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uniformizer; skoki filtracji; grupy rozgalezienia

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jgarnek 2022-11-24 17:59:07 +00:00
parent 0cccfac7cf
commit 6edd5f9c40
15 changed files with 15860 additions and 41 deletions
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15611
sage/.run.term-0.term
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0
sage/as_covers/as_cover/uniformizer.sage Normal file
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78
sage/as_covers/as_cover_class.sage
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@ -8,6 +8,15 @@ class as_cover:
p = C.characteristic
self.characteristic = p
self.prec = prec
#group acting
n = self.height
from itertools import product
pr = [list(GF(p)) for _ in range(n)]
group = []
for a in product(*pr):
group += [a]
self.group = group
#########
f = C.polynomial
m = C.exponent
r = f.degree()
@ -229,6 +238,75 @@ class as_cover:
return forms
def uniformizer(self, i = 0):
'''Return uniformizer of curve self at i-th place at infinity.'''
p = self.characteristic
n = self.height
variable_names = 'x, y'
for j in range(n):
variable_names += ', z' + str(j)
Rxyz = PolynomialRing(F, n+2, variable_names)
x, y = Rxyz.gens()[:2]
z = Rxyz.gens()[2:]
fx = as_function(self, x)
z = [as_function(self, zi) for zi in z]
# We create a list of functions. We add there all variables...
list_of_fcts = [fx]+z
vfx = fx.valuation(i)
vz = [zi.valuation(i) for zi in z]
# Then we subtract powers of variables with the same valuation (so that 1/t^(kp) cancels) and add to this list.
for j1 in range(n):
for j2 in range(n):
if j1>j2:
a = gcd(vz[j1] , vz[j2])
vz1 = vz[j1]/a
vz2 = vz[j2]/a
for b in GF(p):
if (z[j1]^(vz2) - b*z[j2]^(vz1)).valuation(i) > (z[j2]^(vz1)).valuation(i):
list_of_fcts += [z[j1]^(vz2) - b*z[j2]^(vz1)]
for j1 in range(n):
a = gcd(vz[j1], vfx)
vzj = vz[j1] /a
vfx = vfx/a
for b in GF(p):
if (fx^(vzj) - b*z[j1]^(vfx)).valuation(i) > (z[j1]^(vfx)).valuation(i):
list_of_fcts += [fx^(vzj) - b*z[j1]^(vfx)]
#Finally, we check if on the list there are two elements with the same valuation.
for f1 in list_of_fcts:
for f2 in list_of_fcts:
d, a, b = xgcd(f1.valuation(i), f2.valuation(i))
if d == 1:
return f1^a*f2^b
raise ValueError("My method of generating fcts with relatively prime valuation failed.")
def ith_ramification_gp(self, i, place = 0):
'''Find ith ramification group at place at infty of nb place.'''
G = self.group
t = self.uniformizer(place)
Gi = [G[0]]
for g in G:
if g != G[0]:
tg = t.group_action(g)
v = (tg - t).valuation(place)
if v >= i+1:
Gi += [g]
return Gi
def ramification_jumps(self, place = 0):
'''Return list of lower ramification jumps at at place at infty of nb place.'''
G = self.group
ramification_jps = []
i = 0
while len(G) > 1:
Gi = self.ith_ramification_gp(i+1, place)
if len(Gi) < len(G):
ramification_jps += [i]
G = Gi
i+=1
return ramification_jps
def holomorphic_combinations(S):
"""Given a list S of pairs (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt."""
C_AS = S[0][0].curve

