title: A heuristic for the distribution of point counts for random curves over a finite field.
abstract: How many rational points are there on a random algebraic curve of large genus g over a given finite field Fq? 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.
authors: Department of Mathematics, Colorado State University, Fort Collins, CO, USA.
authors: Jeffrey D
authors: http://orcid.org/0000-0001-8492-8532
authors: JD
authors: Achter
authors: Department of Mathematics, University of Wisconsin, Madison, WI, USA.
authors: Daniel
authors: D
authors: Erman
authors: Department of Mathematics, University of California, San Diego, CA, USA kedlaya@ucsd.edu.
authors: Kiran S
authors: http://orcid.org/0000-0001-8700-8758
authors: KS
authors: Kedlaya
authors: Department of Mathematics, University of Wisconsin, Madison, WI, USA American Institute of Mathematics, San Jose, CA, USA.
authors: Melanie Matchett
authors: MM
authors: Wood
authors: Department of Mathematics, Emory University, Atlanta, GA, USA.
authors: David
authors: D
authors: Zureick-Brown
keywords: Grothendieck–Lefschetz trace formula; algebraic curves; finite fields; moduli spaces; stable cohomology