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135
README.md
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@ -1,4 +1,4 @@
# SAGEMATH module: superelliptic curves and their Artin-Schreier covers
# SAGEMATH module: superelliptic curves and their abelian p-group covers
## Basic information
@ -9,47 +9,160 @@ The main file is init.sage. In order to use it, type:
```sage: load('init.sage')```
The main two "packages" are intended for:
- superelliptic curves,
- $(\mathbb Z/p)^n$-covers of superelliptic curves.
See below and the file examples.sage for examples.
## Superelliptic curves
In order to define a superelliptic curve $C : y^4 = x^6 + 1$ over the finite field with 9 elements,
In order to define a superelliptic curve $C : y^4 = x^6 + 1$ over the finite field with 25 elements,
use the following commands:
```
F.<a> = GF(9, 'a')
F.<a> = GF(25, 'a')
Rx.<x> = PolynomialRing(F)
f = x^6 + 1
C = superelliptic(f, 4)
```
The class $C$ has an optional argument *prec*, which gives the precision of precomputed
expansions at infinity of the functions of the curve $C$. Note that curve of the form $y^m = f(x)$ has $\delta := GCD(\deg f, m)$
points at infinity and that $f(x)$ must be separable in order for $C$ to be smooth.
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).
For example, in order to define the function $x + y$ on our curve $C$ we can define it like this:
For example, in order to define the function $x + 2y + 1$ on our curve $C$ we can define it like this:
```
Rxy.<x, y> = PolynomialRing(F, 2)
fct = superelliptic_function(C, x + y)
fct = superelliptic_function(C, x + 2*y + 1)
```
or simpler:
```
fct = C.x + C.y
fct = C.x + 2*C.y + C.one
```
Similarly, in order to define the form $\omega = y \cdot dx$ we may use:
```
omega = superelliptic_form(C, y)
```
or simpler:
```
omega = C.y * C.dx
```
The cech cocycles are given as triples:
$$ (\omega_0, f, \omega_{\infty}), $$
## Troubleshooting
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.
One can also look for magical elements:
```
print(AS.magical_element())
```
## Common errors:
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 $z_0^p - z_0 = f^p - f$).
- precision
- threshold
- no root in the field
- basis -- coordinates.

2
as_covers/as_auxilliary.sage
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@ -1,4 +1,4 @@
def magmathis(A, B, text = False, prefix="", sufix=""):
def magma_module_decomposition(A, B, text = False, prefix="", sufix=""):
"""Find decomposition of Z/p^2-module given by matrices A, B into indecomposables using magma.
If text = True, print the command for Magma. Else - return the output of Magma free."""
q = parent(A).base_ring().order()

2
as_covers/as_cover_class.sage
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@ -145,7 +145,7 @@ class as_cover:
print("I haven't found all forms, only ", len(forms), " of ", self.genus())
return holomorphic_differentials_basis(self, threshold = threshold + 1)
if len(forms) > self.genus():
print("Increase precision.")
raise ValueError("Increase precision.")
return forms
def cartier_matrix(self, prec=50):

3
as_covers/group_action_matrices.sage
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@ -16,7 +16,8 @@ def group_action_matrices_holo(AS):
ei = n*[0]
ei[i] = 1
generators += [ei]
return group_action_matrices(AS.holomorphic_differentials_basis(), generators, basis = AS.holomorphic_differentials_basis())
basis = AS.holomorphic_differentials_basis()
return group_action_matrices(basis, generators, basis = basis)
def group_action_matrices_dR(AS, threshold=8):
n = AS.height

13
as_covers/tests/group_action_matrices_test.sage
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@ -1,17 +1,18 @@
p = 7
p = 3
m = 2
F = GF(p)
Rx.<x> = PolynomialRing(F)
f = x^3 + 1
f = x^3 + x
C_super = superelliptic(f, m)
Rxy.<x, y> = PolynomialRing(F, 2)
f1 = superelliptic_function(C_super, x^2*y)
f2 = superelliptic_function(C_super, x^3)
f1 = C_super.x^2*C_super.y
f2 = C_super.x^3
AS = as_cover(C_super, [f1, f2], prec=1000)
A, B = group_action_matrices_holo(AS)
n = A.dimensions()[0]
print(A*B == B*A)
print(A^p == identity_matrix(n))
print(B^p == identity_matrix(n))
print(B^p == identity_matrix(n))
print(magma_module_decomposition(A, B))

74
example.sage Normal file
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@ -0,0 +1,74 @@
print('Remember to load init.sage!')
