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

Vishal Arul

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

Algebraic Geometry 2020-08-10 v4

Authors: Vishal Arul

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 $D$ .

Keywords

superelliptic curves jacobians division by 1-zeta hyperelliptic curves mumford representation divisors

Cite

@article{arxiv.1810.07299,
  title  = {Division by $1-\zeta$ on superelliptic curves and jacobians},
  author = {Vishal Arul},
  journal= {arXiv preprint arXiv:1810.07299},
  year   = {2020}
}

Comments

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

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

Abstract

Keywords

Cite

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