manYPoints – Table of Curves with Many Points
Entry details for q =
2
7
= 128
, g =
3
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Lower bound
N
min
= 192
Submitted by
Gerrit Oomens
Date
01-01-1900
Reference
S. Sémirat
Problèmes de nombres de classes pour les corps de fonctions et applications
Thèse, Université Pierre et Marie Curie, Paris, 2000.
Comments
Tags
Towers of curves with many points
User comments
Explicit example
Everett Howe
05-20-2010 22:13
An explicit example is given by the plane quartic
X^2 + X*Y + Y^2 + r^9*Y + 1 = 0
where X = x^2 + x and Y = y^2 + y, and where r^7 + r + 1 = 0.
Upper bound
N
max
= 192
Submitted by
Everett Howe
Date
06-11-2010
Reference
Kristin Lauter
Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields
J. Algebraic Geom. 10 (2001) 19–36
Comments
Let m = Floor(2*Sqrt(q)) = 22. The Honda-Tate theorem shows that there does not exist an abelian threefold with Weil polynomial (x^2 + m*x + q)^3, so defect 0 is impossible.
Defect 1 is impossible as well due to the "resultant 1" method.
Now one can check that all the possibilities of Table 1 in the article for defect 2 are excluded (some because there is no elliptic curve with Weil polynomial x^2 + m*x + q, some because there is no abelian surface with Weil polynomial (x^2 + m*x + q)^2, some because of the "resultant 1" method and the last one because of the value of the fractional part of 2*Sqrt(q)).
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