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
  • John Cremona (2017): implementation of Billerey’s algorithm to find all reducible primes

REFERENCES:

[Serre72]Jean-Pierre 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.
sage.schemes.elliptic_curves.gal_reps_number_field.Billerey_B_bound(E, max_l=200, num_l=8, small_prime_bound=0, debug=False)

Compute Billerey’s bound \(B\).

We compute \(B_l\) for \(l\) up to max_l (at most) until num_l nonzero values are found (at most). Return the list of primes dividing all \(B_l\) computed, excluding those dividing 6 or ramified or of bad reduction or less than small_prime_bound. If no non-zero values are found return [0].

INPUT:

  • E – an elliptic curve over a number field \(K\).
  • max_l (int, default 200) – maximum size of primes l to check.
  • num_l (int, default 8) – maximum number of primes l to check.
  • small_prime_bound (int, default 0) – remove primes less than this from the output.
  • debug (bool, default False) – if True prints details.

Note

The purpose of the small_prime_bound is that it is faster to deal with these using the local test; by ignoring them here, we enable the algorithm to terminate sooner when there are no large reducible primes, which is always the case in practice.

EXAMPLES:

sage: K = NumberField(x**2 - 29, 'a'); a = K.gen()
sage: E = EllipticCurve([1, 0, ((5 + a)/2)**2, 0, 0])
sage: from sage.schemes.elliptic_curves.gal_reps_number_field import Billerey_B_bound
sage: Billerey_B_bound(E)
[5]

If we do not use enough primes \(l\), extraneous primes will be included which are not reducible primes:

sage: Billerey_B_bound(E, num_l=6)
[5, 7]

Similarly if we do not use large enough primes \(l\):

sage: Billerey_B_bound(E, max_l=50, num_l=8)
[5, 7]
sage: Billerey_B_bound(E, max_l=100, num_l=8)
[5]

This curve does have a rational 5-isogeny:

sage: len(E.isogenies_prime_degree(5))
1
sage.schemes.elliptic_curves.gal_reps_number_field.Billerey_B_l(E, l, B=0)

Return Billerey’s \(B_l\), adapted from the definition in [Bil2011], after (9).

INPUT:

  • E – an elliptic curve over a number field \(K\)
  • l (int) – a rational prime
  • B (int) – 0 or LCM of previous \(B_l\): the prime-to-B part of this \(B_l\) is ignored.

EXAMPLES:

sage: K = NumberField(x**2 - 29, 'a'); a = K.gen()
sage: E = EllipticCurve([1, 0, ((5 + a)/2)**2, 0, 0])
sage: from sage.schemes.elliptic_curves.gal_reps_number_field import Billerey_B_l
sage: [Billerey_B_l(E,l) for l in primes(15)]
[1123077552537600,
227279663773903886745600,
0,
0,
269247154818492941287713746693964214802283882086400,
0]
sage.schemes.elliptic_curves.gal_reps_number_field.Billerey_P_l(E, l)

Return Billerey’s \(P_l^*\) as defined in [Bil2011], equation (9).

INPUT:

  • E – an elliptic curve over a number field \(K\)
  • l – a rational prime

EXAMPLES:

sage: K = NumberField(x**2 - 29, 'a'); a = K.gen()
sage: E = EllipticCurve([1, 0, ((5 + a)/2)**2, 0, 0])
sage: from sage.schemes.elliptic_curves.gal_reps_number_field import Billerey_P_l
sage: [Billerey_P_l(E,l) for l in primes(10)]
[x^2 + 8143*x + 16777216,
x^2 + 451358*x + 282429536481,
x^4 - 664299076*x^3 + 205155493652343750*x^2 - 39595310449600219726562500*x + 3552713678800500929355621337890625,
x^4 - 207302404*x^3 - 377423798538689366394*x^2 - 39715249826471656586987520004*x + 36703368217294125441230211032033660188801]
sage.schemes.elliptic_curves.gal_reps_number_field.Billerey_R_bound(E, max_l=200, num_l=8, small_prime_bound=None, debug=False)

Compute Billerey’s bound \(R\).

