More Points Than Expected on Curves over Finite Field Extensions

  • Authors:

  • Bradley W. Brock;Andrew Granville

  • Affiliations:

  • Natural Science Division, Pepperdine University, Malibu, California, 90263-4321, f1bbrock@pepperdine.eduf1;Department of Mathematics, University of Georgia, Athens, Georgia, 30602-7403, f2andrew@math.uga.eduf2

  • Venue:
  • Finite Fields and Their Applications
  • Year:
  • 2001

Quantified Score

Hi-index 0.00

Visualization

Abstract

On average, there are q^r+o(q^r^/^2) F"q"^"r-rational points on curves of genus g defined over F"q"^"r. This is also true if we restrict our average to genus g curves defined over F"q, provided r is odd or r2g. However, if r=2,4,6,... or 2g then the average is q^r+q^r^/^2+o(q^r^/^2). We give a number of proofs of the existence of these q^r^/^2 extra points, and in some cases give a precise formula, but we are unable to provide a satisfactory explanation for this phenomenon.