74 lines
2.6 KiB
Python
74 lines
2.6 KiB
Python
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print('Remember to load init.sage!')
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print('Define the superelliptic curve C : y^4 = x^6 + 1 over GF(5)')
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F = GF(5)
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Rx.<x> = PolynomialRing(F)
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f = x^6 + 1
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C = superelliptic(f, 4)
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print(C)
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print('Is is smooth?')
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print(C.is_smooth())
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print('----------------------\n')
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print('Define the function x + 2y + 1 on our curve C:')
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Rxy.<x, y> = PolynomialRing(F, 2)
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fct1 = superelliptic_function(C, x + 2*y + 1)
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fct2 = C.x + 2*C.y + C.one
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print('In one way:', fct1, 'In another way:', fct2)
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print('----------------------\n')
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print('define the form omega = y * dx on C:')
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omega1 = superelliptic_form(C, y)
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omega2 = C.y * C.dx
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print('In one way:', omega1, 'In another way:', omega2)
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print('----------------------\n')
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print('The holomorphic differentials basis of C:')
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print(C.holomorphic_differentials_basis())
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print('Let us compute now coordinates of some differential form.')
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omega = (2*C.y^2 - C.y + C.one)/C.y^3 * C.dx
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print('First method:', omega.coordinates())
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basis = C.holomorphic_differentials_basis()
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print('Second method (faster):', omega.coordinates(basis = basis))
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print('Compute the Laurent expansion of omega, first at one place at infinity and then at the second:')
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print(omega.expansion_at_infty(place = 0))
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print(omega.expansion_at_infty(place = 1))
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print('----------------------\n')
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print('The basis of de Rham cohomology of C:')
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print(C.de_rham_basis())
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print('Elements of de Rham cohomology are Cech cocycles -- triples:')
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eta = C.de_rham_basis()[-1]
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print(eta)
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print('Let us check that the cocycle condition omega_0 - omega_{\infty} = df is satisfied:')
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print(eta.omega0 - eta.omega8 == eta.f.diffn())
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print('----------------------\n')
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#
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#
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F = GF(3)
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Rx.<x> = PolynomialRing(F)
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f = x^3 + x
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C = superelliptic(f, 2)
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f1 = C.x^2*C_super.y
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f2 = C.x^3
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AS = as_cover(C, [f1, f2], prec=1000)
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print(AS)
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print('----------------------\n')
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print('Compute the group action of $(\mathbb Z/p)^n$ on a form:')
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omega = AS.holomorphic_differentials_basis()[1]
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print('Form:', omega)
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print('Group action by [1, 0]:', omega.group_action([1, 0]))
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print('Group action by [0, 1]:', omega.group_action([0, 1]))
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print('Let us compute the matrices of the group action:')
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p = 3
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A, B = group_action_matrices_holo(AS)
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print(A, '\n', B)
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n = A.dimensions()[0]
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print('Let us check that they commute and are of order p')
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print(A*B == B*A)
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print(A^p == identity_matrix(n))
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print(B^p == identity_matrix(n))
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print('We decompose it into indecomposable $(\mathbb Z/p)^2$-modules:')
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print(magma_module_decomposition(A, B))
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print('----------------------\n')
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print('Let us look for magical elements:')
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z = AS.magical_element()
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print(z)
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print(z.valuation())
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