print('Define the superelliptic curve C : y^4 = x^6 + 1 over GF(5)')
F = GF(5)
Rx.<x> = PolynomialRing(F)
f = x^6 + 1
C = superelliptic(f, 4)
print(C)
print('Is is smooth?')
print(C.is_smooth())
print('----------------------\n')
print('Define the function x + 2y + 1 on our curve C:')
Rxy.<x, y> = PolynomialRing(F, 2)
fct1 = superelliptic_function(C, x + 2*y + 1)
fct2 = C.x + 2*C.y + C.one
print('In one way:', fct1, 'In another way:', fct2)
print('----------------------\n')
print('define the form omega = y * dx on C:')
omega1 = superelliptic_form(C, y)
omega2 = C.y * C.dx
print('In one way:', omega1, 'In another way:', omega2)
print('----------------------\n')
print('The holomorphic differentials basis of C:')
print(C.holomorphic_differentials_basis())
print('Let us compute now coordinates of some differential form.')
omega = (2*C.y^2 - C.y + C.one)/C.y^3 * C.dx
print('First method:', omega.coordinates())
basis = C.holomorphic_differentials_basis()
print('Second method (faster):', omega.coordinates(basis = basis))
print('Compute the Laurent expansion of omega, first at one place at infinity and then at the second:')
print(omega.expansion_at_infty(place = 0))
print(omega.expansion_at_infty(place = 1))
print('----------------------\n')
print('The basis of de Rham cohomology of C:')
print(C.de_rham_basis())
print('Elements of de Rham cohomology are Cech cocycles -- triples:')
eta = C.de_rham_basis()[-1]
print(eta)
print('Let us check that the cocycle condition omega_0 - omega_{\infty} = df is satisfied:')
print(eta.omega0 - eta.omega8 == eta.f.diffn())
print('----------------------\n')
#
#
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)
print(AS)
print('----------------------\n')
print('Compute the group action of $(\mathbb Z/p)^n$ on a form:')
omega = AS.holomorphic_differentials_basis()[1]
print('Form:', omega)
print('Group action by [1, 0]:', omega.group_action([1, 0]))
print('Group action by [0, 1]:', omega.group_action([0, 1]))
print('Let us compute the matrices of the group action:')
p = 3
A, B = group_action_matrices_holo(AS)
print(A, '\n', B)
n = A.dimensions()[0]
print('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))
print('We decompose it into indecomposable $(\mathbb Z/p)^2$-modules:')
print(magma_module_decomposition(A, B))
print('----------------------\n')
print('Let us look for magical elements:')
z = AS.magical_element()
print(z)
print(z.valuation())

8
init.sage
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@ -27,10 +27,4 @@ load('auxilliaries/hensel.sage')
load('auxilliaries/linear_combination_polynomials.sage')
load('auxilliaries/laurent_analytic_part.sage')
##############
##############
#load('drafty/convert_superelliptic_into_AS.sage')
load('drafty/draft.sage')
#load('drafty/draft_klein_covers.sage')
#load('drafty/draft_klein_covers.sage')
#load('drafty/2gpcovers.sage')
load('drafty/pole_numbers.sage')
##############

9
superelliptic/superelliptic_form_class.sage
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@ -206,4 +206,11 @@ class superelliptic_form:
omega_regular = omega.regular_form()
C = omega.curve
p = C.characteristic
return (omega_regular.dx)^p*C.x^(p-1)*C.dx + (omega_regular.dy)^p*C.y^(p-1)*C.y.diffn()
return (omega_regular.dx)^p*C.x^(p-1)*C.dx + (omega_regular.dy)^p*C.y^(p-1)*C.y.diffn()
def valuation(self, place = 0):
'''Return valuation at i-th place at infinity.'''
C = self.curve
F = C.base_ring
Rt.<t> = LaurentSeriesRing(F)
return Rt(self.expansion_at_infty(place = place)).valuation()

9
superelliptic/superelliptic_function_class.sage
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@ -149,4 +149,11 @@ class superelliptic_function:
auxilliary_form = auxilliary_form.cartier()
auxilliary_form = C.x * auxilliary_form
auxilliary_form = auxilliary_form.form
return superelliptic_function(C, auxilliary_form)
return superelliptic_function(C, auxilliary_form)
def valuation(self, place = 0):
'''Return valuation at i-th place at infinity.'''
C = self.curve
F = C.base_ring
Rt.<t> = LaurentSeriesRing(F)
return Rt(self.expansion_at_infty(place = place)).valuation()