9
sage/as_covers/as_form_class.sage
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@ -122,9 +122,9 @@ class as_form:
z = Rxyz.gens()[2:]
RxyzQ = FractionField(Rxyz)
result = as_form(C, 0)
for i in range(0, p):
for j in range(0, p):
result += self.group_action([i, j])
G = C.group
for a in G:
result += self.group_action(a)
result = result.form
Rxy.<x, y> = PolynomialRing(F, 2)
Qxy = FractionField(Rxy)
@ -138,11 +138,12 @@ def artin_schreier_transform(power_series, prec = 10):
z^p - z = power_series, where z = 1/t_new^(jump) and express z in terms of the new uniformizer."""
correction = 0
F = power_series.parent().base()
p = F.characteristic()
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
RtQ = FractionField(Rt)
power_series = RtQ(power_series)
if power_series.valuation() == +Infinity:
return(0,0,t,0)
raise ValueError("Precision is too low.")
while(power_series.valuation() % p == 0 and power_series.valuation() < 0):
M = -power_series.valuation()/p
coeff = power_series.list()[0] #wspolczynnik a_(-p) w f_AS

22
sage/as_covers/as_function_class.sage
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@ -39,6 +39,17 @@ class as_function:
g2 = other.function
return as_function(C, g1*g2)
def __truediv__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
return as_function(C, g1/g2)
def __pow__(self, exponent):
C = self.curve
g1 = self.function
return as_function(C, g1^(exponent))
def expansion_at_infty(self, i = 0):
C = self.curve
delta = C.nb_of_pts_at_infty
@ -116,10 +127,15 @@ class as_function:
z = Rxyz.gens()[2:]
RxyzQ = FractionField(Rxyz)
result = as_function(C, 0)
for i in range(0, p):
for j in range(0, p):
result += self.group_action([i, j])
G = C.group
for a in G:
result += self.group_action(a)
result = result.function
Rxy.<x, y> = PolynomialRing(F, 2)
Qxy = FractionField(Rxy)
return superelliptic_function(C_super, Qxy(result))
def valuation(self, i = 0):
'''Return valuation at i-th place at infinity.'''
return self.expansion_at_infty(i).valuation()

8
sage/as_covers/dual_element.sage
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@ -2,7 +2,7 @@ def dual_elt(AS, zmag):
'''Find the trace dual of a given elt zmag in the function field of an Artin-Schreier cover AS.'''
p = AS.characteristic
n = AS.height
group_elts = [(i, j) for i in range(p) for j in range(p)]
G = AS.group
variable_names = 'x, y'
for i in range(n):
variable_names += ', z' + str(i)
@ -13,12 +13,12 @@ def dual_elt(AS, zmag):
M = matrix(RxyzQ, p^n, p^n)
for i in range(p^n):
for j in range(p^n):
M[i, j] = (zmag.group_action(group_elts[i])*zmag.group_action(group_elts[j])).trace2()
M[i, j] = (zmag.group_action(G[i])*zmag.group_action(G[j])).trace2()
main_det = M.determinant()
zvee = as_function(AS, 0)
for i in range(p^n):
Mprim = matrix(RxyzQ, M)
Mprim[:, i] = vector([(j == 0) for j in range(p^2)])
Mprim[:, i] = vector([(j == 0) for j in range(p^n)])
fi = Mprim.determinant()/main_det
zvee += fi*zmag.group_action(group_elts[i])
zvee += fi*zmag.group_action(G[i])
return zvee

9
sage/as_covers/ith_magical_component.sage
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@ -1,9 +1,8 @@
def ith_magical_component(omega, zvee, i):
'''Given a form omega on AS cover and normal basis element zmag, find the decomposition
sum_g g(zmag) omega_g and return omega_g, where g is the ith element of the group.'''
def ith_magical_component(omega, zvee, g):
'''Given a form omega on AS cover, element g of group AS.group and normal basis element zmag, find the decomposition
sum_g g(zmag) omega_g and return omega_g.'''
AS = omega.curve
p = AS.characteristic
group_elts = [(j1, j2) for j1 in range(p) for j2 in range(p)]
z_vee_fct = zvee.group_action(group_elts[i]).function
z_vee_fct = zvee.group_action(g).function
new_form = as_form(AS, z_vee_fct*omega.form)
return new_form.trace2()

