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.
## Abelian covers of superelliptic curves
This module allows to define $(\mathbb Z/p)^n$-covers of superelliptic curves in characteristic $p$ that
are **ramified over the points of infinity**.
We define now a $(\mathbb Z/3)^2$ cover of curve $C : y^2 = x^3 + x$, given by the equations $z_0^3 - z_0 = x^2 * y$,
$z_1^3 - z_1 = x^3$.
```
F = GF(3)
Rx.<x> = PolynomialRing(F)
f = x^3 + x
C = superelliptic(f, 2)
f1 = C.x^2*C_super.y
f2 = C.x^3
AS = as_cover(C, [f1, f2], prec=1000)
```
Note that defining abelian cover may take quite a long time, since several parameters are computed. Again *prec* parameter is optional
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.
```
omega = AS.holomorphic_differentials_basis()[1]
print(omega.group_action([1, 0])) #group action by element [1, 0]
print(omega.group_action([0, 1])) #group action by element [0, 1]
```
In order to compute the matrices of the action, use *group_action_matrices_holo* and *group_action_matrices_dR*:
```
p = 3
A, B = group_action_matrices_holo(AS)
n = A.dimensions()[0]
#Let us check that they commute and are of order p:
print(A*B == B*A)
print(A^p == identity_matrix(n))
print(B^p == identity_matrix(n))
```
One can decompose it into indecomposable $(\mathbb Z/p)^2$-modules, using
*magma_module_decomposition*:
```
print(magma_module_decomposition(A, B))
```
Note that this won't work for large genus of AS, as it uses free Magma with limited input.
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 $z_0^p - z_0 = f^p - f$).
