Galois representations for elliptic curves over number fields.¶

This file contains the code to compute for which primes the Galois representation attached to an elliptic curve (over an arbitrary number field) is surjective. The functions in this file are called by the is_surjective and non_surjective methods of an elliptic curve over a number field.

EXAMPLES:

sage: K = NumberField(x**2 - 29, 'a'); a = K.gen()
sage: E = EllipticCurve([1, 0, ((5 + a)/2)**2, 0, 0])
sage: rho = E.galois_representation()
sage: rho.is_surjective(29) # Cyclotomic character not surjective.
False
sage: rho.is_surjective(31) # See Section 5.10 of [Serre72].
True
sage: rho.non_surjective()  # long time (4s on sage.math, 2014)
[3, 5, 29]

sage: E = EllipticCurve_from_j(1728).change_ring(K) # CM
sage: E.galois_representation().non_surjective()  # long time (2s on sage.math, 2014)
[0]

AUTHORS:

Eric Larson (2012-05-28): initial version.
Eric Larson (2014-08-13): added isogeny_bound function.
John Cremona (2016, 2017): various efficiency improvements to _semistable_reducible_primes

REFERENCES:

[Serre72]

Serre. Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Inventiones mathematicae, 1972.

[Sutherland12]

Sutherland. A local-global principle for rational isogenies of prime degree. Journal de Théorie des Nombres de Bordeaux, 2012.

class sage.schemes.elliptic_curves.gal_reps_number_field.GaloisRepresentation(E)¶

Bases: sage.structure.sage_object.SageObject

The compatible family of Galois representation attached to an elliptic curve over a number field.

Given an elliptic curve \(E\) over a number field \(K\) and a rational prime number \(p\), the \(p^n\)-torsion \(E[p^n]\) points of \(E\) is a representation of the absolute Galois group \(G_K\) of \(K\). As \(n\) varies we obtain the Tate module \(T_p E\) which is a representation of \(G_K\) on a free \(\ZZ_p\)-module of rank \(2\). As \(p\) varies the representations are compatible.

EXAMPLES:

sage: K = NumberField(x**2 + 1, 'a')
sage: E = EllipticCurve('11a1').change_ring(K)
sage: rho = E.galois_representation()
sage: rho
Compatible family of Galois representations associated to the Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-10)*x + (-20) over Number Field in a with defining polynomial x^2 + 1

elliptic_curve()¶

Return the elliptic curve associated to this representation.

EXAMPLES:

sage: K = NumberField(x**2 + 1, 'a'); a = K.gen()
sage: E = EllipticCurve_from_j(a)
sage: rho = E.galois_representation()
sage: rho.elliptic_curve() == E
True

is_surjective(p, A=100)¶

Return True if the mod-p representation is (provably) surjective onto \(Aut(E[p]) = GL_2(\mathbb{F}_p)\). Return False if it is (probably) not.

INPUT:

p - int - a prime number.
A - int - a bound on the number of traces of Frobenius to use

while trying to prove surjectivity.

EXAMPLES:

sage: K = NumberField(x**2 - 29, 'a'); a = K.gen()
sage: E = EllipticCurve([1, 0, ((5 + a)/2)**2, 0, 0])
sage: rho = E.galois_representation()
sage: rho.is_surjective(29) # Cyclotomic character not surjective.
False
sage: rho.is_surjective(7) # See Section 5.10 of [Serre72].
True

If \(E\) is defined over \(\QQ\), then the exceptional primes for \(E_{/K}\) are the same as the exceptional primes for \(E\), except for those primes that are ramified in \(K/\QQ\) or are less than \([K:\QQ]\):

sage: K = NumberField(x**2 + 11, 'a')
sage: E = EllipticCurve([2, 14])
sage: rhoQQ = E.galois_representation()
sage: rhoK = E.change_ring(K).galois_representation()
sage: rhoQQ.is_surjective(2) == rhoK.is_surjective(2)
False
sage: rhoQQ.is_surjective(3) == rhoK.is_surjective(3)
True
sage: rhoQQ.is_surjective(5) == rhoK.is_surjective(5)
True

For CM curves, the mod-p representation is never surjective:

sage: K.<a> = NumberField(x^2-x+1)
sage: E = EllipticCurve([0,0,0,0,a])
sage: E.has_cm()
True
sage: rho = E.galois_representation()
sage: any(rho.is_surjective(p) for p in [2,3,5,7])
False

isogeny_bound(A=100)¶

Return a list of primes \(p\) including all primes for which the image of the mod-\(p\) representation is contained in a Borel.

Note

For the actual list of primes \(p\) at which the representation is reducible see reducible_primes().

INPUT:

A – int (a bound on the number of traces of Frobenius to use while trying to prove the mod-\(p\) representation is not contained in a Borel).

OUTPUT:

list - A list of primes which contains (but may not be equal to) all \(p\) for which the image of the mod-\(p\) representation is contained in a Borel subgroup. At any prime not in this list, the image is definitely not contained in a Borel. If E has \(CM\) defined over \(K\), the list [0] is returned.

