A heuristic for the distribution of point counts for random curves over a finite field-Reference-Cited by-同舟云学术

A heuristic for the distribution of point counts for random curves over a finite field

Published:2015-04-28 Issue:2040 Volume:373 Page:20140310
ISSN:1364-503X
Container-title:Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
language:en
Short-container-title:Phil. Trans. R. Soc. A.

Author:

Achter Jeffrey D.¹^ORCID,Erman Daniel²,Kedlaya Kiran S.³^ORCID,Wood Melanie Matchett²⁴,Zureick-Brown David⁵

Affiliation:

1. Department of Mathematics, Colorado State University, Fort Collins, CO, USA

2. Department of Mathematics, University of Wisconsin, Madison, WI, USA

3. Department of Mathematics, University of California, San Diego, CA, USA

4. American Institute of Mathematics, San Jose, CA, USA

5. Department of Mathematics, Emory University, Atlanta, GA, USA

Abstract

How many rational points are there on a random algebraic curve of large genus g over a given finite field

? We propose a heuristic for this question motivated by a (now proven) conjecture of Mumford on the cohomology of moduli spaces of curves; this heuristic suggests a Poisson distribution with mean q +1+1/( q −1). We prove a weaker version of this statement in which g and q tend to infinity, with q much larger than g .

Publisher

The Royal Society

Subject

General Physics and Astronomy,General Engineering,General Mathematics

Link

https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2014.0310

Reference34 articles.

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2. How many rational points does a random curve have?

3. The fluctuations in the number of points on a hyperelliptic curve over a finite field

4. Statistics for Traces of Cyclic Trigonal Curves over Finite Fields

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