2003. Curves of genus 3 over small finite fields. Indagationes Mathematicae, Vol. 14, Issue. 2, p. 275.

Hurt, Norman E. 2003. Many Rational Points. p. 317.

Choi, So-Young and Koo, Ja-Kyung 2007. AN AFFINE MODEL OF X0(mn). Bulletin of the Korean Mathematical Society, Vol. 44, Issue. 2, p. 379.

Nart, Enric and Ritzenthaler, Christophe 2008. Jacobians in isogeny classes of supersingular abelian threefolds in characteristic 2. Finite Fields and Their Applications, Vol. 14, Issue. 3, p. 676.

Kodama, Tetsuo Top, Jaap and Washio, Tadashi 2009. Maximal hyperelliptic curves of genus three. Finite Fields and Their Applications, Vol. 15, Issue. 3, p. 392.

Alekseenko, E. Aleshnikov, S. Markin, N. and Zaytsev, A. 2011. Optimal curves over finite fields with discriminant −19. Finite Fields and Their Applications, Vol. 17, Issue. 4, p. 350.

Zaytsev, Alexey 2014. An improvement of the Hasse–Weil–Serre bound for curves over some finite fields. Finite Fields and Their Applications, Vol. 30, Issue. , p. 88.

Hirschfeld, J.W.P. and Thas, J.A. 2015. Open problems in finite projective spaces. Finite Fields and Their Applications, Vol. 32, Issue. , p. 44.

Tian, Song Yu, Wei Li, Bao and Wang, Kunpeng 2015. Information Security Practice and Experience. Vol. 9065, Issue. , p. 181.

Zaytsev, Alexey 2016. Optimal curves of low genus over finite fields. Finite Fields and Their Applications, Vol. 37, Issue. , p. 203.

2016. Lectures on N_X(p). p. 111.

Jordan, Bruce W. Keeton, Allan G. Poonen, Bjorn Rains, Eric M. Shepherd-Barron, Nicholas and Tate, John T. 2018. Abelian varieties isogenous to a power of an elliptic curve. Compositio Mathematica, Vol. 154, Issue. 5, p. 934.

Lercier, Reynald Ritzenthaler, Christophe Rovetta, Florent Sijsling, Jeroen and Smith, Benjamin 2019. Distributions of Traces of Frobenius for Smooth Plane Curves Over Finite Fields. Experimental Mathematics, Vol. 28, Issue. 1, p. 39.

Marseglia, Stefano 2022. Computing base extensions of ordinary abelian varieties over finite fields. International Journal of Number Theory, Vol. 18, Issue. 09, p. 1957.

Centeleghe, Tommaso Giorgio and Stix, Jakob 2023. Categories of abelian varieties over finite fields II: Abelian varieties over $${\mathbb{F}_q}$$ and Morita equivalence. Israel Journal of Mathematics, Vol. 257, Issue. 1, p. 103.

Ferreira, Walteir de Paula and Speziali, Pietro 2023. Optimal plane curves of degree q − 1 over a finite field. Finite Fields and Their Applications, Vol. 91, Issue. , p. 102258.

Bergström, Jonas Howe, Everett W. Lorenzo García, Elisa and Ritzenthaler, Christophe 2024. Refinements of Katz–Sarnak theory for the number of points on curves over finite fields. Canadian Journal of Mathematics, p. 1.

Article contents

The Maximum or Minimum Number of Rational Points on Genus Three Curves over Finite Fields

Abstract

Keywords

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Article contents

The Maximum or Minimum Number of Rational Points on Genus Three Curves over Finite Fields

Abstract

Keywords

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests