16 KiB
16 KiB
Theory
Let $C : y^m = f(x)$. Then:
the basis of $H^0(C, \Omega_{C/k})$ is given by: $$x^{i-1} dx/y^j,$$ where $1 \le i \le r-1$, $1 \le j \le m-1$, $-mi + rj \ge \delta$ and $\delta := GCD(m, r)$, $r := \deg f$.
the above forms along with $$\lambda_{ij} = \left[ \left( \frac{\psi_{ij} , dx}{m x^{i+1} y^{m - j}}, \frac{-\phi_{ij} , dx}{m x^{i+1} y^{m - j}}, \frac{y^j}{x^i} \right) \right]$$ (where $s_{ij} = jx f'(x) - mi f(x)$, $\psi_{ij}(x) = s_{ij}^{\ge i+1}$, $\phi_{ij}(x) = s_{ij}^{< i+1}$) form a basis of $H^1_{dR}(C/K)$.
# The program computes the basis of holomorphic differentials of y^m = f(x) in char p.
# The coefficient j means that we compute the j-th eigenpart, i.e.
# forms y^j * f(x) dx. Output is [f(x), 0]
def baza_holo(m, f, j, p):
R.<x> = PolynomialRing(GF(p))
f = R(f)
r = f.degree()
delta = GCD(m, r)
baza = {}
k = 0
for i in range(1, r):
if (r*j - m*i >= delta):
baza[k] = [x^(i-1), R(0), j]
k = k+1
return baza# The program computes the basis of de Rham cohomology of y^m = f(x) in char p.
# We treat them as pairs [omega, f], where omega is regular on the affine part
# and omega - df is regular on the second atlas.
# The coefficient j means that we compute the j-th eigenpart, i.e.
# [f(x) dx/y^j, y^(m-j)*g(x)]. Output is [f(x), g(x)]
def baza_dr(m, f, j, p):
R.<x> = PolynomialRing(GF(p))
f = R(f)
r = f.degree()
delta = GCD(m, r)
baza = {}
holo = baza_holo(m, f, j, p)
for k in range(0, len(holo)):
baza[k] = holo[k]
k = len(baza)
for i in range(1, r):
if (r*(m-j) - m*i >= delta):
s = R(m-j)*R(x)*R(f.derivative()) - R(m)*R(i)*f
psi = R(obciecie(s, i, p))
baza[k] = [psi, R(m)/x^i, j]
k = k+1
return baza#auxiliary programs
def stopnie_bazy_holo(m, f, j, p):
baza = baza_holo(m, f, j, p)
stopnie = {}
for k in range(0, len(baza)):
stopnie[k] = baza[k][0].degree()
return stopnie
def stopnie_bazy_dr(m, f, j, p):
baza = baza_dr(m, f, j, p)
stopnie = {}
for k in range(0, len(baza)):
stopnie[k] = baza[k][0].degree()
return stopnie
def stopnie_drugiej_wspolrzednej_bazy_dr(m, f, j, p):
baza = baza_dr(m, f, j, p)
stopnie = {}
for k in range(0, len(baza)):
if baza[k][1] != 0:
stopnie[k] = baza[k][1].denominator().degree()
return stopnie
def obciecie(f, i, p):
R.<x> = PolynomialRing(GF(p))
f = R(f)
coeff = f.coefficients(sparse = false)
return sum(x^(j-i-1) * coeff[j] for j in range(i+1, f.degree() + 1))
#Any element [f dx, g] is represented as a combination of the basis vectors.
