Higher-order Weierstrass weights of branch points on superelliptic curves

TL;DR

This paper derives exact formulas for higher-order Weierstrass weights of branch points on superelliptic curves, providing asymptotic ratios and conditions for equality, advancing understanding of algebraic curve invariants.

Contribution

It introduces explicit formulas for the q-weights of branch points and points at infinity on superelliptic curves, and analyzes their asymptotic behavior as parameters grow.

Findings

Exact formulas for q-weights of affine branch points

Formula for q-weight of points at infinity when gcd(n,d)=1

Asymptotic ratio of branch point weights to total weights

Abstract

In this paper we consider the problem of calculating the higher-order Weierstrass weight of the branch points of a superelliptic curve $C$. For any $q>1$, we give an exact formula for the $q$-weight of an affine branch point. We also find a formula for the $q$-weight of a point at infinity in the case where $n$ and $d$ are relatively prime. With these formulas, for any fixed $n$, we obtain an asymptotic formula for the ratio of the $q$-weight of the branch points, denoted $BW_q$, to the total $q$-weight of points on the curve: \[ \liminf_{d\to\infty}\frac{BW_q}{g(g-1)^2(2q-1)^2}\geq \frac{n+1}{3(n-1)^2(2q-1)^2},\] with equality when the limit is taken such that $\gcd(n,d)=1$.