We compute \(R_q\) for \(q\) dividing primes \(\ell\) up to max_l (at most) until num_l nonzero values are found (at most). Return the list of primes dividing all R_q computed, excluding those dividing 6 or ramified or of bad reduction or less than small_prime_bound. If no non-zero values are found return [0].

INPUT:

  • E – an elliptic curve over a number field \(K\).
  • max_l (int, default 200) – maximum size of rational primes l for which the primes q above l are checked.
  • num_l (int, default 8) – maximum number of rational primes l for which the primes q above l are checked.
  • small_prime_bound (int, default 0) – remove primes less than this from the output.
  • debug (bool, default False) – if True prints details.

Note

The purpose of the small_prime_bound is that it is faster to deal with these using the local test; by ignoring them here, we enable the algorithm to terminate sooner when there are no large reducible primes, which is always the case in practice.

EXAMPLES:

sage: K = NumberField(x**2 - 29, 'a'); a = K.gen()
sage: E = EllipticCurve([1, 0, ((5 + a)/2)**2, 0, 0])
sage: from sage.schemes.elliptic_curves.gal_reps_number_field import Billerey_R_bound
sage: Billerey_R_bound(E)
[5]

We may get no bound at all if we do not use enough primes:

sage: Billerey_R_bound(E, max_l=2, debug=False)
[0]

Or we may get a bound but not a good one if we do not use enough primes:

sage: Billerey_R_bound(E, num_l=1, debug=False)
[5, 17, 67, 157]

In this case two primes is enough to restrict the set of possible reducible primes to just \(\{5\}\). This curve does have a rational 5-isogeny:

sage: Billerey_R_bound(E, num_l=2, debug=False)
[5]
sage: len(E.isogenies_prime_degree(5))
1
sage.schemes.elliptic_curves.gal_reps_number_field.Billerey_R_q(E, q, B=0)

Return Billerey’s \(R_q\), adapted from the definition in [Bil2011], Theorem 2.8.

INPUT:

  • E – an elliptic curve over a number field \(K\)
  • q – a prime ideal of \(K\)
  • B (int) – 0 or LCM of previous \(R_q\): the prime-to-B part of this \(R_q\) is ignored.

EXAMPLES:

sage: K = NumberField(x**2 - 29, 'a'); a = K.gen()
sage: E = EllipticCurve([1, 0, ((5 + a)/2)**2, 0, 0])
sage: from sage.schemes.elliptic_curves.gal_reps_number_field import Billerey_R_q
sage: [Billerey_R_q(E,K.prime_above(l)) for l in primes(10)]
[1123077552537600,
227279663773903886745600,
51956919562116960000000000000000,
252485933820556361829926400000000]
sage.schemes.elliptic_curves.gal_reps_number_field.Frobenius_filter(E, L, patience=100)

Determine which primes in L might have an image contained in a Borel subgroup, by checking of traces of Frobenius.

Note

This function will sometimes return primes for which the image is not contained in a Borel subgroup. This issue cannot always be fixed by increasing patience as it may be a result of a failure of a local-global principle for isogenies.

INPUT:

  • E – EllipticCurve - over a number field.
  • L – list - a list of prime numbers.
  • patience (int), default 100– a positive integer bounding the number of traces of Frobenius to use while trying to prove irreducibility.

OUTPUT:

  • list – The list of all primes \(\ell\) in L for which the mod \(\ell\) image might be contained in a Borel subgroup of \(GL_2(\mathbf{F}_{\ell})\).

EXAMPLES:

sage: E = EllipticCurve('11a1') # has a 5-isogeny
sage: sage.schemes.elliptic_curves.gal_reps_number_field.Frobenius_filter(E,primes(40))
[5]

Example to show that the output may contain primes where the representation is in fact reducible. Over \(\QQ\) the following is essentially the unique such example by [Sutherland12]:

sage: E = EllipticCurve_from_j(2268945/128)
sage: sage.schemes.elliptic_curves.gal_reps_number_field.Frobenius_filter(E, [7, 11])
[7]

This curve does possess a 7-isogeny modulo every prime of good reduction, but has no rational 7-isogeny:

sage: E.isogenies_prime_degree(7)
[]

