Division by $1-\zeta$ on superelliptic curves and jacobians

Abstract

In 2016, Yuri Zarhin gave formulas for "dividing a point on a hyperelliptic curve by 2." Given a point $P$ on a hyperelliptic curve $\mathcal{C}$, Zarhin gives the Mumford's representation of every degree $g$ divisor $D$ such that $2(D-g\infty )\sim P-\infty$. The aim of this paper is to generalize Zarhin's result to the superelliptic situation; instead of dividing by 2, we divide by $1-\zeta$. Even though there is no Mumford's representation for superelliptic curves, we give a formula for functions which cut out $D$.

To appear in Int. Math. Res. Not. IMRN; reorganized and improved proof of main theorem; added more details in "Varying the choice of r_i" section and intersection multiplicity section; added more references