There are three auxilliary classes: superelliptic_function (for functions defined on superelliptic curves), superelliptic_form (for forms defined on superelliptic curves) and superelliptic_cech (for cech cocycles for the de Rham cohomology on superelliptic curves).
where $\omega_0$ is a form regular on $U_0$ (i.e. on the affine curve $y^m = f(x)$),
$\omega_{\infty}$ is a form regular on $U_{\infty}$, the affine curve containing the points at infinity (explicitly given by $w^{\delta} = g(v^M \cdot w^b)$, $g(x) = x^{\deg f} \cdot f(1/x)$, $\delta := GCD(m, \deg f)$, $br - am = \delta$, $M := m/\delta$) and $f$ is a function regular on $U_0 \cap U_{\infty}$ such that $\omega_0 - \omega_{\infty} = df$. See e.g. [Section 2 in article of Kock and Tait](https://arxiv.org/pdf/1709.03422.pdf). In order to access the arguments omega_0, f, omega_{\infty} of a cocyle *eta* we use the arguments *eta.omega0*, *eta.f*, *eta.omega8* respectively. Thus, let us check that the cocycle condition omega_0 - omega_{\infty} = df is satisfied for an exemplary cocycle:
```
eta = C.de_rham_basis()[-1] # we pick one of the forms in the de Rham basis of C
print(eta.omega0 - eta.omega8 == eta.f.diffn())
```
The module allows to compute the basis of of holomorphic differential forms:
```
print(C.holomorphic_differentials_basis())
```
One may also compute the coordinates of a given holomorphic differential form. On default,
the coordinates are computed with respect to *C.holomorphic_differentials_basis()*.
One may also give a basis as an optional argument. Note that this speeds up computation, since
the basis is not calculated several times.
```
omega = (2*C.y^2 - C.y + C.one)/C.y^3 * C.dx
print(omega.coordinates())
basis = C.holomorphic_differentials_basis()
print(omega.coordinates(basis = basis))
```
The method *expansion_at_infty()* allows to compute the Laurent expansion of a given function at a place at infinity.
The parameter *place* is optional. It is a number from 0 to $\delta - 1$, giving a place at infinity in which
the expansion should be computed.
```
print(omega.expansion_at_infty(place=0))
print(omega.expansion_at_infty(place=1))
```
One can check valuation of form/function at given place at infinity, using *valuation()* method.
This module allows to define $p$\-group covers of superelliptic curves in characteristic $p$ that
are **ramified over the points of infinity**. For now the following $p$\-groups are possible: $\mathbb Z/p^n$, $(\mathbb Z/p)^n$, $E(p^3)$ \(the Heisenberg group mod $p$ of order $p^3$\), $Q_8$ \(the Heisenberg group $8$\). We define now a $(\mathbb Z/3)^2$ cover of curve $C : y^2 = x^3 + x$, given by the equations $z_0^3 - z_0 = x^2 y$,
and is required to compute some parameters of the cover. Note that the functions f1, f2 **must be polynomials in x and y** so that AS
has ramification points at infinity.
Similarly, the are classes _as\_function, as\_form, as\_cech_ and one can write _AS.x, AS.dx_, etc. There are also methods _holomorphic\_differentials\_basis\(\)_, _de\_rham\_basis\(\)_, _coordinates\(\)_, _expansion\_at\_infty\(\)_, *valuation()* etc.
Note that some functions \(e.g. _holomorphic\_differential\_basis_\) have optional _threshold_ parameter. Increase it in case of problems.
In order to compute the group action of $(\mathbb Z/p)^n$ on a given function/form/cocycle, use *group_action()*, e.g.
Note that this won't work for large genus of AS, as it uses free Magma with limited input. You may however use it with argument *text=True*, to obtain Magma command on output.
print(AS.holo_polydifferentials_basis(2, threshold = 15)) #we increase the threshold if needed, see below
```
The class *as_symmetric_product_forms* may be used to define an element of $\textrm{Sym}^n \, H^0(\Omega_X)$. The command *as_symmetric_power_basis* returns a basis of $\textrm{Sym}^n \, H^0(\Omega_X)$.
The method *canonical_ideal* computes the elements of $\textrm{Sym}^n \, H^0(\Omega_X)$ that are in the kernel of the multiplication map with codomain in $H^0(\Omega_X^{\otimes n})$ (i.e. the n-th homogeneous part of the canonical ideal). The method *canonical_ideal_polynomials* computes the corresponding polynomials and *group_action_canonical_ideal* the matrices of the group action on the n-th homogeneous part of the canonical ideal.
Some of the above methods are also implemented for quaternion covers in characteristic $2$. Those are defined by the equations $z_0^2 + z_0 = f_0, z_1^2 + z_1 = f_1, z_2^2 + z_2 = f_2 + z_0f_0 + z_1 (f_0 + f_1)$. The arguments of *quaternion_cover* are: the covered superelliptic curve $C$ and the functions $f_0$, $f_1$, $f_2$ on $C$.
1.*Increase precision.* - Increase the *prec* argument of the curve.
1.*I haven't found all forms, only x of y* - Increase threshold when computing a basis.
1.*no 12 -th root; divide by 2* - when defining AS cover, one needs to compute roots of some numbers. This error means that a number is not in the field. You can either enlarge the base field, or divide one of the functions by given number and study the modified curve.
1.*unsupported operand parent(s) for %: 'The Infinity Ring' and 'The Infinity Ring'* - One of the power series turned out to be zero. Probably the AS cover that you've given is not connected (for example it is of the form $z_0^p - z_0 = f^p - f$).
