Homological Stability Methods in the Arithmetic Statistics of Curves

Abstract

We study the asymptotic distribution of Fq-rational point counts and of class numbers of Fq-curves of genus g in the limit as g→∞. We do so by using the Grothendieck-Lefschetz trace formula to count the Fq-points of various moduli spaces over Mg, the moduli stack of genus g curves. This reduces the problem to understanding the cohomology of these moduli spaces. We exposit this method following Achter et. al., and discuss some background on the cohomology of Mg. The main new contribution is for the question of how the class numbers behave as g→∞. We show that the stable cohomology for the universal Jacobian stack over Mg gives an asymptotic growth rate for the average class number over Mg. Conditional on the stable cohomology being the main term, we compute this growth rate exactly, and show it achieves the infimum growth rate for families of curves as in the framework of Tsfasman-Vladut. As an application, we show that this growth rate implies a local density of at least 12 for the distribution of zeros for zeta functions of Fq-curves on the circle. Finally, we give some explicit class number computations for the family of modular curves of prime level p, and discuss both their growth rate for p<104 and their ℓ-divisibility for p<105 for various small primes ℓ.