naprawione regular form; superelliptyczne maja C.x_series itd

sage/drafty/draft.sage

@@ -1,14 +1,14 @@
 p = 5
 m = 2
 F = GF(p)
-Rx.<x> = PolynomialRing(F)
+Rxx.<x> = PolynomialRing(F)
 #f = (x^3 - x)^3 + x^3 - x
 f = x^3 + x
 f1 = f(x = x^5 - x)
 C = superelliptic(f, m)
-C1 = superelliptic(f1, m)
+C1 = superelliptic(f1, m, prec = 500)
 B = C.crystalline_cohomology_basis(prec = 100, info = 1)
-B1 = C1.crystalline_cohomology_basis(prec = 500, info = 1)
+B1 = C1.crystalline_cohomology_basis(prec = 100, info = 1)
 
 def crystalline_matrix(C, prec = 50):
     B = C.crystalline_cohomology_basis(prec = prec)
@@ -20,6 +20,12 @@ def crystalline_matrix(C, prec = 50):
         M[i, :] = vector(autom(b).coordinates(basis = B))
     return M
 
+for b in B:
+    print(b.regular_form())
+    
+for b in B1:
+    print(b.regular_form())
+
 #M = crystalline_matrix(C, prec = 150)
 #print(M)
 #print(M^3)

sage/superelliptic/superelliptic_class.sage

@@ -1,7 +1,7 @@
 class superelliptic:
     """Class of a superelliptic curve. Given a polynomial f(x) with coefficient field F, it constructs
        the curve y^m = f(x)"""
-    def __init__(self, f, m):
+    def __init__(self, f, m, prec = 100):
         Rx = f.parent()
         x = Rx.gen()
         F = Rx.base()        
@@ -20,6 +20,31 @@ class superelliptic:
         self.y = superelliptic_function(self, Rxy(y))
         self.dx = superelliptic_form(self, Rxy(1))
         self.one = superelliptic_function(self, Rxy(1))
+        # We compute now expansions at infinity of x and y.
+        Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
+        RptW.<W> = PolynomialRing(Rt)
+        RptWQ = FractionField(RptW)
+        Rxy.<x, y> = PolynomialRing(F)
+        RxyQ = FractionField(Rxy)
+        delta, a, b = xgcd(m, r)
+        a = -a
+        M = m/delta
+        R = r/delta
+        while a<0:
+            a += R
+            b += M
+        g = (x^r*f(x = 1/x))
+        gW = RptWQ(g(x = t^M * W^b)) - W^(delta)
+        self.x_series = []
+        self.y_series = []
+        self.dx_series = []
+        for place in range(delta):
+            ww = naive_hensel(gW, F, start = root_of_unity(F, delta)^place, prec = prec)
+            xx = Rt(1/(t^M*ww^b))
+            yy = 1/(t^R*ww^a)
+            self.x_series += [xx]
+            self.y_series += [yy]
+            self.dx_series += [xx.derivative()]
         
     def __repr__(self):
         f = self.polynomial

sage/superelliptic/superelliptic_form_class.sage

@@ -137,10 +137,10 @@ class superelliptic_form:
         g = self.form
         C = self.curve
         g = superelliptic_function(C, g)
-        g = g.expansion_at_infty(place = place, prec=prec)
+        F = C.base_ring
-        x_series = C.x.expansion_at_infty(place = place, prec=prec)
+        Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
-        dx_series = x_series.derivative()
+        g = Rt(g.expansion_at_infty(place = place, prec=prec))
-        return g*dx_series
+        return g*C.dx_series[place]
     
     def expansion(self, pt, prec = 50):
         '''Expansion in the completed ring of the point pt. If pt is an integer, it means the corresponding place at infinity.'''

sage/superelliptic/superelliptic_function_class.sage

@@ -108,31 +108,11 @@ class superelliptic_function:
     
