Dual pairs of algebras over the rationals¶

Dual pairs of algebras over the rational numbers.

class dual_pairs.dual_pair_rational.DualPair_rational(*args: Any, **kwargs: Any)¶

A dual pair of algebras over the field \(\mathbf{Q}\).

TESTS:

sage: R.<x> = QQ[]
sage: from dual_pairs import FiniteFlatAlgebra, DualPair
sage: A = FiniteFlatAlgebra(QQ, [x, x, x^2 + 17])
sage: Phi = Matrix(QQ, [[1/4,  1/4,  1/2,   0],
....:                   [1/4,  1/4, -1/2,   0],
....:                   [1/2, -1/2,    0,   0],
....:                   [  0,    0,    0, -17]])
sage: D = DualPair(A, Phi)
sage: type(D)
<class 'dual_pairs.dual_pair_rational.DualPair_rational'>

dirichlet_character()¶

Return the Dirichlet character corresponding to the determinant of self.

EXAMPLES:

sage: from dual_pairs.dual_pair_import import dual_pair_import
sage: D = dual_pair_import('example_data/D4_mod_3.gp')
sage: chi = D.dirichlet_character(); chi
Dirichlet character modulo 39 of conductor 39 mapping 14 |--> 2, 28 |--> 2
sage: p = random_prime(1000)
sage: D.dirichlet_character()(p) == D.frobenius_charpoly(p).constant_coefficient()
True
sage: D = dual_pair_import('example_data/GL2_mod_3.gp')
sage: D.dirichlet_character()
Dirichlet character modulo 3 of conductor 3 mapping 2 |--> 2
sage: D = dual_pair_import('example_data/GL2_mod_5.gp')
sage: D.dirichlet_character()
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> 3

frobenius_traces(B=100)¶

Return the traces of Frobenius elements at all unramified primes below B.

EXAMPLES:

sage: from dual_pairs import FiniteFlatAlgebra, DualPair
sage: R.<t> = QQ[]
sage: A = FiniteFlatAlgebra(QQ, [t, t^3 - t + 1])
sage: phi = 1/4*Matrix([[1,  3, 0,  2],
....:                   [3, -3, 0, -2],
....:                   [0,  0, 4, -6],
....:                   [2, -2, -6, 0]])
sage: D = DualPair(A, phi)
sage: D.frobenius_traces()
[(3, 1),
 (5, 0),
 (7, 0),
 (11, 0),
 (13, 1),
 (17, 0),
 (19, 0),
 (29, 1),
 (31, 1),
 (37, 0),
 (41, 1),
 (43, 0),
 (47, 1),
 (53, 0),
 (59, 0),
 (61, 0),
 (67, 0),
 (71, 1),
 (73, 1),
 (79, 0),
 (83, 0),
 (89, 0),
 (97, 0)]

Verify numerically that the dual pair from GL2_mod_3.gp corresponds to the 3-torsion of the elliptic curve 11a3:

sage: from dual_pairs.dual_pair_import import dual_pair_import
sage: D = dual_pair_import('example_data/GL2_mod_3.gp')
sage: E = EllipticCurve('11a3')
sage: all(Mod(E.ap(p), 3) == t for p, t in D.frobenius_traces())
True

group_structure_algebraic_closure()¶

Return the group of points of self over an algebraic closure.

EXAMPLES:

sage: R.<x> = QQ[]
sage: from dual_pairs import FiniteFlatAlgebra, DualPair
sage: A = FiniteFlatAlgebra(QQ, [x, x, x^2 + 17])
sage: Phi = Matrix(QQ, [[1/4,  1/4,  1/2,   0],
....:                   [1/4,  1/4, -1/2,   0],
....:                   [1/2, -1/2,    0,   0],
....:                   [  0,    0,    0, -17]])
sage: D = DualPair(A, Phi)
sage: D.group_structure_algebraic_closure()[0]
Additive abelian group isomorphic to Z/2 + Z/2

ramified_primes()¶

Return the set of ramified primes of self.

EXAMPLES:

sage: R.<x> = QQ[]
sage: from dual_pairs import FiniteFlatAlgebra, DualPair
sage: A = FiniteFlatAlgebra(QQ, [x, x, x^2 + 17])
sage: Phi = Matrix(QQ, [[1/4,  1/4,  1/2,   0],
....:                   [1/4,  1/4, -1/2,   0],
....:                   [1/2, -1/2,    0,   0],
....:                   [  0,    0,    0, -17]])
sage: D = DualPair(A, Phi)
sage: D.ramified_primes()
{2, 17}

splitting_field(names)¶

Return a splitting field for self.

EXAMPLES:

sage: from dual_pairs import FiniteFlatAlgebra, DualPair
sage: R.<x> = QQ[]
sage: A = FiniteFlatAlgebra(QQ, [x, x^2 - 7])
sage: B = FiniteFlatAlgebra(QQ, [x, x^2 + 21])
sage: Phi = Matrix(QQ, [[1/3,  2/3,   0],
....:                   [2/3, -2/3,   0],
....:                   [  0,    0, 14]])
sage: D = DualPair(A, B, Phi)
sage: D.splitting_field('a')
Number Field in a with defining polynomial x^4 + 28*x^2 + 784

splitting_field_polynomial()¶

Return a defining polynomial for the splitting field of self.

EXAMPLES:

sage: from dual_pairs import FiniteFlatAlgebra, DualPair
sage: R.<x> = QQ[]
sage: A = FiniteFlatAlgebra(QQ, [x, x, x^2 + 17])
sage: Phi = Matrix(QQ, [[1/4,  1/4,  1/2,   0],
....:                   [1/4,  1/4, -1/2,   0],
....:                   [1/2, -1/2,    0,   0],
....:                   [  0,    0,    0, -17]])
sage: D = DualPair(A, Phi)
sage: D.splitting_field_polynomial()
x^2 + 17

torsor_class_group(S)¶

Return the group of isomorphism classes of torsors for self.

INPUT:

S – a finite set of primes

EXAMPLES:

sage: from dual_pairs import FiniteFlatAlgebra, DualPair
sage: R.<x> = QQ[]
sage: A = FiniteFlatAlgebra(QQ, [x, x^2 - 7])
sage: B = FiniteFlatAlgebra(QQ, [x, x^2 + 21])
sage: Phi = Matrix(QQ, [[1/3,  2/3,   0],
....:                   [2/3, -2/3,   0],
....:                   [  0,    0, 14]])
sage: D = DualPair(A, B, Phi)
sage: H = D.torsor_class_group([])
sage: H
Group of isomorphism classes of G-torsors where G is defined by
Dual pair of algebras over Rational Field
A = Finite flat algebra of degree 3 over Rational Field, product of:
Number Field in a0 with defining polynomial x
Number Field in a1 with defining polynomial x^2 - 7
B = Finite flat algebra of degree 3 over Rational Field, product of:
Number Field in a0 with defining polynomial x
Number Field in a1 with defining polynomial x^2 + 21
sage: H.group_structure()
(Trivial Abelian group,
 [],
 <function TorsorClassGroup.group_structure.<locals>.exp at 0x...>,
 <function TorsorClassGroup.group_structure.<locals>.log at 0x...>)

dual_pairs.dual_pair_rational.lift_to_prime(a)¶

Return the smallest prime in the residue class a.

EXAMPLES:

sage: from dual_pairs.dual_pair_rational import lift_to_prime
sage: [lift_to_prime(x) for x in Zmod(10) if x.is_unit()]
[11, 3, 7, 19]

Dual pairs of algebras over the rationals¶

Dual pairs of algebras

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