DeRhamComputation/sage/superelliptic/superelliptic_form_class.sage

class superelliptic_form:
    def __init__(self, C, g):
        F = C.base_ring
        Rxy.<x, y> = PolynomialRing(F, 2)
        Fxy = FractionField(Rxy)
        g = Fxy(reduction_form(C, g))
        self.form = g
        self.curve = C
    
    def __eq__(self, other):
        if self.reduce().form == other.reduce().form:
            return True
        return False
    
    def __add__(self, other):
        C = self.curve
        g1 = self.form
        g2 = other.form
        g = reduction(C, g1 + g2)
        return superelliptic_form(C, g)
    
    def __sub__(self, other):
        C = self.curve
        g1 = self.form
        g2 = other.form
        g = reduction(C, g1 - g2)
        return superelliptic_form(C, g)
    
    def __neg__(self):
        C = self.curve
        g = self.form
        return superelliptic_form(C, -g)
    
    def __repr__(self):
        g = self.form
        if len(str(g)) == 1:
            return str(g) + ' dx'
        return '('+str(g) + ') dx'

    def __rmul__(self, constant):
        C = self.curve
        omega = self.form
        return superelliptic_form(C, constant*omega)
    
    def cartier(self):
        '''Computes Cartier operator on the form. Idea: y^m = f(x) -> y^(p^r - 1) = f(x)^M, where r = ord_p(m),
        M = (p^r - 1)/m. Thus h(x)/y^j dx = h(x) f(x)^(M*j)/y^(p^r * j) dx. Thus C(h(x)/y^j dx) = 1/y^(p^(r-1)*j) C(h(x) f(x)^(M*j) dx).'''
        C = self.curve
        m = C.exponent
        p = C.characteristic
        f = C.polynomial
        F = C.base_ring
        Rx.<x> = PolynomialRing(F)
        Fx = FractionField(Rx)
        FxRy.<y> = PolynomialRing(Fx)
        Fxy = FractionField(FxRy)
        result = 0*C.dx
        mult_order = Integers(m)(p).multiplicative_order()
        M = Integer((p^(mult_order)-1)/m)
        
        for j in range(0, m):
            fct_j = self.jth_component(j)
            h = Fx(fct_j*f^(M*j))
            h_denom = h.denominator()
            h *= (h_denom)^(p)
            h = Rx(h)
            j1 = (p^(mult_order-1)*j)%m
            B = floor(p^(mult_order-1)*j/m)
            result += superelliptic_form(C, polynomial_part(p, h)/(f^B*y^(j1)*h_denom))
        return result   

    def serre_duality_pairing(self, fct, prec=20):
        '''Compute Serre duality pairing of the form with a cohomology class in H1(X, OX) represented by function fct.'''
        result = 0
        C = self.curve
        delta = C.nb_of_pts_at_infty
        for i in range(delta):
            result += (fct*self).expansion_at_infty(place=i, prec=prec)[-1]
        return -result
    
    def coordinates(self, basis = 0):
        """Find coordinates of the given holomorphic form self in terms of the basis forms in a list holo."""
        C = self.curve
        if basis == 0:
            basis = C.holomorphic_differentials_basis()
        Fxy, Rxy, x, y = C.fct_field
        # We need to have only polynomials to use monomial_coefficients in linear_representation_polynomials,
        # and sometimes basis elements have denominators. Thus we multiply by them.
        denom = LCM([denominator(omega.form) for omega in basis])
        basis = [denom*omega.form for omega in basis]
        self_with_no_denominator = denom*self.form
        return linear_representation_polynomials(Rxy(self_with_no_denominator), [Rxy(omega) for omega in basis])
    
