Preliminaries

1. Preliminaries#

Many works on hyperelliptic functions and Riemann surfaces use similar notation, which can lead to potential naming collisions. To avoid such collisions, a legend of notations used in this notebook is provided below.

1.1. Legend#

  • \(\mathscr{C}\) - an algebraic curve

  • \(X\) - a Riemann surface

  • \(\lambda_i\) - coefficients of the equation of \(\mathscr{C}\)

  • \(e_i\) - branch points

  • \(g\) - a genus of \(\mathscr{C}\)

  • \(du\) - first kind (or holomorphic) differential on \(\mathscr{C}\)

  • \(dr\) - second kind differential on \(\mathscr{C}\)

  • \(\mathfrak{a}_i, \mathfrak{b}_i\) - not normalised canonical homology cycles

  • \(\omega\) - first kind integrals (first kind not normalised \(\mathfrak{a}\), and \(\mathfrak{b}\)-period matrices)

  • \(\eta\) - second kind integrals (second kind not normalised \(\mathfrak{a}\), and \(\mathfrak{b}\)-period matrices)

  • \(\mathrm{Jac}(\mathscr{C}) = \mathbb{C}^g/\{\omega, \omega' \}\) - Jacobian variety of the curve \(\mathscr{C}\)

  • \(D\) - Divisor on the Riemann surface \(X\)

  • \(\mathcal{A}(P)\), \(\mathcal{A}(D)\) - Abel image (or first kind integral) of a point \(P\) and a divisor \(D\)

  • \(\Sigma\) - the theta-divisor defined by \(\{ \mathbf{u}\in \mathrm{Jac}(\mathscr{C})| \sigma(\mathbf{u})=0 \}\)

  • \(\mathfrak{U}(\mathscr{C})\) - differential field of \(\wp\)-functions on \(\mathrm{Jac}(\mathscr{C}) \setminus \Sigma\)

  • \(\mathbf{\varepsilon}\) - a characteristic vectors

  • \(\mathbf{K}\) - a vector of Riemann constants

1.2. Hyperelliptic curve#

In studies on hyperelliptic functions, various conventions for defining a curve, and consequently the hyperelliptic functions, can be found. In this notebook, we adopt the convention presented in [1], [4]and [4].

Additionally, conventions for curves in both general and canonical forms are also discussed in dedicated notebooks: canonical-name.ipynb and general-name.ipynb. Please refer to these notebooks for further details.

Definition 1.1

The hyperelliptic curve is defined by the equation

\[\mathscr{C} = \{ (x,y)\in \mathbb{C}^2 \mid f(x,y)=0 \},\]

where

\[f(x,y) = -y^2 + x^{2g+1} + \sum_{i=0}^{2g}\lambda_{2i+2}x^{2g-i}, \quad \lambda_{k\leq0}=0,\; \lambda_k \in \mathbb{R}.\]

Note

We assume \(\lambda_k \in \mathbb{R}\), instead of \(\lambda_k \in \mathbb{C}\) which can be found in various texts, because we would like to apply these functions to physical equations first.