20
sage/as_covers/tests/ith_ramification_gp_test.sage Normal file
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@ -0,0 +1,20 @@
p = 3
m = 1
F = GF(p)
Rx.<x> = PolynomialRing(F)
f = x
C_super = superelliptic(f, m)
Rxy.<x, y> = PolynomialRing(F, 2)
f1 = superelliptic_function(C_super, x^7)
f2 = superelliptic_function(C_super, x^4)
AS = as_cover(C_super, [f1, f2], prec=1000)
n = AS.height
d_test = (p^n - 1)
Gi = AS.group
i = 1
while(len(Gi) > 1):
Gi = AS.ith_ramification_gp(i)
d_test += len(Gi) - 1
i+=1
print(d_test == AS.exponent_of_different())

25
sage/as_covers/tests/ramification_jumps_test.sage Normal file
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@ -0,0 +1,25 @@
p = 5
m = 1
F = GF(p)
Rx.<x> = PolynomialRing(F)
f = x
C_super = superelliptic(f, m)
Rxy.<x, y> = PolynomialRing(F, 2)
f1 = superelliptic_function(C_super, x^2)
f2 = superelliptic_function(C_super, x^3)
AS = as_cover(C_super, [f1, f2], prec=500)
m = AS.jumps[0]
m1 = m[0]
m2 = m[1]
#We compute jumps from jumps in the tower of two Z/p-covers.
if m1 >= m2:
M1 = m2
M2 = m2 + p*(m1 - m2)
else:
M1 = m1
M2 = m2
theoretical_jumps = [M1, M2]
theoretical_jumps.sort()
print(theoretical_jumps == AS.ramification_jumps())

12
sage/as_covers/tests/uniformizer_test.sage Normal file
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@ -0,0 +1,12 @@
p = 3
m = 1
F = GF(p)
Rx.<x> = PolynomialRing(F)
f = x
C_super = superelliptic(f, m)
Rxy.<x, y> = PolynomialRing(F, 2)
f1 = superelliptic_function(C_super, x^7)
f2 = superelliptic_function(C_super, x^4)
AS = as_cover(C_super, [f1, f2], prec=1000)
print(AS.unifomizer().valuation() == 1)

37
sage/draft.sage → sage/drafty/draft.sage
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@ -1,4 +1,4 @@
p = 5
p = 3
m = 1
F = GF(p)
Rx.<x> = PolynomialRing(F)
@ -6,14 +6,16 @@ f = x
C_super = superelliptic(f, m)
Rxy.<x, y> = PolynomialRing(F, 2)
f1 = superelliptic_function(C_super, x^3)
f2 = superelliptic_function(C_super, x^11)
AS = as_cover(C_super, [f1, f2], prec=500)
f1 = superelliptic_function(C_super, x^2)
f2 = superelliptic_function(C_super, x^4)
f3 = superelliptic_function(C_super, x^5)
AS = as_cover(C_super, [f1, f2, f3], prec=1000)
print(AS)
zmag = AS.magical_element(threshold = 20)[0]
zvee = dual_elt(AS, zmag)
### DEFINE THE POLYNOMIALS
n = 2
n = AS.height
variable_names = 'x, y'
for i in range(n):
variable_names += ', z' + str(i)
@ -32,26 +34,23 @@ def val_of_components(omega, zvee):
result += [val]
return result
#############
print(zvee.expansion_at_infty().valuation())
G = AS.group
g = AS.genus()
print(AS.exponent_of_different_prim())
#for i in range(g):
# om = AS.holomorphic_differentials_basis(threshold = 30)[i]
# print(AS.exponent_of_different_prim(), val_of_components(om, zvee))
v_x = as_function(AS, x).expansion_at_infty().valuation()
v_z0 = as_function(AS, z[0]).expansion_at_infty().valuation()
v_z1 = as_function(AS, z[1]).expansion_at_infty().valuation()
n = 2
n = AS.height
v_z = [as_function(AS, z[i]).expansion_at_infty().valuation() for i in range(n)]
omega = AS.holomorphic_differentials_basis()[-16]
from itertools import product
pr = [list(range(p)) for _ in range(n)]
for i in range(0, 30):
for k in product(*pr):
v_w = i*v_x+k[0]*v_z0+k[1]*v_z1
v_w = i*v_x+ sum(k[i]*v_z[i] for i in range(n))
if (v_w < - AS.exponent_of_different_prim() + 10 and v_w > - AS.exponent_of_different_prim()):
w = as_function(AS, x^i * prod(z[i1]^(k[i1]) for i1 in range(n)))
tr_wz = (zvee*w).trace()
val = tr_wz.expansion_at_infty().valuation()
#val *= p^2
#print(val)
result = as_form(AS, 0)
for g in G:
component = ith_magical_component(omega, zvee, g).form
result += as_form(AS, w.group_action(g).function*component)
print(result.expansion_at_infty().valuation(), w.expansion_at_infty().valuation() - zmag.expansion_at_infty().valuation())

