Galois representations attached to elliptic curves¶

If $E$ is an elliptic curve over a global field $K$ , currently only implemented over $\QQ$ .

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 a representation of $G_K$ on a free $\ZZ_p$ -module of rank $2$ . As $p$ varies the representations are compatible.

Currently sage can decide whether the Galois module $E[p]$ is reducible, i.e. if $E$ admits an isogeny of degree $p$ , and whether the image of the representation on $E[p]$ is surjective onto $Aut(E[p]) = GL_2(\mathbb{F}_p)$ .

EXAMPLES:

sage: E = EllipticCurve('196a1')
sage: rho = E.galois_representation()
sage: rho.is_irreducible(7)
True
sage: rho.is_reducible(3)
True
sage: rho.is_irreducible(2)
True
sage: rho.is_surjective(2)
False
sage: rho.is_surjective(3)
False
sage: rho.is_surjective(5)
True
sage: rho.reducible_primes()
[3]
sage: rho.non_surjective()
[2, 3]

For semi-stable curve it is known that the representation is surjective if and only if it is irreducible:

sage: E = EllipticCurve('11a1')
sage: rho = E.galois_representation()
sage: rho.non_surjective()
[5]
sage: rho.reducible_primes()
[5]

For cm curves it is not true that there are only finitely many primes for which the Galois representation mod p is surjective onto $GL_2(\mathbb{F}_p)$ :

sage: E = EllipticCurve('27a1')
sage: rho = E.galois_representation()
sage: rho.non_surjective()
[0]
sage: rho.reducible_primes()
[3]
sage: E.has_cm()
True

REFERENCES:

[Se1]

Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15 (1972), no. 4, 259–331.

[Se2]

Jean-Pierre Serre, Sur les représentations modulaires de degré 2 de Gal`(overlineQQ/QQ)`. Duke Math. J. 54 (1987), no. 1, 179–230.

[Co]	Alina Carmen Cojocaru, On the surjectivity of the Galois representations associated to non-CM elliptic curves. With an appendix by Ernst Kani. Canad. Math. Bull. 48 (2005), no. 1, 16–31.

AUTHORS:

chris wuthrich (02/10) - moved from ell_rational_field.py.

class sage.schemes.elliptic_curves.gal_reps.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 a representation of $G_K$ on a free $\ZZ_p$ -module of rank $2$ . As $p$ varies the representations are compatible.

EXAMPLES:

sage: rho = EllipticCurve('11a1').galois_representation()
sage: rho
Compatible family of Galois representations associated to the Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field

elliptic_curve()¶

The elliptic curve associated to this representation.

EXAMPLES:

sage: E = EllipticCurve('11a1')
sage: rho = E.galois_representation()
sage: rho.elliptic_curve() == E
True

is_irreducible(p)¶

Return True if the mod p representation is irreducible.

EXAMPLES:

sage: rho = EllipticCurve('37b').galois_representation()
sage: rho.is_irreducible(2)
True
sage: rho.is_irreducible(3)
False
sage: rho.is_reducible(2)
False
sage: rho.is_reducible(3)
True

is_reducible(p)¶

Return True if the mod-p representation is reducible. This is equivalent to the existence of an isogeny of degree $p$ from the elliptic curve.

INPUT:

p - a prime number

Note

The answer is cached.

EXAMPLES:

sage: rho = EllipticCurve('121a').galois_representation()
sage: rho.is_reducible(7)
False
sage: rho.is_reducible(11)
True
sage: EllipticCurve('11a').galois_representation().is_reducible(5)
True
sage: rho = EllipticCurve('11a2').galois_representation()
sage: rho.is_reducible(5)
True
sage: EllipticCurve('11a2').torsion_order()
1

is_surjective(p, A=1000)¶

Return True if the mod-p representation is surjective onto $Aut(E[p]) = GL_2(\mathbb{F}_p)$ .

False if it is not, or None if we were unable to determine whether it is or not.

Note

The answer is cached.

INPUT:

p - int (a prime number)
A - int (a bound on the number of a_p to use)

OUTPUT:

boolean. True if the mod-p representation is surjective: and False if (probably) not

EXAMPLES:

sage: rho = EllipticCurve('37b').galois_representation()
sage: rho.is_surjective(2)
True
sage: rho.is_surjective(3)
False

REMARKS:

If $p = 5$ then the mod-p representation is surjective if and only if the p-adic representation is surjective. When $p = 2, 3$ there are counterexamples. See a very recent paper of Elkies for more details when $p=3$ .
When p = 3 this function always gives the correct result irregardless of A, since it explicitly determines the p-division polynomial.

non_surjective(A=1000)¶

Returns a list of primes p such that the mod-p representation might not be surjective (this list usually contains 2, because of shortcomings of the algorithm). If $p$ is not in the returned list, then the mod-p representation is provably surjective.

By a theorem of Serre, there are only finitely many primes in this list, except when the curve has complex multiplication.

If the curve has CM, we simply return the sequence [0] and do no further computation.

INPUT:

A - an integer

OUTPUT:

list - if curve has CM, returns [0]. Otherwise, returns 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.

EXAMPLES:

sage: E = EllipticCurve([0, 0, 1, -38, 90])  # 361A
sage: E.galois_representation().non_surjective()   # CM curve
[0]

sage: E = EllipticCurve([0, -1, 1, 0, 0]) # X_1(11)
sage: E.galois_representation().non_surjective()
[5]

sage: E = EllipticCurve([0, 0, 1, -1, 0]) # 37A
sage: E.galois_representation().non_surjective()
[]

sage: E = EllipticCurve([0,-1,1,-2,-1])   # 141C
sage: E.galois_representation().non_surjective()
[13]

ALGORITHM: When $p = 3$ use division polynomials. For $5 = p = B$ , where $B$ is Cojocaru’s bound, use the results in Section 2 of

reducible_primes()¶

Returns a list of the primes $p$ such that the mod $p$ representation is reducible. For all other primes the representation is irreducible.

EXAMPLES:

sage: rho = EllipticCurve('225a').galois_representation()
sage: rho.reducible_primes()
[3]

Galois representations attached to elliptic curves¶

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