Achter Jeffrey D, Erman Daniel, Kedlaya Kiran S, Wood Melanie Matchett, Zureick-Brown David
Department of Mathematics, Colorado State University, Fort Collins, CO, USA.
Department of Mathematics, University of Wisconsin, Madison, WI, USA.
Philos Trans A Math Phys Eng Sci. 2015 Apr 28;239(2040). doi: 10.1098/rsta.2014.0310.
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.
在给定的有限域(F_q)上，大亏格(g)的随机代数曲线上有多少个有理点？我们受芒福德关于曲线模空间上同调的一个（现已证明的）猜想启发，针对这个问题提出了一种启发式方法；该启发式方法表明是均值为(q + 1 + \frac{1}{q - 1})的泊松分布。我们证明了这个陈述的一个较弱版本，其中(g)和(q)趋于无穷，且(q)远大于(g)。
