naprawiony problem z uniformizatorem w superelliptic; decomposition_omega8_hpdh dziala

sage/init.sage

@@ -18,12 +18,14 @@ load('superelliptic_drw/decomposition_into_g0_g8.sage')
 load('superelliptic_drw/superelliptic_witt.sage')
 load('superelliptic_drw/superelliptic_drw_form.sage')
 load('superelliptic_drw/superelliptic_drw_cech.sage')
+load('superelliptic_drw/superelliptic_drw_auxilliaries.sage')
 load('superelliptic_drw/regular_form.sage')
 load('superelliptic_drw/de_rham_witt_lift.sage')
 load('superelliptic_drw/automorphism.sage')
 load('auxilliaries/reverse.sage')
 load('auxilliaries/hensel.sage')
 load('auxilliaries/linear_combination_polynomials.sage')
+load('auxilliaries/laurent_analytic_part.sage')
 ##############
 ##############
 load('drafty/convert_superelliptic_into_AS.sage')

sage/superelliptic/superelliptic_class.sage

@@ -201,10 +201,22 @@ class superelliptic:
                     basis += [superelliptic_function(self, Fxy(m*y^(m-j)/x^i))]
         return basis
 
-#Auxilliary. Given a superelliptic curve C : y^m = f(x) and a polynomial g(x, y)
+    def uniformizer(self):
-#it replaces repeteadly all y^m's in g(x, y) by f(x). As a result
+        m = self.exponent
-#you obtain \sum_{i = 0}^{m-1} y^i g_i(x).
+        r = self.polynomial.degree()
+        delta, a, b = xgcd(m, r)
+        a = -a
+        M = m/delta
+        R = r/delta
+        while a<0:
+            a += R
+            b += M
+        return (C.x)^a/(C.y)^b
+
 def reduction(C, g):
+    '''Auxilliary. Given a superelliptic curve C : y^m = f(x) and a polynomial g(x, y)
+    it replaces repeteadly all y^m's in g(x, y) by f(x). As a result
+    you obtain \sum_{i = 0}^{m-1} y^i g_i(x).'''
     p = C.characteristic
     F = C.base_ring
     Rxy.<x, y> = PolynomialRing(F, 2)

sage/superelliptic/superelliptic_function_class.sage

@@ -122,13 +122,12 @@ class superelliptic_function:
         fct = RxyQ(fct)
         r = f.degree()
         delta, a, b = xgcd(m, r)
-        b = -b
+        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)
         ww = naive_hensel(gW, F, start = root_of_unity(F, delta)^place, prec = prec)

sage/superelliptic_drw/decomposition_into_g0_g8.sage

@@ -57,11 +57,4 @@ def decomposition_omega0_omega8(omega, prec=50):
     #Rx.<x> = PolynomialRing(F)
     #aux_fct = (g0.form)*y
     else:
-        raise ValueError("Something went wrong for "+str(omega) +". Result would be "+str(g0)+ " and " + str(g8))
+        raise ValueError("Something went wrong for "+str(omega) +". Result would be "+str(g0)+ " and " + str(g8))
-        
-        
-def decomposition_g0_g8_pth_power(fct):
-    '''Decompose fct as g0 - g8 + A^p, if possible. Output: (g0, g8, A).'''
-    coor = fct.coordinates()
-    C = fct.curve
-    return 0

sage/superelliptic_drw/superelliptic_drw_auxilliaries.sage

@@ -0,0 +1,49 @@
+def decomposition_g0_pth_power(fct):
+    '''Decompose fct as g0 + A^p, if possible. Output: (g0, A).'''
+    omega = fct.diffn().regular_form()
+    g0 = omega.int()
+    A = (fct - g0).pth_root()
+    return (g0, A)
+
+def decomposition_g0_p2th_power(fct):
+    '''Decompose fct as g0 + A^(p^2), if possible. Output: (g0, A).'''
+    g0, A = decomposition_g0_pth_power(fct)
+    A0, A1 = decomposition_g0_pth_power(A)
+    return (g0 + A0^p, A1)
+
+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)'''
+    omega1 = omega.cartier().cartier()
+    omega1 = omega1.inv_cartier().inv_cartier()
+    fct = (omega.cartier() - omega1.cartier()).int()
+    return (omega1, fct)
+
+def decomposition_omega8_hpdh(omega, prec = 50):
+    '''Decompose omega = (regular on U8) + h^(p-1) dh, so that Cartier(omega) = (regular on U8) + dh.
+    Result: (regular on U8, h)'''
+    C = omega.curve
+    g = C.genus()
+    Fxy, Rxy, x, y = C.fct_field
+    F = C.base_ring
+    p = C.characteristic
+    Rt.<t> = LaurentSeriesRing(F)
+    RT.<T> = PolynomialRing(F)
+    FT = FractionField(RT)
+    omega_analytic = FT(laurent_analytic_part(omega.expansion_at_infty(prec = prec))(t = T))
+    print('omega_analytic', omega_analytic)
+    Cv = C.uniformizer()
+    v = Fxy(Cv.function)
+    omega_analytic = Fxy(omega_analytic(T = v))
+    print('expansions', superelliptic_function(C, omega_analytic).expansion_at_infty(prec = prec), '\n', Cv.diffn().expansion_at_infty(prec = prec), 
+         '\n', (superelliptic_function(C, omega_analytic)*Cv.diffn()).expansion_at_infty(prec = prec))
+    omega_analytic = superelliptic_function(C, omega_analytic)*Cv.diffn()
+    print('omega_analytic.expansion_at_infty()', omega_analytic.expansion_at_infty(prec = prec))
+    print('omega_analytic', omega_analytic)
+    omega8 = omega - omega_analytic
+    print('omega8', omega8)
+    dh = omega.cartier() - omega8.cartier()
+    print('dh', dh)
+    h = dh.int()
+    print('omega8.expansion_at_infty()', omega8.expansion_at_infty(prec = prec))
+    return (omega8, h)

