fixed fiber problem for elementary covers
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README.md
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README.md
@ -97,11 +97,10 @@ print(omega.expansion_at_infty(place=1))
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One can check valuation of form/function at given place at infinity, using *valuation()* method.
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## Abelian covers of superelliptic curves
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## $p$-group covers of superelliptic curves
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This module allows to define $(\mathbb Z/p)^n$\-covers of superelliptic curves in characteristic $p$ that
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are **ramified over the points of infinity**.
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We define now a $(\mathbb Z/3)^2$ cover of curve $C : y^2 = x^3 + x$, given by the equations $z_0^3 - z_0 = x^2 y$,
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This module allows to define $p$\-group covers of superelliptic curves in characteristic $p$ that
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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 $z_0^3 - z_0 = x^2 y$,
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$z_1^3 - z_1 = x^3$.
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```
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@ -112,10 +111,29 @@ C = superelliptic(f, 2)
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f1 = C.x^2*C.y
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f2 = C.x^3
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AS = as_cover(C, [f1, f2], prec=1000)
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AS = elementary_cover([f1, f2], prec=1000)
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```
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Note that defining abelian cover may take quite a long time, since several parameters are computed. Again *prec* parameter is optional
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Similarly, one defines $\mathbb Z/p^n$, $E(p^3)$ and $Q_8$-covers:
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```
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sage: F = GF(3)
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sage: Rx.<x> = PolynomialRing(F)
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sage: P1 = superelliptic(x, 1)
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sage: AS = witt_cover([P1.x, P1.x^2], prec = 600)
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sage: AS;
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(Z/p^n)-cover of Superelliptic curve with the equation y^1 = x over Finite Field of size 3 with the equations:
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z0^p - z0 = (x)
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z1^p - z1 = -z0^7 + z0^5 + (x^2)
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sage: AS = heisenberg_cover([P1.x^2, P1.x, P1.x], prec = 600)
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sage: AS;
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(E(p^3))-cover of Superelliptic curve with the equation y^1 = x over Finite Field of size 3 with the equations:
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z0^p - z0 = (x^2)
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z1^p - z1 = (x)
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z2^p - z2 = z0*(x) - z1*(x) + (x)
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```
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Note that defining a cover may take quite a long time, since several parameters are computed. Again *prec* parameter is optional
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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
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has ramification points at infinity.
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@ -125,6 +143,8 @@ Note that some functions \(e.g. _holomorphic\_differential\_basis_\) have option
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In order to compute the group action of $(\mathbb Z/p)^n$ on a given function/form/cocycle, use *group_action()*, e.g.
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```
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G = AS.group() #p-group acting on curve
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print(G.elts())
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omega = AS.holomorphic_differentials_basis()[1]
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print(omega.group_action([1, 0])) #group action by element [1, 0]
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print(omega.group_action([0, 1])) #group action by element [0, 1]
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@ -134,7 +154,15 @@ In order to compute the matrices of the action, use *group_action_matrices_holo*
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```
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p = 3
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A, B = group_action_matrices_holo(AS)
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F = GF(3)
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Rx.<x> = PolynomialRing(F)
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f = x^3 + x
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C = superelliptic(f, 2)
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f1 = C.x^2*C.y
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f2 = C.x^3
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AS = elementary_cover([f1, f2], prec=1000)
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A, B = AS.group_action_matrices_holo()
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n = A.dimensions()[0]
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#Let us check that they commute and are of order p:
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print(A*B == B*A)
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@ -337,17 +337,18 @@ class as_cover:
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def fiber(self, place = 0):
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'Gives representatives for the quotient G/G_P for given place. Those are in bijection with the fiber.'