EXAMPLES:

sage: K = NumberField(x**2 - 29, 'a'); a = K.gen()
sage: E = EllipticCurve([1, 0, ((5 + a)/2)**2, 0, 0])
sage: rho = E.galois_representation()
sage: rho.isogeny_bound() # See Section 5.10 of [Serre72].
[3, 5]
sage: K = NumberField(x**2 + 1, 'a')
sage: EllipticCurve_from_j(K(1728)).galois_representation().isogeny_bound() # CM over K
[0]
sage: EllipticCurve_from_j(K(0)).galois_representation().isogeny_bound() # CM NOT over K
[2, 3]
sage: E = EllipticCurve_from_j(K(2268945/128)) # c.f. [Sutherland12]
sage: E.galois_representation().isogeny_bound() # No 7-isogeny, but...
[7]

For curves with rational CM, there are infinitely many primes \(p\) for which the mod-\(p\) representation is reducible, and [0] is returned:

sage: K.<a> = NumberField(x^2-x+1)
sage: E = EllipticCurve([0,0,0,0,a])
sage: E.has_rational_cm()
True
sage: rho = E.galois_representation()
sage: rho.isogeny_bound()
[0]

An example (an elliptic curve with everywhere good reduction over an imaginary quadratic field with quite large discriminant), which failed until fixed at trac ticket #21776:

sage: K.<a> = NumberField(x^2 - x + 112941801)
sage: E = EllipticCurve([a+1,a-1,a,-23163076*a + 266044005933275,57560769602038*a - 836483958630700313803])
sage: E.conductor().norm()
1
sage: GR = E.galois_representation()
sage: GR.isogeny_bound()
[]

non_surjective(A=100)¶

Return a list of primes \(p\) including all primes for which the mod-\(p\) representation might not be surjective.

INPUT:

A – int (a bound on the number of traces of Frobenius to use while trying to prove surjectivity).

OUTPUT:

list – A list of primes where mod-\(p\) representation is very likely not surjective. At any prime not in this list, the representation is definitely surjective. If \(E\) has CM, the list [0] is returned.

EXAMPLES:

sage: K = NumberField(x**2 - 29, 'a'); a = K.gen()
sage: E = EllipticCurve([1, 0, ((5 + a)/2)**2, 0, 0])
sage: rho = E.galois_representation()
sage: rho.non_surjective() # See Section 5.10 of [Serre72].
[3, 5, 29]
sage: K = NumberField(x**2 + 3, 'a'); a = K.gen()
sage: E = EllipticCurve([0, -1, 1, -10, -20]).change_ring(K) # X_0(11)
sage: rho = E.galois_representation()
sage: rho.non_surjective()  # long time (4s on sage.math, 2014)
[3, 5]
sage: K = NumberField(x**2 + 1, 'a'); a = K.gen()
sage: E = EllipticCurve_from_j(1728).change_ring(K) # CM
sage: rho = E.galois_representation()
sage: rho.non_surjective()
[0]
sage: K = NumberField(x**2 - 5, 'a'); a = K.gen()
sage: E = EllipticCurve_from_j(146329141248*a - 327201914880) # CM
sage: rho = E.galois_representation()
sage: rho.non_surjective() # long time (3s on sage.math, 2014)
[0]

reducible_primes()¶

Return a list of primes \(p\) for which the mod-\(p\) representation is reducible, or [0] for CM curves.

OUTPUT:

list - A list of those primes \(p\) for which the mod-\(p\) representation is contained in a Borel subgroup, i.e. is reducible. If E has CM defined over K, the list [0] is returned (in this case the representation is reducible for infinitely many primes).

EXAMPLES:

sage: K = NumberField(x**2 - 29, 'a'); a = K.gen()
sage: E = EllipticCurve([1, 0, ((5 + a)/2)**2, 0, 0])
sage: rho = E.galois_representation()
sage: rho.isogeny_bound() # See Section 5.10 of [Serre72].
[3, 5]
sage: rho.reducible_primes()
[3, 5]

sage: K = NumberField(x**2 + 1, 'a')
sage: EllipticCurve_from_j(K(1728)).galois_representation().isogeny_bound() # CM over K
[0]
sage: EllipticCurve_from_j(K(0)).galois_representation().reducible_primes() # CM but NOT over K
[2, 3]
sage: E = EllipticCurve_from_j(K(2268945/128)) # c.f. [Sutherland12]
sage: rho = E.galois_representation()
sage: rho.isogeny_bound() # ... but there is no 7-isogeny ...
[7]
sage: rho.reducible_primes()
[]

For curves with rational CM, there are infinitely many primes \(p\) for which the mod-\(p\) representation is reducible, and [0] is returned:

sage: K.<a> = NumberField(x^2-x+1)
sage: E = EllipticCurve([0,0,0,0,a])
sage: E.has_rational_cm()
True
sage: rho = E.galois_representation()
sage: rho.reducible_primes()
[0]

sage.schemes.elliptic_curves.gal_reps_number_field.deg_one_primes_iter(K, principal_only=False)¶

Return an iterator over degree 1 primes of K.

INPUT:

K – a number field
principal_only – bool; if True, only yield principal primes

OUTPUT:

An iterator over degree 1 primes of \(K\) up to the given norm, optionally yielding only principal primes.

EXAMPLES:

sage: K.<a> = QuadraticField(-5)
sage: from sage.schemes.elliptic_curves.gal_reps_number_field import deg_one_primes_iter
sage: it = deg_one_primes_iter(K)
sage: [next(it) for _ in range(6)]
[Fractional ideal (2, a + 1),
 Fractional ideal (3, a + 1),
 Fractional ideal (3, a + 2),
 Fractional ideal (-a),
 Fractional ideal (7, a + 3),
 Fractional ideal (7, a + 4)]
sage: it = deg_one_primes_iter(K, True)
sage: [next(it) for _ in range(6)]
[Fractional ideal (-a),
 Fractional ideal (-2*a + 3),
 Fractional ideal (2*a + 3),
 Fractional ideal (a + 6),
 Fractional ideal (a - 6),
 Fractional ideal (-3*a + 4)]

Galois representations for elliptic curves over number fields.¶

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