A number field example:

sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([1+i, -i, i, -399-240*i,  2627+2869*i])
sage: sage.schemes.elliptic_curves.gal_reps_number_field.Frobenius_filter(E, primes(20))
[2, 3]

Here the curve really does possess isogenies of degrees 2 and 3:

sage: [len(E.isogenies_prime_degree(l)) for l in [2,3]]
[1, 1]
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 \(Aut(E[p]) = GL_2(\GF{p})\). 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)]
sage.schemes.elliptic_curves.gal_reps_number_field.reducible_primes_Billerey(E, num_l=None, max_l=None, verbose=False)

Return a finite set of primes \(\ell\) containing all those for which \(E\) has a \(K\)-rational ell-isogeny, where \(K\) is the base field of \(E\): i.e., the mod-\(\ell\) representation is irreducible for all \(\ell\) outside the set returned.

INPUT:

  • E – an elliptic curve defined over a number field \(K\).
  • max_l (int or None (default)) – the maximum prime \(\ell\) to use for the B-bound and R-bound. If None, a default value will be used.
  • num_l (int or None (default)) – the number of primes \(\ell\) to use for the B-bound and R-bound. If None, a default value will be used.

Note

If E has CM then [0] is returned. In this case use the function sage.schemes.elliptic_curves.isogeny_class.possible_isogeny_degrees

We first compute Billeray’s B_bound using at most num_l primes of size up to max_l. If that fails we compute Billeray’s R_bound using at most num_q primes of size up to max_q.

Provided that one of these methods succeeds in producing a finite list of primes we check these using a local condition, and finally test that the primes returned actually are reducible. Otherwise we return [0].

EXAMPLES:

sage: from sage.schemes.elliptic_curves.gal_reps_number_field import reducible_primes_Billerey
sage: K = NumberField(x**2 - 29, 'a'); a = K.gen()
sage: E = EllipticCurve([1, 0, ((5 + a)/2)**2, 0, 0])
sage: reducible_primes_Billerey(E)
[3, 5]
sage: K = NumberField(x**2 + 1, 'a')
sage: E = EllipticCurve_from_j(K(1728)) # CM over K
sage: reducible_primes_Billerey(E)
[0]
sage: E = EllipticCurve_from_j(K(0)) # CM but NOT over K
sage: reducible_primes_Billerey(E)
[2, 3]

An example where a prime is not reducible but passes the test:

sage: E = EllipticCurve_from_j(K(2268945/128)).global_minimal_model() # c.f. [Sutherland12]
sage: reducible_primes_Billerey(E)
[7]
sage.schemes.elliptic_curves.gal_reps_number_field.reducible_primes_naive(E, max_l=None, num_P=None, verbose=False)

Return locally reducible primes \(\ell\) up to max_l.

The list of primes \(\ell\) returned consists of all those up to max_l such that \(E\) mod \(P\) has an \(\ell\)-isogeny, where \(K\) is the base field of \(E\), for num_P primes \(P\) of \(K\). In most cases \(E\) then has a \(K\)-rational \(\ell\)-isogeny, but there are rare exceptions.

INPUT:

  • E – an elliptic curve defined over a number field \(K\)
  • max_l (int or None (default)) – the maximum prime \(\ell\) to test.
  • num_P (int or None (default)) – the number of primes \(P\) of \(K\) to use in testing each \(\ell\).

EXAMPLES:

sage: from sage.schemes.elliptic_curves.gal_reps_number_field import reducible_primes_naive
sage: K.<a> = NumberField(x^4 - 5*x^2 + 3)
sage: E = EllipticCurve(K, [a^2 - 2, -a^2 + 3, a^2 - 2, -50*a^2 + 35, 95*a^2 - 67])
sage: reducible_primes_naive(E,num_P=10)
[2, 5, 53, 173, 197, 241, 293, 317, 409, 557, 601, 653, 677, 769, 773, 797]
sage: reducible_primes_naive(E,num_P=15)
[2, 5, 197, 557, 653, 769]
sage: reducible_primes_naive(E,num_P=20)
[2, 5]
sage: reducible_primes_naive(E)
[2, 5]
sage: [phi.degree() for phi in E.isogenies_prime_degree()]
[2, 2, 2, 5]