def zapis_w_bazie_dr(elt, m, f, p):
j = elt[2]
R.<x> = PolynomialRing(GF(p))
RR = FractionField(R)
f = R(f)
r = f.degree()
delta = GCD(m, r)
baza = baza_dr(m, f, j, p)
wymiar = len(baza)
zapis = vector([GF(p)(0) for i in baza])
stopnie = stopnie_bazy_dr(m, f, j, p)
inv_stopnie = {v: k for k, v in stopnie.items()}
stopnie_holo = stopnie_bazy_holo(m, f, j, p)
inv_stopnie_holo = {v: k for k, v in stopnie_holo.items()}
## zmiana
if elt[0]== 0 and elt[1] == 0:
return zapis
if elt[1] == 0:
elt[0] = R(elt[0])
d = elt[0].degree()
a = elt[0].coefficients(sparse = false)[d]
k = inv_stopnie_holo[d] #ktory element bazy jest stopnia d? ten o indeksie k
a1 = baza[k][0].coefficients(sparse = false)[d]
elt1 = [R(0),R(0),j]
elt1[0] = elt[0] - a/a1 * baza[k][0]
elt1[1] = R(0)
return zapis_w_bazie_dr(elt1, m, f, p) + vector([a/a1*GF(p)(i == k) for i in range(0, len(baza))])
g = elt[1]
a = wspolczynnik_wiodacy(g)
d = -stopien_roznica(g)
Rr = r/delta
Mm = m/delta
stopnie2 = stopnie_drugiej_wspolrzednej_bazy_dr(m, f, j, p)
inv_stopnie2 = {v: k for k, v in stopnie2.items()}
if (d not in stopnie2.values()):
if d> 0:
j1 = m-j
elt1 = [elt[0], RR(elt[1]) - a*1/R(x^d), j]
else:
d1 = -d
j1 = m-j
elt1 = [elt[0] - a*(j1*x^(d1) * f.derivative()/m + d1*f*x^(d1 - 1)), RR(elt[1]) - a*R(x^(d1)), j]
return zapis_w_bazie_dr(elt1, m, f, p)
k = inv_stopnie2[d]
b = wspolczynnik_wiodacy(baza[k][1])
elt1 = [R(0), R(0), j]
elt1[0] = elt[0] - a/b*baza[k][0]
elt1[1] = elt[1] - a/b*baza[k][1]
return zapis_w_bazie_dr(elt1, m, f, p) + vector([a*GF(p)(i == k) for i in range(0, len(baza))])
def zapis_w_bazie_holo(elt, m, f, p):
j = elt[2]
R.<x> = PolynomialRing(GF(p))
f = R(f)
r = f.degree()
delta = GCD(m, r)
baza = baza_holo(m, f, j, p)
wymiar = len(baza)
zapis = vector([GF(p)(0) for i in baza])
stopnie = stopnie_bazy_holo(m, f, j, p)
inv_stopnie = {v: k for k, v in stopnie.items()}
if elt[0] == 0:
return zapis
d = elt[0].degree()
a = elt[0].coefficients(sparse = false)[d]
k = inv_stopnie[d] #ktory element bazy jest stopnia d? ten o indeksie k
a1 = baza[k][0].coefficients(sparse = false)[d]
elt1 = [R(0),R(0), j]
elt1[0] = elt[0] - a/a1 * baza[k][0]
return zapis_w_bazie_holo(elt1, m, f, p) + vector([a/a1 * GF(p)(i == k) for i in range(0, len(baza))])
We have: $V(\omega, f) = (C(\omega), 0)$ and $F(\omega, f) = (0, f^p)$, where C denotes the Cartier operator. Moreover:
let $t = multord_m(p)$, $M := (p^t - 1)/m$. Then: $y^{p^t - 1} = f(x)^M$ and $1/y = f(x)^M/y^{p^t}$. Thus:
$$ C(P(x) , dx / y^j) = C(P(x) , f(x)^{M \cdot j} , dx /y^{p^t \cdot j}) = \frac{1}{y^{p^{t - 1} \cdot j}} C(P(x) , f(x)^{M \cdot j} , dx) = \frac{1}{y^{(p^{t - 1} \cdot j) , mod , m}} \cdot \frac{1}{f(x)^{[p^{t - 1} \cdot j/m]}} \cdot C(P(x) , f(x)^{M \cdot j} , dx)$$
def czesc_wielomianu(p, h):
R.<x> = PolynomialRing(GF(p))
h = R(h)
wynik = R(0)
for i in range(0, h.degree()+1):
if (i%p) == p-1:
potega = Integer((i-(p-1))/p)
wynik = wynik + Integer(h[i]) * x^(potega)
return wynik
def cartier_dr(p, m, f, elt, j): #Cartier na y^m = f dla elt = [forma rozniczkowa, fkcja]
R.<x> = PolynomialRing(GF(p))
f = R(f)
r = f.degree()
