added manual and tests for polydifferential forms

2024-01-29 10:37:21 +00:00 · 2024-01-29 10:37:21 +00:00 · 4533eec102
parent a8e8dbbaa3
commit 4533eec102
5 changed files with 126 additions and 11 deletions
--- a/README.md
+++ b/README.md
@ -110,7 +110,7 @@ Rx.<x> = PolynomialRing(F)
 f = x^3 + x
 C = superelliptic(f, 2)

-f1 = C.x^2*C_super.y
+f1 = C.x^2*C.y
 f2 = C.x^3
 AS = as_cover(C, [f1, f2], prec=1000)
 ```
@ -149,7 +149,7 @@ One can decompose it into indecomposable $(\mathbb Z/p)^2$-modules, using
 print(magma_module_decomposition(A, B))
 ```

-Note that this won't work for large genus of AS, as it uses free Magma with limited input.
+Note that this won't work for large genus of AS, as it uses free Magma with limited input. You may however use it with argument *text=True*, to obtain Magma command on output.

 One can also look for magical elements:

@ -157,6 +157,71 @@ One can also look for magical elements:
 print(AS.magical_element())
 ```

+## Polydifferential forms on abelian covers
+
+For any $(\mathbb Z/p)^n$\-cover as above, one can define a polydifferential form (i.e. a section of $\Omega^{\otimes n}$) as follows:
+```
+F = GF(3)
+Rx.<x> = PolynomialRing(F)
+f = x^3 + x
+C = superelliptic(f, 2)
+AS = as_cover(C, [C.x^2*C.y, C.x^3], prec=1000)
+omega = as_polyform(AS.x^5, 3) # the first argument is a function on AS, the second is the multiplicity of polyform
+print(omega)
+print(omega.expansion_at_infty())
+```
+Using the command *holo_polydifferentials_basis* one may compute the basis of $H^0(\Omega^{\otimes n})$:
+```
+p = 5
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+C = superelliptic(x, 1)
+AS = as_cover(C, [C.x^8], prec = 200)
+print(AS.holo_polydifferentials_basis(2, threshold = 15)) #we increase the threshold if needed, see below
+```
+
+The class *as_symmetric_product_forms* may be used to define an element of $\textrm{Sym}^n \, H^0(\Omega_X)$. The command *as_symmetric_power_basis* returns a basis of $\textrm{Sym}^n \, H^0(\Omega_X)$.
+```
+p = 5
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+C = superelliptic(x, 1)
+AS = as_cover(C, [C.x^8], prec = 200)
+omega = as_symmetric_product_forms([[2, AS.dx, AS.z[0]*AS.dx], [-1, AS.x*AS.dx, AS.z[0]*AS.dx]])
+#note that the argument is a list of lists (first coefficient, then forms that occur in the given simple tensor)
+print(omega)
+print(as_symmetric_power_basis(AS, 2))
+```
+The method *canonical_ideal* computes the elements of $\textrm{Sym}^n \, H^0(\Omega_X)$ that are in the kernel of the multiplication map with codomain in $H^0(\Omega_X^{\otimes n})$ (i.e. the n-th homogeneous part of the canonical ideal). The method *canonical_ideal_polynomials* computes the corresponding polynomials and *group_action_canonical_ideal* the matrices of the group action on the n-th homogeneous part of the canonical ideal.
+```
+p = 5
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+C = superelliptic(x, 1)
+AS = as_cover(C, [C.x^8], prec = 200)
+print(AS.canonical_ideal(2, threshold = 15))
+print(AS.canonical_ideal_polynomials(2, threshold = 15))
+print(AS.group_action_canonical_ideal(2, threshold = 15))
+```
+
+## Quaternion covers
+
+Some of the above methods are also implemented for quaternion covers in characteristic $2$. Those are defined by the equations $z_0^2 + z_0 = f_0, z_1^2 + z_1 = f_1, z_2^2 + z_2 = f_2 + z_0f_0 + z_1 (f_0 + f_1)$. The arguments of *quaternion_cover* are: the covered superelliptic curve $C$ and the functions $f_0$, $f_1$, $f_2$ on $C$.