43
sage/drafty/draft2.sage Normal file
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@ -0,0 +1,43 @@
p = 3
m = 1
F = GF(p)
Rx.<x> = PolynomialRing(F)
f = x
C_super = superelliptic(f, m)
Rxy.<x, y> = PolynomialRing(F, 2)
f1 = superelliptic_function(C_super, x^7)
f2 = superelliptic_function(C_super, x^4)
AS = as_cover(C_super, [f1, f2], prec=1000)
print(AS.uniformizer())
def rep_jumps(AS, threshold = 30):
ile = 1
i = 1
result = []
flag = 0
while(flag == 0):
if len(AS.at_most_poles(i, threshold = threshold)) > ile:
ile = len(AS.at_most_poles(i, threshold = threshold))
result += [i]
if i%p != 0:
flag = 1
i+=1
return result
#print(rep_jumps(AS, threshold = 30))
def uniformizer(AS, threshold = 30):
p = AS.characteristic
n = AS.height
variable_names = 'x, y'
for i in range(n):
variable_names += ', z' + str(i)
Rxyz = PolynomialRing(F, n+2, variable_names)
x, y = Rxyz.gens()[:2]
z = Rxyz.gens()[2:]
zmag = AS.magical_element()[0]
v_x = as_function(AS, x).expansion_at_infty().valuation()
dprim = AS.exponent_of_different_prim()
d, a, b = xgcd(v_x, -dprim)
return as_function(AS, x^a*zmag.function^b)

7
sage/drafty/draft3.sage Normal file
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@ -0,0 +1,7 @@
Rx.<x> = PolynomialRing(GF(5))
C = superelliptic(x^3 + 1, 2)
Rxy.<x, y> = PolynomialRing(GF(5), 2)
f1 = superelliptic_function(C, x*y)
CAS = as_cover(C, [f1])
g = CAS.uniformizer()
AA = g

4
sage/init.sage
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@ -12,3 +12,7 @@ load('as_covers/combination_components.sage')
load('as_covers/group_action_matrices.sage')
load('auxilliaries/reverse.sage')
load('auxilliaries/hensel.sage')
##############
##############
load('drafty/draft2.sage')
#load('drafty/draft3.sage')

10
sage/tests.sage
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@ -4,5 +4,11 @@
#load('as_covers/tests/group_action_matrices_test.sage')
#print("dual_element_test:")
#load('as_covers/tests/dual_element_test.sage')
print("ith_component_test:")
load('as_covers/tests/ith_component_test.sage')
#print("ith_component_test:")
#load('as_covers/tests/ith_component_test.sage')
#print("ith ramification group test:")
#load('as_covers/tests/ith_ramification_gp_test.sage')
#print("uniformizer test:")
#load('as_covers/tests/uniformizer_test.sage')
print("ramification jumps test:")
load('as_covers/tests/ramification_jumps_test.sage')