Vol. 12, No. 5, 2019

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The number of rational points of hyperelliptic curves over subsets of finite fields

Kristina Nelson, József Solymosi, Foster Tom and Ching Wong

Vol. 12 (2019), No. 5, 755–765
Abstract

We prove two related concentration inequalities concerning the number of rational points of hyperelliptic curves over subsets of a finite field. In particular, we investigate the probability of a large discrepancy between the numbers of quadratic residues and nonresidues in the image of such subsets over uniformly random hyperelliptic curves of given degrees. We find a constant probability of such a high difference and show the existence of sets with an exceptionally large discrepancy.

Keywords
hyperelliptic curves, finite fields
Mathematical Subject Classification 2010
Primary: 68Q87, 68R05
Milestones
Received: 19 January 2018
Revised: 21 June 2018
Accepted: 28 July 2018
Published: 22 May 2019

Communicated by Kenneth S. Berenhaut
Authors
Kristina Nelson
Department of Mathematics
University of California
Berekeley, CA
United States
József Solymosi
Department of Mathematics
University of British Columbia
Vancouver, BC
Canada
Foster Tom
Department of Mathematics
University of British Columbia
Vancouver, BC
Canada
Ching Wong
Department of Mathematics
University of British Columbia
Vancouver, BC
Canada