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On lower bounds for the Ihara constants $A(2)$ and $A(3)$

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  03 May 2013

Iwan Duursma  and
Kit-Ho Mak
Iwan Duursma
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 273 Altgeld Hall, MC-382, 1409 W. Green Street, Urbana, Illinois 61801, USA email duursma@math.uiuc.edu
Kit-Ho Mak
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 273 Altgeld Hall, MC-382, 1409 W. Green Street, Urbana, Illinois 61801, USA email mak4@illinois.edu
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Abstract

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Let $ \mathcal{X} $ be a curve over ${ \mathbb{F} }_{q} $ and let $N( \mathcal{X} )$, $g( \mathcal{X} )$ be its number of rational points and genus respectively. The Ihara constant $A(q)$ is defined by $A(q)= {\mathrm{lim~sup} }_{g( \mathcal{X} )\rightarrow \infty } N( \mathcal{X} )/ g( \mathcal{X} )$. In this paper, we employ a variant of Serre’s class field tower method to obtain an improvement of the best known lower bounds on $A(2)$ and $A(3)$.

Keywords

curves with many pointsasymptotic boundIhara constant

MSC classification

Secondary: 11G20: Curves over finite and local fields 14G15: Finite ground fields 14G05: Rational points
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 7 , July 2013 , pp. 1108 - 1128
Copyright
© The Author(s) 2013 

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