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result = [(0, 0, 0)]
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result = [AS.group.one]
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p = self.characteristic
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G = self.group
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H = self.stabilizer(place = place)
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for g in self.group.elts:
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flag = 1
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for v in result:
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if (G.elt(g)*(-G.elt(v))).as_tuple in H:
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flag = 0
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if flag:
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result += [g]
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if g != AS.group.one:
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flag = 1
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for v in result:
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if (G.elt(g)*(-G.elt(v))).as_tuple in H:
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flag = 0
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if flag:
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result += [g]
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return result
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def ith_ramification_gp(self, i, place = 0, quasiuniformizer = 0, threshold = 20):
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@ -622,16 +623,14 @@ class as_cover:
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for i in range(0, threshold*r):
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for j in range(0, m):
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for k in product(*pr):
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eta = heisenberg_form(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))/y^j)
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eta = as_form(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))/y^j)
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eta_exp = eta.expansion_at_infty()
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S += [(eta, eta_exp)]
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forms = holomorphic_combinations_forms(S, pole_orders[(0, (0, 0, 0))])
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print('iteration')
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forms = holomorphic_combinations_forms(S, pole_orders[(0, self.group.one)])
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for i in range(delta):
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for g in self.fiber(place = i):
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if i!=0 or g != self.group.one:
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print('iteration')
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forms = [(omega, omega.group_action(g).expansion_at_infty(place = i)) for omega in forms]
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forms = holomorphic_combinations_forms(forms, pole_orders[(i, g)])
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return forms
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@ -132,15 +132,18 @@ class as_function:
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aux = as_reduction(self.curve, self.function)
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return as_function(self.curve, aux)
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def trace(self, super=True):
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def trace(self, super=True, subgp = 0):
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C = self.curve
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C_super = C.quotient
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n = C.height
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F = C.base_ring
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if isinstance(subgp, Integer) or isinstance(subgp, int):
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subgp = C.group.elts
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else:
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super = False
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RxyzQ, Rxyz, x, y, z = C.fct_field
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result = as_function(C, 0)
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G = C.group.elts
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for a in G:
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for a in subgp:
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result += self.group_action(a)
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result = result.function
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Rxy.<x, y> = PolynomialRing(F, 2)
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@ -148,9 +151,9 @@ class as_function:
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result = as_reduction(C, result)
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if super:
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return superelliptic_function(C_super, Qxy(result))
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return as_function(AS, Qxy(result))
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RxyzQ, Rxyz, x, y, z = C.fct_field
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return as_function(C, RxyzQ(result))
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def coordinates(self, prec = 100, basis = 0):
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"Return coordinates in H^1(X, OX)."
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AS = self.curve
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@ -33,6 +33,10 @@ class group_elt:
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result_as_tuple = self.group.mult(self.as_tuple, other.as_tuple)
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return group_elt(result_as_tuple, self.group)
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def __rmul__(self, other):
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result_as_tuple = self.group.mult(self.as_tuple, other)
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return group_elt(result_as_tuple, self.group)
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def __neg__(self):
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result_as_tuple = self.group.inv(self.as_tuple)
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return group_elt(result_as_tuple, self.group)
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@ -72,10 +76,10 @@ def elementary_gp(p, n):
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from itertools import product
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elts = []
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for a in product(*pr):
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elts += [a]
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elts += [tuple(a)]
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one = elts[0]
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mult = lambda i1, i2: [(i1[j] + i2[j]) % p for j in range(n)]
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inv = lambda i: [(-i[j]) % p for j in range(n)]
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mult = lambda i1, i2: tuple([(i1[j] + i2[j]) % p for j in range(n)])
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inv = lambda i: tuple([(-i[j]) % p for j in range(n)])
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gens = []
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for i in range(n):
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e = n*[0]
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@ -1,6 +1,6 @@
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def ith_magical_component(omega, zvee, g):
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def ith_magical_component(omega, zvee, g, super=True):
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'''Given a form omega on AS cover, element g of group AS.group and normal basis element zmag, find the decomposition
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sum_g g(zmag) omega_g and return omega_g.'''
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z_vee_g = zvee.group_action(g)
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new_form = z_vee_g*omega
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return new_form.trace()
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return new_form.trace(super=super)
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