class as_function: def __init__(self, C, g): self.curve = C F = C.base_ring n = C.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:] RxyzQ = FractionField(Rxyz) self.function = RxyzQ(g) #self.function = as_reduction(AS, RxyzQ(g)) def __repr__(self): return str(self.function) def __add__(self, other): C = self.curve g1 = self.function g2 = other.function return as_function(C, g1 + g2) def __sub__(self, other): C = self.curve g1 = self.function g2 = other.function return as_function(C, g1 - g2) def __rmul__(self, constant): C = self.curve g = self.function return as_function(C, constant*g) def __mul__(self, other): if isinstance(other, as_function): C = self.curve g1 = self.function g2 = other.function return as_function(C, g1*g2) if isinstance(other, as_form): C = self.curve g1 = self.function g2 = other.form return as_form(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 F = C.base_ring x_series = C.x_series[i] y_series = C.y_series[i] z_series = C.z_series[i] n = C.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:] RxyzQ = FractionField(Rxyz) prec = C.prec Rt. = LaurentSeriesRing(F, default_prec=prec) g = self.function g = RxyzQ(g) sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)} return g.substitute(sub_list) def group_action(self, ZN_tuple): C = self.curve n = C.height F = C.base_ring 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:] RxyzQ = FractionField(Rxyz) sub_list = {x : x, y : y} | {z[j] : z[j]+ZN_tuple[j] for j in range(n)} g = self.function return as_function(C, g.substitute(sub_list)) def trace(self): C = self.curve C_super = C.quotient n = C.height F = C.base_ring 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:] RxyzQ = FractionField(Rxyz) result = as_function(C, 0) G = C.group for a in G: result += self.group_action(a) result = result.function Rxy. = PolynomialRing(F, 2) Qxy = FractionField(Rxy) result = as_reduction(AS, result) return superelliptic_function(C_super, Qxy(result)) def diffn(self): C = self.curve C_super = C.quotient n = C.height RxyzQ, Rxyz, x, y, z = C.fct_field fcts = C.functions f = self.function y_super = superelliptic_function(C_super, y) dy_super = y_super.diffn().form dz = [] for i in range(n): dfct = fcts[i].diffn().form dz += [-dfct] result = f.derivative(x) result += f.derivative(y)*dy_super for i in range(n): result += f.derivative(z[i])*dz[i] return as_form(C, result) def valuation(self, i = 0): '''Return valuation at i-th place at infinity.''' C = self.curve F = C.base_ring Rt. = LaurentSeriesRing(F) return Rt(self.expansion_at_infty(i)).valuation()