**From hyperelliptic to superelliptic curves**

The book will hopefully be finished by Fall 2018. I am looking for a publisher

## Part 1:

- Curves and their automorphisms
- Curves and their automorphisms
- Automorphism groups of hyperelliptic curves
- Superelliptic curves over characteristic 0 fields
- Superelliptic curves over positive characteristic fields

- Superelliptic curves and binary forms
- An introduction to binary forms
- Transvections and root differences
- Ring of invariants (Gordan, Noether, Hilbert)
- Computational invariant theory

- Weierstrass points
- Linear systems, inflection points, Wronskian
- Weierstrass points
- Hurwitz theorem
- Weierstrass points of superelliptic curves

- Superelliptic loci in the moduli space of curves
- Moduli space of covers and Hurwitz spaces
- Signatures of automorphism groups
- Superelliptic loci in M_g

- Equations of curves with fixed automorphism group
- Equations of hyperelliptic curves with fixed automorphism group
- Equations of superelliptic curves with fixed automorphism group
- Equations over fields of positive characteristic

- Theta functions
- Theta functions of curves
- Frobenious formula
- Thomae's formula for cyclic coverings
- Thetanulls of superelliptic curves

- Superelliptic Jacobians
- Jacobians of curves
- Hyperelliptic Jacobians
- Superelliptic Jacobians
- Decomposable Jacobians

Part 2: Minimal models

- Curves and their automorphisms
- Automorphism groups of hyperelliptic curves
- Superelliptic curves over characteristic 0 fields
- Superelliptic curves over positive characteristic fields

- An introduction to binary forms
- Transvections and root differences
- Ring of invariants (Gordan, Noether, Hilbert)
- Computational invariant theory

- Linear systems, inflection points, Wronskian
- Weierstrass points
- Hurwitz theorem
- Weierstrass points of superelliptic curves

- Moduli space of covers and Hurwitz spaces
- Signatures of automorphism groups
- Superelliptic loci in M_g

- Equations of hyperelliptic curves with fixed automorphism group
- Equations of superelliptic curves with fixed automorphism group
- Equations over fields of positive characteristic

- Theta functions of curves
- Frobenious formula
- Thomae's formula for cyclic coverings
- Thetanulls of superelliptic curves

- Jacobians of curves
- Hyperelliptic Jacobians
- Superelliptic Jacobians
- Decomposable Jacobians

- On the field of moduli of curves
- Field of moduli versus field of definition
- Hyperelliptic curves over a minimal field of definition
- Equation of superelliptic curves over a minimal field of definition
- Equation of superelliptic curves with extra automorphisms
- Neron models
- Neron models of curves
- Neron models for hyperelliptic curves
- neron models for superelliptic curves
- Reduction theory of binary forms
- An introduction to reduction theory
- Reduction of quadratics
- Julia quadratic and reduction of cubics
- Higher degree binary forms
- Weighted projective spaces
- An introduction to weighted projective spaces
- Weighted moduli space of binary forms
- Heights on weighted projective spaces
- Twists of curves
- Twists of algebraic curves
- Hyperelliptic and superelliptic twists
- Twists of bounded height

Part 3: Rational points