delta = GCD(m, r)
rzad = Integers(m)(p).multiplicative_order()
M = Integer((p^(rzad)-1)/m)
W = R(elt[0])
h = R(W*f^(M*j))
B = floor(p^(rzad-1)*j/m)
g = czesc_wielomianu(p, h)/f^B
jj = (p^(rzad-1)*j)%m
#jj = Integers(m)(j/p)
return [g, 0, jj] #jest to w czesci indeksowanej jj
def macierz_cartier_dr(p, m, f, j):
baza = baza_dr(m, f, j, p)
A = matrix(GF(p), len(baza), len(baza))
for k in range(0, len(baza)):
cart = cartier_dr(p, m, f, baza[k], j)
v = zapis_w_bazie_dr(cart, m, f, p)
A[k, :] = matrix(v)
return A.transpose()$F((\omega, P(x) \cdot y^j)) = (0, P(x)^p \cdot y^{p \cdot j}) = (0, P(x)^p \cdot f(x)^{[p \cdot j/m]} \cdot y^{(p \cdot j) , mod , m})$
def frobenius_dr(p, m, f, elt, j): #Cartier na y^m = f dla elt = [forma rozniczkowa, fkcja]
R.<x> = PolynomialRing(GF(p))
RR = FractionField(R)
f = R(f)
j1 = m-j
M = floor(j1*p/m)
return [0, f^M * RR(elt[1])^p, (j1*p)%m] #eigenspace = j1*p mod m
def macierz_frob_dr(p, m, f, j):
baza = baza_dr(m, f, j, p)
A = matrix(GF(p), len(baza), len(baza))
for k in range(0, len(baza)):
frob = frobenius_dr(p, m, f, baza[k], j)
v = zapis_w_bazie_dr(frob, m, f, p)
A[k, :] = matrix(v)
return A.transpose()
def wspolczynnik_wiodacy(f):
R.<x> = PolynomialRing(GF(p))
RR = FractionField(R)
f = RR(f)
f1 = f.numerator()
f2 = f.denominator()
d1 = f1.degree()
d2 = f2.degree()
a1 = f1.coefficients(sparse = false)[d1]
a2 = f2.coefficients(sparse = false)[d2]
return(a1/a2)
def stopien_roznica(f):
R.<x> = PolynomialRing(GF(p))
RR = FractionField(R)
f = RR(f)
f1 = f.numerator()
f2 = f.denominator()
d1 = f1.degree()
d2 = f2.degree()
return(d1 - d2)
def czy_w_de_rhamie(elt, m, f, j, p):
j1 = m - j
R.<x> = PolynomialRing(GF(p))
RR = FractionField(R)
f = R(f)
elt = [RR(elt[0]), RR(elt[1])]
auxiliary = elt[0] - j1/m*elt[1]*f.derivative() - f*elt[1].derivative()
deg = stopien_roznica(auxiliary)
r = f.degree()
delta = GCD(r, m)
Rr = r/delta
Mm = m/delta
return(j*Rr - deg*Mm >= 0)
def full_cartier(m, f, p):
R.<x> = PolynomialRing(GF(p))
f = R(f)
r = f.degree()
delta = GCD(m, r)
g = 1/2*((m-1)*(r-1) - delta)
print(g)
wymiary = [0]+[len(baza_holo(m, f, j, p)) for j in range(1, m)]
print(wymiary)
for j1 in range(1, m):
for j2 in range(1, m):
print(j1, j2)
print(matrix(GF(p), wymiary[j1], wymiary[j2]))
lista = [[matrix(GF(p), wymiary[j1], wymiary[j2]) for j1 in range(0, m)] for j2 in range(0, m)]
rzad = Integers(m)(p).multiplicative_order()
for j in range(1, m):
jj = (p^(rzad-1)*j)%m
print(j, jj)
print('wymiary', macierz_cartier_dr(p, m, f, j).dimensions(), wymiary[j], wymiary[jj])
lista[j][jj] = macierz_cartier_dr(p, m, f, j)
return lista
return block_matrix(lista)m = 4
p = 5
f = x^3 + x+2
lista = full_cartier(m, f, p)5/2 [0, 0, 1, 2] 1 1 [] 1 2 [] 1 3 [] 2 1 [] 2 2 [0] 2 3 [0 0] 3 1 [] 3 2 [0] [0] 3 3 [0 0] [0 0] 1 1 wymiary (2, 2) 0 0 2 2 wymiary (2, 2) 1 1 3 3 wymiary (2, 2) 2 2
lista[2][
[2 3] [0]
[], [], [0 0], [0]
]macierz_cartier_dr(p, m, f, 1)[0 0] [0 0]
baza_dr(m, f, 0, p){0: [3*x, 4/x, 0], 1: [4, 4/x^2, 0]}p = 5
R.<x> = PolynomialRing(GF(p))
f = x^3 + x + 2
m = 7
baza_dr(m, f, 3, p){0: [1, 0, 3], 1: [0, 2/x, 3]}