sage/superelliptic_drw/superelliptic_drw_cech.sage

@@ -67,30 +67,34 @@ class superelliptic_drw_cech:
         aux.omega0 -= aux_f_t_0.teichmuller().diffn()
         aux.omega8 = aux.omega0 - aux.f.diffn()
         #
+        # Now omega = (regular on U0) + h^(p-1) dh, so that Cartier(omega) = (regular on U0) + dh.
+        # We replace omega by regular on U0
         omega = aux.omega0.omega
-        omega1 = omega.cartier().cartier()
+        aux.omega0.omega, fct = decomposition_omega0_hpdh(aux.omega0.omega)
-        omega1 = omega1.inv_cartier().inv_cartier()
-        fct = (omega.cartier() - omega1.cartier()).int()
         aux.omega0.h2 += fct^p
-        aux.omega0.omega = omega1
+        # Now we have to ensure that aux.omega0.h2.function in Rxy...
-        if aux.omega0.h2.function in Rxy:
+        # In other words, we decompose h2 = (regular on U0) + A^(p^2).
-            aux.f -= aux.omega0.h2.verschiebung()
+        aux.omega0.h2 = decomposition_g0_p2th_power(aux.omega0.h2)[0]
-            aux.omega0.h2 = 0*C.x
+        # Now we can reduce: (... + dV(h2), V(f), ...) --> (..., V(f - h2), ...)
-            if aux.omega8.h2.expansion_at_infty().valuation() >= 0:
+        aux.f -= aux.omega0.h2.verschiebung()
-                aux.f += aux.omega8.h2.verschiebung()
+        aux.omega0.h2 = 0*C.x
-                aux.omega8.h2 = 0*C.x
+        
-                print('aux', aux)
+        if aux.omega8.h2.expansion_at_infty().valuation() >= 0:
-                # Now aux should be of the form (V(omega1), V(f), V(omega2))
+            aux.f += aux.omega8.h2.verschiebung()
-                # Thus aux = p*(Cartier(omega1), p-th_root(f), Cartier(omega2))
+            aux.omega8.h2 = 0*C.x
-                aux_divided_by_p = superelliptic_cech(C, aux.omega0.omega.cartier(), aux.f.f.pth_root())
+            print('aux', aux)
-                print('aux_divided_by_p', aux_divided_by_p)
+            # Now aux should be of the form (V(omega1), V(f), V(omega2))
-                print('is regular', aux_divided_by_p.omega0.is_regular_on_U0(), aux_divided_by_p.omega8.is_regular_on_Uinfty())
+            # Thus aux = p*(Cartier(omega1), p-th_root(f), Cartier(omega2))
-                print('aux.omega0.omega.cartier() - aux.f.f.pth_root().diffn() == aux.omega8.omega.cartier()', aux.omega0.omega.cartier() - aux.f.f.pth_root().diffn() == aux.omega8.omega.cartier())
+            aux_divided_by_p = superelliptic_cech(C, aux.omega0.omega.cartier(), aux.f.f.pth_root())
-                return aux_divided_by_p
+            print('aux_divided_by_p', aux_divided_by_p)
-            else:
+            print('is regular', aux_divided_by_p.omega0.is_regular_on_U0(), aux_divided_by_p.omega8.is_regular_on_Uinfty())
-                raise ValueError("aux.omega8.h2.expansion_at_infty().valuation() < 0:", aux.omega8.h2.expansion_at_infty())
+            print('aux.omega0.omega.cartier() - aux.f.f.pth_root().diffn() == aux.omega8.omega.cartier()', aux.omega0.omega.cartier() - aux.f.f.pth_root().diffn() == aux.omega8.omega.cartier())
+            return aux_divided_by_p
         else:
-            raise ValueError("aux.omega0.h2.function not in Rxy:", aux.omega0.h2.function)
+            print('aux.omega8.omega', aux.omega8.omega)
+            print('aux.omega8.h2', aux.omega8.h2)
+            print('second_patch(aux.omega8.h2.diffn()).is_regular_on_U0()', second_patch(aux.omega8.h2.diffn()).is_regular_on_U0())
+            raise ValueError("aux.omega8.h2.expansion_at_infty().valuation() < 0:", aux.omega8.h2.expansion_at_infty())
         
     def coordinates(self, basis = 0):
         C = self.curve
@@ -113,4 +117,4 @@ class superelliptic_drw_cech:
         print(self.omega0.r().is_regular_on_U0(), self.omega8.r().is_regular_on_Uinfty(), self.omega0.frobenius().is_regular_on_U0(),     self.omega8.frobenius().is_regular_on_Uinfty())
         eq1 = self.omega0.r().is_regular_on_U0() and self.omega8.r().is_regular_on_Uinfty()
         eq2 = self.omega0.frobenius().is_regular_on_U0() and self.omega8.frobenius().is_regular_on_Uinfty()
-        return eq1 and eq2
+        return eq1 and eq2