     def expansion_at_infty(self, place = 0, prec=20):
         C = self.curve
-        f = C.polynomial
-        m = C.exponent
-        F = C.base_ring
-        Rx.<x> = PolynomialRing(F)
-        f = Rx(f)
-        Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
-        RptW.<W> = PolynomialRing(Rt)
-        RptWQ = FractionField(RptW)
-        Rxy.<x, y> = PolynomialRing(F)
-        RxyQ = FractionField(Rxy)
         fct = self.function
-        fct = RxyQ(fct)
+        F = C.base_ring
-        r = f.degree()
+        Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
-        delta, a, b = xgcd(m, r)
+        xx = C.x_series[place]
-        a = -a
+        yy = C.y_series[place]
-        M = m/delta
-        R = r/delta
-        while a<0:
-            a += R
-            b += M
-        g = (x^r*f(x = 1/x))
-        gW = RptWQ(g(x = t^M * W^b)) - W^(delta)
-        ww = naive_hensel(gW, F, start = root_of_unity(F, delta)^place, prec = prec)
-        xx = Rt(1/(t^M*ww^b))
-        yy = 1/(t^R*ww^a)
         return Rt(fct(x = Rt(xx), y = Rt(yy)))
     
     def expansion(self, pt, prec = 50):

sage/superelliptic_drw/de_rham_witt_lift.sage

@@ -34,10 +34,10 @@ def de_rham_witt_lift(cech_class, prec = 50):
 
 def crystalline_cohomology_basis(self, prec = 50, info = 0):
     result = []
+    prec1 = prec
     for i, a in enumerate(self.de_rham_basis()):
         if info:
             print("Computing " + str(i) +". basis element")
-        prec1 = prec
         while True:
             try:
                 result += [de_rham_witt_lift(a, prec = prec1)]

sage/superelliptic_drw/regular_form.sage

@@ -74,12 +74,14 @@ class superelliptic_regular_drw_form:
         
 def regular_drw_form(omega):
     C = omega.curve
+    p = C.characteristic
     omega_aux = omega.r()
     omega_aux = omega_aux.regular_form()
     aux = omega - omega_aux.dx.teichmuller()*C.x.teichmuller().diffn() - omega_aux.dy.teichmuller()*C.y.teichmuller().diffn()
     aux.omega, fct = decomposition_omega0_hpdh(aux.omega)
     aux.h2 += fct^p
-    aux.h2 = decomposition_g0_p2th_power(aux.h2)[0]
+    aux.h2, A = decomposition_g0_pth_power(aux.h2)
+    aux.omega += (A.diffn()).inv_cartier()
     result = superelliptic_regular_drw_form(omega_aux.dx, omega_aux.dy, aux.omega.regular_form(), aux.h2)
     return result
 

sage/superelliptic_drw/superelliptic_drw_auxilliaries.sage

@@ -1,4 +1,8 @@
 def decomposition_g0_pth_power(fct):
+    C = fct.curve
+    Fxy, Rxy, xy, y = C.fct_field
+    if fct.function in Rxy:
+        return (fct, 0*C.x)
     '''Decompose fct as g0 + A^p, if possible. Output: (g0, A).'''
     omega = fct.diffn().regular_form()
     g0 = omega.int()
@@ -7,6 +11,8 @@ def decomposition_g0_pth_power(fct):
 
 def decomposition_g0_p2th_power(fct):
     '''Decompose fct as g0 + A^(p^2), if possible. Output: (g0, A).'''
+    C = fct.curve
+    p = C.characteristic
     g0, A = decomposition_g0_pth_power(fct)
     A0, A1 = decomposition_g0_pth_power(A)
     return (g0 + A0^p, A1)
@@ -14,6 +20,9 @@ def decomposition_g0_p2th_power(fct):
 def decomposition_omega0_hpdh(omega):
     '''Decompose omega = (regular on U0) + h^(p-1) dh, so that Cartier(omega) = (regular on U0) + dh.
     Result: (regular on U0, h)'''
+    C = omega.curve
+    if omega.is_regular_on_U0():
+        return (omega, 0*C.x)
     omega1 = omega.cartier().cartier()
     omega1 = omega1.inv_cartier().inv_cartier()
     fct = (omega.cartier() - omega1.cartier()).int()
@@ -27,6 +36,9 @@ def decomposition_omega8_hpdh(omega, prec = 50):
     Fxy, Rxy, x, y = C.fct_field
     F = C.base_ring
     p = C.characteristic
+    C = omega.curve
+    if omega.is_regular_on_Uinfty():
+        return (omega, 0*C.x)
     Rt.<t> = LaurentSeriesRing(F)
     omega_analytic = Rt(laurent_analytic_part(omega.expansion_at_infty(prec = prec)))
     Cv = C.uniformizer()
@@ -44,6 +56,8 @@ def decomposition_g8_pth_power(fct, prec = 50):
     F = C.base_ring
     Rt.<t> = LaurentSeriesRing(F)
     Fxy, Rxy, x, y = C.fct_field
+    if fct.expansion_at_infty().valuation() >= 0:
+        return (fct, 0*C.x)
     A = laurent_analytic_part(fct.expansion_at_infty(prec = prec))
     Cv = C.uniformizer()
     v = Cv.function

sage/superelliptic_drw/superelliptic_drw_cech.sage

@@ -56,13 +56,19 @@ class superelliptic_drw_cech:
         C = self.curve
         return superelliptic_cech(C, omega0.h1*C.dx, f.t)
     