+```
+p = 2
+F.<a> = GF(p^2)
+Rx.<x> = PolynomialRing(F)
+C = superelliptic(x, 1)
+Q = quaternion_cover(C, [C.x^3, a*C.x^3, 0*C.x], prec = 300)
+print(Q)
+print(Q.genus())
+print(Q.holomorphic_differentials_basis())
+print(Q.canonical_ideal_polynomials(2))
+Qi, Qj = QuaternionGroup().gens()
+omega = Q.z[2]*Q.dx
+print(omega.group_action(Qi), omega.group_action(Qj))
+```
+
 ## Common errors:

 1. *Increase precision.* - Increase the *prec* argument of the curve. 
--- a/as_covers/as_polyforms.sage
+++ b/as_covers/as_polyforms.sage
@ -1,5 +1,6 @@
 class as_polyform:
    def __init__(self, form, mult):
+        '''Elements of H^0(Ω^⊗n). Usage: mult is n, form should be as_function.'''
        self.form = form
        self.curve = form.curve
        self.mult =  mult
@ -28,6 +29,7 @@ class as_polyform:
    
    
 def as_holo_polydifferentials_basis(AS, mult, threshold = 8):
+    '''Give the basis of H^0(Ω^⊗n) for n = mult.'''
    v = AS.dx.valuation()
    result = AS.at_most_poles(mult*v, threshold=threshold)
    result = [as_polyform(omega, mult) for omega in result]
@ -40,6 +42,7 @@ def as_holo_polydifferentials_basis(AS, mult, threshold = 8):
 as_cover.holo_polydifferentials_basis = as_holo_polydifferentials_basis

 def as_symmetric_power_basis(AS, n, threshold = 8):
+    '''Give the basis of H^0(Ω)^⊙n for n = mult.'''
    g = AS.genus()
    B0 = AS.holomorphic_differentials_basis(threshold=threshold)
    from itertools import product
@ -57,6 +60,7 @@ def as_symmetric_power_basis(AS, n, threshold = 8):
    return result

 def as_canonical_ideal(AS, n, threshold=8):
+    '''Return the n-th homogeneous part of the canonical ideal.'''
    B0 = AS.holomorphic_differentials_basis(threshold=threshold)
    F = AS.base_ring
    g = AS.genus()
@ -83,13 +87,14 @@ def as_canonical_ideal(AS, n, threshold=8):
 as_cover.canonical_ideal = as_canonical_ideal

 def as_canonical_ideal_polynomials(AS, n, threshold=8):
+    '''Return the polynomials defining n-th homogeneous part of the canonical ideal.'''
    return [a.polynomial() for a in AS.canonical_ideal(n, threshold=threshold)]

 as_cover.canonical_ideal_polynomials = as_canonical_ideal_polynomials

 class as_symmetric_product_forms:
    def __init__(self, forms_and_coeffs):
-        '''Elements of forms_and_coeffs are of the form [coeff, form1, ..., formn]'''
+        '''Elements of forms_and_coeffs are of the form [coeff, form_1, ..., form_n]'''
        self.n = len(forms_and_coeffs[0]) - 1
        forms_and_coeffs1 = []
        for atuple in forms_and_coeffs:
@ -148,6 +153,7 @@ class as_symmetric_product_forms:
            return as_symmetric_product_forms(aux_tuples)
    
    def multiply(self):
+        '''The map H^0(Ω)^⊙n ---> H^0(Ω^⊗n)'''
        n = self.n
        AS = self.curve
        RxyzQ, Rxyz, x, y, z = AS.fct_field
@ -163,6 +169,7 @@ class as_symmetric_product_forms:
        return result

    def polynomial(self):
+        '''Return the associated polynomial.'''
        AS = self.curve
        F = AS.base_ring
        g = AS.genus()
@ -193,6 +200,7 @@ def non_decreasing(L):
    return all(x<=y for x, y in zip(L, L[1:]))

 def as_matrices_group_action_canonical_ideal(AS, mult, threshold = 8):
