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

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.