Local statistics and average rank of genus $g$ hyperelliptic curves with a Weierstrass point

Abstract

In this paper, we determine the probability that a genus $g$ hyperelliptic curve with a Weierstrass point over a number field has good reduction at a given prime of residue characteristic $>2g+1$. We also obtain analogous probability formulas for several other reduction types, including cases with positive toric or unipotent rank. As an application, assuming the Hasse--Weil conjecture and the generalized Riemann hypothesis, we derive an explicit upper bound for the average analytic rank of genus $g$ hyperelliptic curves with a Weierstrass point.