-    def div_by_p(self):
+    def div_by_p(self, info = 0):
         '''Given a regular cocycle of the form (V(omega) + dV(h), [f] + V(t), ...), where [f] = 0 in H^1(X, OX),
         find de Rham cocycle (xi0, f, xi8) such that (V(omega) + dV(h), [f] + V(t), ...) = p*(xi0, f, xi8).'''
+        #
+        if info: print("Computing " + str(self) + " divided by p.")
+        #
         C = self.curve
         aux = self
         Fxy, Rxy, x, y = C.fct_field
         aux_f_t_0 = decomposition_g0_g8(aux.f.t, prec=50)[0]
+        #
+        if info: print("Computed decomposition_g0_g8 of self.f.t.")
+        #
         aux.f.t = 0*C.x
         aux.omega0 -= aux_f_t_0.teichmuller().diffn()
         aux.omega8 = aux.omega0 - aux.f.diffn()
@@ -72,9 +78,17 @@ class superelliptic_drw_cech:
         omega = aux.omega0.omega
         aux.omega0.omega, fct = decomposition_omega0_hpdh(aux.omega0.omega)
         aux.omega0.h2 += fct^p
+        #
+        if info: print("Computed decomposition_omega0_hpdh of self.omega0.omega.")
+        #
         # Now we have to ensure that aux.omega0.h2.function in Rxy...
-        # In other words, we decompose h2 = (regular on U0) + A^(p^2).
+        # In other words, we decompose h2 = (regular on U0) + A^p.
-        aux.omega0.h2 = decomposition_g0_p2th_power(aux.omega0.h2)[0]
+        # Then we replace h2 by (regular on U0) and omega by omega + inverse Cartier(dA)
+        aux.omega0.h2, A = decomposition_g0_pth_power(aux.omega0.h2)
+        aux.omega0.omega += (A.diffn()).inv_cartier()
+        #
+        if info: print("Computed decomposition_g0_p2th_power(aux.omega0.h2).")
+        #
         # Now we can reduce: (... + dV(h2), V(f), ...) --> (..., V(f - h2), ...)
         aux.f -= aux.omega0.h2.verschiebung()
         aux.omega0.h2 = 0*C.x
@@ -87,18 +101,21 @@ class superelliptic_drw_cech:
         aux_divided_by_p = superelliptic_cech(C, aux.omega0.omega.cartier(), aux.f.f.pth_root())
         return aux_divided_by_p
         
-    def coordinates(self, basis = 0, prec = 50):
+    def coordinates(self, basis = 0, prec = 50, info = 0):
+        if info: print("Computing coordinates of " + str(self))
         C = self.curve
         g = C.genus()
         coord_mod_p = self.r().coordinates()
+        if info: print("Computed coordinates mod p.")
         coord_lifted = [lift(a) for a in coord_mod_p]
         if basis == 0:
             basis = C.crystalline_cohomology_basis()
         aux = self
         for i, a in enumerate(basis):
             aux -= coord_lifted[i]*a
-        aux_divided_by_p = aux.div_by_p()
+        aux_divided_by_p = aux.div_by_p(info = info)
         coord_aux_divided_by_p = aux_divided_by_p.coordinates()
+        if info: print("Computed coordinates mod p of (self - lift)/p.")
         coord_aux_divided_by_p = [ZZ(a) for a in coord_aux_divided_by_p]
         coordinates = [ (coord_lifted[i] + p*coord_aux_divided_by_p[i])%p^2 for i in range(2*g)]
         return coordinates

sage/tests.sage

@@ -1,6 +1,6 @@
-load('init.sage')
+#load('init.sage')
-#print("Expansion at infty test:")
+print("Expansion at infty test:")
-#load('superelliptic/tests/expansion_at_infty.sage')
+load('superelliptic/tests/expansion_at_infty.sage')
 #print("superelliptic form coordinates test:")
 #load('superelliptic/tests/form_coordinates_test.sage')
 #print("p-th root test:")