    def jth_component(self, j):
        '''If self = sum_j h_j(x)/y^j dx, output is h_j(x).'''
        g = self.form
        C = self.curve
        m = C.exponent
        F = C.base_ring
        Rx.<x> = PolynomialRing(F)
        Fx = FractionField(Rx)
        FxRy.<y> = PolynomialRing(Fx)
        g = reduction(C, y^m*g)
        g = FxRy(g)
        if j == 0:
            return g.monomial_coefficient(y^(0))/C.polynomial
        return g.monomial_coefficient(y^(m-j))
    
    def is_regular_on_U0(self):
        C = self.curve
        F = C.base_ring
        m = C.exponent
        Rx.<x> = PolynomialRing(F)
        for j in range(0, m):
            if self.jth_component(j) not in Rx:
                return False
        return True
    
    def is_regular_on_Uinfty(self):
        C = self.curve
        F = C.base_ring
        m = C.exponent
        f = C.polynomial
        r = f.degree()
        delta = GCD(m, r)
        M = m/delta
        R = r/delta
        
        for j in range(1, m):
            A = self.jth_component(j)
            d = degree_of_rational_fctn(A, F)
            if(-d*M + j*R -(M+1)<0):
                return False
        return True

    def expansion_at_infty(self, place = 0, prec=10):
        g = self.form
        C = self.curve
        g = superelliptic_function(C, g)
        g = g.expansion_at_infty(place = place, prec=prec)
        x_series = C.x.expansion_at_infty(place = place, prec=prec)
        dx_series = x_series.derivative()
        return g*dx_series
    
    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.'''
        if pt in ZZ:
            return self.expansion_at_infty(place=pt, prec=prec)
        C = self.curve
        dx_series = C.x.expansion(pt = pt, prec=prec).derivative()
        aux_fct = superelliptic_function(C, self.form)
        return aux_fct.expansion(pt=pt, prec=prec)*dx_series
        
    def residue(self, place = 0, prec=30):
        return self.expansion_at_infty(place = place, prec=prec)[-1]
    
    def reduce(self):
        fct = self.form
        C = self.curve
        fct = reduction(C, fct)
        return superelliptic_form(C, fct)
    
    def reduce2(self):
        fct = self.form
        C = self.curve
        m = C.exponent
        F = C.base_ring
        Rxy.<x, y> = PolynomialRing(F, 2)
        Fxy = FractionField(Rxy)
        fct = reduction(C, Fxy(y^m*fct))
        return superelliptic_form(C, fct/y^m)
    
    def int(self):
        '''Computes an "integral" of a form dg. Idea: y^m = f(x) -> y^(p^r - 1) = f(x)^M, where r = ord_p(m),
        M = (p^r - 1)/m. Thus h(x)/y^j dx = h(x) f(x)^(M*j)/y^(p^r * j) dx. Thus int(h(x)/y^j dx) = 1/y^(p^(r-1)*j) int(h(x) f(x)^(M*j) dx).'''
        C = self.curve
        m = C.exponent
        p = C.characteristic
        f = C.polynomial
        F = C.base_ring
        Rx.<x> = PolynomialRing(F)
        Fx = FractionField(Rx)
        FxRy.<y> = PolynomialRing(Fx)
        Fxy = FractionField(FxRy)
        result = 0*C.x
        mult_order = Integers(m)(p).multiplicative_order()
        M = Integer((p^(mult_order)-1)/m)
        
        for j in range(0, m):
            fct_j = self.jth_component(j)
            h = Fx(fct_j*f^(M*j))
            h_denom = h.denominator()
            h *= (h_denom)^(p)
            h = Rx(h)
            j1 = (p^(mult_order)*j)%m
            B = floor(p^(mult_order)*j/m)
            result += superelliptic_function(C, h.integral()/(f^(B)*y^(j1)*h_denom^p))
        return result   
    
    def inv_cartier(omega):
        '''If omega is regular, return form eta such that Cartier(eta) = omega'''
        omega_regular = omega.regular_form()
        C = omega.curve
        p = C.characteristic
        return (omega_regular.dx)^p*C.x^(p-1)*C.dx + (omega_regular.dy)^p*C.y^(p-1)*C.y.diffn()