SAGEMATH module: superelliptic curves and their p-group covers

Usage

The main file is init.sage. In order to use it, navigate to the folder with the module (cd yourpath/DeRhamComputation) and type:

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. Also, folders with name tests contain tests, which might be useful for learning.

Superelliptic curves

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(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, we can define the function x + 2y + 1 on our curve C like this:

Rxy.<x, y> = PolynomialRing(F, 2)
fct = superelliptic_function(C, x + 2*y + 1)

or simpler:

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}), 

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. 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.

$p$-group covers of superelliptic curves

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, 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.y
f2 = C.x^3
AS = elementary_cover([f1, f2], prec=1000)

Similarly, one defines \mathbb Z/p^n, E(p^3) and $Q_8$-covers:

sage: F = GF(3)
sage: Rx.<x> = PolynomialRing(F)
sage: P1 = superelliptic(x, 1)
sage: AS = witt_cover([P1.x, P1.x^2], prec = 600)
sage: AS;
  (Z/p^n)-cover of Superelliptic curve with the equation y^1 = x over Finite Field of size 3 with the equations: 
  z0^p - z0 = (x)
  z1^p - z1 = -z0^7 + z0^5 + (x^2)
sage: AS = heisenberg_cover([P1.x^2, P1.x, P1.x], prec = 600)
sage: AS;
  (E(p^3))-cover of Superelliptic curve with the equation y^1 = x over Finite Field of size 3 with the equations: 
z0^p - z0 = (x^2)
z1^p - z1 = (x)
z2^p - z2 = z0*(x) - z1*(x) + (x)

Note that defining a 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.

G = AS.group() #p-group acting on curve
print(G.elts())
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
F = GF(3)
Rx.<x> = PolynomialRing(F)
f = x^3 + x
C = superelliptic(f, 2)
f1 = C.x^2*C.y
f2 = C.x^3
AS = elementary_cover([f1, f2], prec=1000)

A, B = AS.group_action_matrices_holo()
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. You may however use it with argument text=True, to obtain Magma command on output.

One can also look for magical elements:

print(AS.magical_element())

Polydifferential forms on abelian covers

For any $(\mathbb Z/p)^n$-cover as above, one can define a polydifferential form (i.e. a section of \Omega^{\otimes n}) as follows:

F = GF(3)
Rx.<x> = PolynomialRing(F)
f = x^3 + x
C = superelliptic(f, 2)
AS = as_cover(C, [C.x^2*C.y, C.x^3], prec=1000)
omega = as_polyform(AS.x^5, 3) # the first argument is a function on AS, the second is the multiplicity of polyform
print(omega)
print(omega.expansion_at_infty())

Using the command holo_polydifferentials_basis one may compute the basis of H^0(\Omega^{\otimes n}):

p = 5
F = GF(p)
Rx.<x> = PolynomialRing(F)
C = superelliptic(x, 1)
AS = as_cover(C, [C.x^8], prec = 200)
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).

p = 5
F = GF(p)
Rx.<x> = PolynomialRing(F)
C = superelliptic(x, 1)
AS = as_cover(C, [C.x^8], prec = 200)
omega = as_symmetric_product_forms([[2, AS.dx, AS.z[0]*AS.dx], [-1, AS.x*AS.dx, AS.z[0]*AS.dx]])
#note that the argument is a list of lists (first coefficient, then forms that occur in the given simple tensor)
print(omega)
print(as_symmetric_power_basis(AS, 2))

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.

p = 5
F = GF(p)
Rx.<x> = PolynomialRing(F)
C = superelliptic(x, 1)
AS = as_cover(C, [C.x^8], prec = 200)
print(AS.canonical_ideal(2, threshold = 15))
print(AS.canonical_ideal_polynomials(2, threshold = 15))
print(AS.group_action_canonical_ideal(2, threshold = 15))

Quaternion covers

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.

p = 2
F.<a> = GF(p^2)
Rx.<x> = PolynomialRing(F)
C = superelliptic(x, 1)
Q = quaternion_cover(C, [C.x^3, a*C.x^3, 0*C.x], prec = 300)
print(Q)
print(Q.genus())
print(Q.holomorphic_differentials_basis())
print(Q.canonical_ideal_polynomials(2))
Qi, Qj = QuaternionGroup().gens()
omega = Q.z[2]*Q.dx
print(omega.group_action(Qi), omega.group_action(Qj))

Common errors:

1. Increase precision. - Increase the prec argument of the curve.
2. I haven't found all forms, only x of y - Increase threshold when computing a basis.
3. 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.
4. 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).