+    '''Return the group action matrices for the n-th homogeneous part of the canonical ideal.'''
    K = as_canonical_ideal(AS, mult, threshold = threshold)
    n = AS.height
    F = AS.base_ring
@ -205,4 +213,6 @@ def as_matrices_group_action_canonical_ideal(AS, mult, threshold = 8):
        for i in range(r):
            M[i, :] = vector(linear_representation_polynomials(K_group_action_polynomials[i], K_polynomials))
        matrices += [M]
-    return matrices
+    return matrices
+
+as_cover.group_action_canonical_ideal = as_matrices_group_action_canonical_ideal
--- a/as_covers/tests/as_polyforms_test.sage
+++ b/as_covers/tests/as_polyforms_test.sage
@ -0,0 +1,27 @@
+p = 2
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+C = superelliptic(x, 1)
+AS = as_cover(C, [C.x^3, C.x^5], prec = 200)
+B = AS.holo_polydifferentials_basis(3)
+print(len(B) == (2 * 3 - 1) * (AS.genus() - 1)) #is the dimension as predicted by Riemann--Roch?
+I2 = AS.canonical_ideal_polynomials(2)
+R = I2[0].parent() #ring, to which the polynomials belong
+J2 = R.ideal(I2) #ideal defined by the set I
+print(J2.is_prime(), len(R.gens()) - J2.dimension() - 1)
+I3 = AS.canonical_ideal_polynomials(3)
+J3 = R.ideal(I2 + I3)
+print(J3.is_prime(), len(R.gens()) - J3.dimension() - 1)
+
+p = 5
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+C = superelliptic(x, 1)
+AS = as_cover(C, [C.x^8], prec = 200)
+I2 = AS.canonical_ideal_polynomials(2, threshold=15)
+R = I2[0].parent() #ring, to which the polynomials belong
+J2 = R.ideal(I2) #ideal defined by the set I
+print(J2.is_prime(), len(R.gens()) - J2.dimension() - 1)
+I3 = AS.canonical_ideal_polynomials(3, threshold=20)
+J3 = R.ideal(I2 + I3)
+print(J3.is_prime(), len(R.gens()) - J3.dimension() - 1)
--- a/quaternion_covers/tests/quaternion_covers_tests.sage
+++ b/quaternion_covers/tests/quaternion_covers_tests.sage
@ -0,0 +1,12 @@
+p = 2
+F.<a> = GF(p^2)
+Rx.<x> = PolynomialRing(F)
+C = superelliptic(x, 1)
+Q = quaternion_cover(C, [C.x^3, a*C.x^3, 0*C.x], prec = 300)
+print(Q.genus() == len(Q.holomorphic_differentials_basis()))
+I2 = Q.canonical_ideal_polynomials(2)
+R = I2[0].parent() #ring, to which the polynomials belong
+J2 = R.ideal(I2) #ideal defined by the set I
+print(J2.is_prime(), len(R.gens()) - J2.dimension() - 1)
+A, B = quaternion_matrices_group_action_canonical_ideal(Q, 2)
+print(A * B == B^3*A, A^2 == B^2, A^4 == identity_matrix(A.dimensions()[0]))
--- a/tests.sage
+++ b/tests.sage
@ -27,11 +27,12 @@
 #load('as_covers/tests/diffn_test.sage')
 #print("Cartier test:")
 #load('as_covers/tests/cartier_test.sage')
-print("Decomposition into g0, g8/ omega0, omega8 test:")
-load('superelliptic_drw/tests/decomposition_into_g0_g8_tests.sage')
-print("Auxilliary decomposition test:")
-load('superelliptic_drw/tests/auxilliary_decompositions_test.sage')
-print("Superelliptic de Rham-Witt test:")
-load('superelliptic_drw/tests/superelliptic_drw_tests.sage')
-
+print("Polydifferentials test:")
+load('as_covers/tests/as_polyforms_test.sage')
+#print("Decomposition into g0, g8/ omega0, omega8 test:")
+#load('superelliptic_drw/tests/decomposition_into_g0_g8_tests.sage')
+#print("Auxilliary decomposition test:")
+#load('superelliptic_drw/tests/auxilliary_decompositions_test.sage')
+#print("Superelliptic de Rham-Witt test:")
+#load('superelliptic_drw/tests/superelliptic_drw_tests.sage')