The abelian variety Af/Q associated to a newform f is isogenous over Qbar to a power (Bf)r of some simple variety Bf, which is called a "building block". This section is concerned with computations related to the field of definition of Bf, and its endomorphism algebra.
The functions in this section take spaces of modular symbols, and must have sign +1. For instance, the modular symbols of level N, weight 2 and sign +1 are obtained from ModularSymbols(N,2,1). (These spaces have the same dimension as the corresponding spaces of modular forms.) Furthermore, the space of symbols is expected to be cuspidal, new and irreducible over Q (in other words, the corresponding space of cusp forms is spanned by a single Galois orbit of newforms).
Acknowledgement: This section was contributed by Jordi Quer. For more details of the theory, and some tables, see [Que09].
Let f=∑n=1^∞an qn∈S2new(N, ε) be a newform of weight 2, level N and Nebentypus ε. Let E be the number field generated by the coefficients an, and let F be the field generated by the numbers μp := (ap2/ε(p)) for primes p not dividing N.
The abelian variety Af/Q attached to f (that is, associated to the space spanned by f and its conjugates) is a variety of dimension equal to the degree [E:Q] with endomorphism algebra EndQ(Af) tensor Q isomorphic to E.
The variety Af has complex multiplication if there is a nontrivial character χ (necessarily of order 2) such that ap=χ(p) ap for all p not dividing N, and in this case the variety Bf is an elliptic curve with complex multiplication by the quadratic field fixed by the kernel of χ.
We assume for the rest of this section that Af has no complex multiplication. Then, the field F is totally real, and E/F is an abelian extension. E is totally real when ε is trivial, and otherwise E is a CM field. The endomorphism algebra End(Bf) tensor Q is a division algebra with centre F. It is either equal to F, and then dim(Bf)=[F:Q] and r=[E:F], or otherwise is a quaternion algebra over F, and then dim(Bf)=2[F:Q] and r=[E:F]/2.
For every element s∈GF there exists a unique primitive Dirichlet character χs, which only depends on the action of s on the field E, such that aps =χs(p) ap for all p not dividing N. The characters χs are called the inner twists of the form f.
Let Fδ be the extension of F generated by the square roots of the elements μp∈F (for p not dividing N). Note that E Fδ=E(Sqrt(ε)), the field obtained by adjoining square roots of the character values, which is either E or a quadratic extension of E. We may associate to each s ∈Gal(Fδ/F) a primitive quadratic Dirichlet character ψs, defined by Sqrt(μp)s=ψs(p) Sqrt(μp) (for p not dividing N). These are called the quadratic degree characters of the form f. They are related with the inner twists of f by the identity χs(p)=ψs(p)Sqrt(ε(p))s/Sqrt(ε(p)) .
Let KP/Q be the field fixed by the kernel of the homomorphism δ : GQ to F * /F * 2 that sends Frobp to μp for all p not dividing N with ap nonzero. (This is enough to determine the homomorphism, because the primes with ap=0 have density zero). This homomorphism is returned by DegreeMap (the syntax uses the notation below).
The Galois group Gal(KP/Q) is abelian of exponent 2; choose a basis σ1, ..., σr. Let ψ1, ..., ψr be the basis of Hom(Gal(KP/Q), Z/2) dual to the basis {σi}, and consider the ψi's as characters on GQ. Also let ti be a quadratic discriminant such that Q(Sqrt(ti)) ⊂KP is the field fixed by the kernel of ψi.
Denote by γε the element of Br(Q)[2] given by the two-cocycle sending σ, τ to Sqrt(ε(σ)) Sqrt(ε(τ)) (Sqrt(ε(στ))) - 1. Then the class in Br(F)[2] of the endomorphism algebra End0(Bf) is the restriction to F of γε ∏(ti, δ(σi)). This is computed by BrauerClass.
In general the smallest fields of definition up to isogeny for Bf are quadratic extensions of KP. It is sometimes possible to "descend" Bf (up to isogeny) to the field KP. The obstruction to this descent lies in Br(KP)[2], and is given by the restriction of γε to KP. This is computed by ObstructionDescentBuildingBlock.
A sequence containing the irreducible subspaces of the modular symbols of weight k and Nebentypus character ε corresponding to non-CM newforms for which the degree [F:Q] of the centre of the endomorphism algebra (of the associated abelian variety) is in the sequence degrees.
Proof: BoolElt Default: false
Return true if and only if the modular abelian variety attached to the given space of modular symbols has complex multiplication. When the level is larger than 100, an unproved bound is used in the computation, unless Proof is set to true.
Proof: BoolElt Default: false
The inner twists of the newform f = ∑an qn corresponding to the given space of modular symbols, or to the given modular abelian variety. This should be irreducible over Q, and f should not have CM, and only spaces of modular symbols with sign +1 are accepted.The inner twists are Dirichlet characters satisfying χs(p) = aps/ap (for primes p not dividing the level), where s is in the absolute Galois group of Q.
Warning: when the level is larger than 100, a non-rigorous bound is used in the computation, unless Proof is set to true. Even in that case, the returned twists are only checked to be inner twists up to precision 10 - 5.
Proof: BoolElt Default: false
The homomorphism δ : GQ to F * /F * 2 (as defined in the introduction), attached to the given space M of modular symbols (which should be new, irreducible over Q, and have sign +1). The second object returned is F. The first object returned is a sequence of tuples <ti, fi>, where ti ∈Q are quadratic discriminants and fi ∈F. The ti determine a basis σi of Gal(KP/Q) as in the introduction, and δ is the map sending σi to fi.
Given a space of modular symbols M corresponding to a newform f, this function computes the Brauer class of the endomorphism algebra of the associated abelian variety Af (or motive Mf). The endomorphism algebra is either a number field F (in which case an empty sequence is returned), or a quaternion algebra over a number field F. The corresponding class in the Brauer group Br(F)[2] is specified by returning the sequence of places of F that ramify in the quaternion algebra.The given space M should be irreducible over Q, and have sign +1, and f should not have CM.
Given a space of modular symbols M corresponding to a newform f, this function computes the obstruction to "descending" the building block Bf to the field KP (for definitions, see the introduction). The obstruction to the existence of a building block over KP isogenous to Bf is an element of the Brauer group Br(KP)[2]. (More precisely, for any field L there exists such a building block over L if and only if L is an extension of KP over which this Brauer element splits.) The Brauer class is specified by returning the sequence of places of KP where the class is not locally trivial.The given space M should be irreducible over Q, and have sign +1, and f should not have CM.
The lowest level where a nontrivial obstruction occurs is 28, with a character of order 6. We find that the space of modular symbols (with sign 1) for this character has dimension 2 over the field of character values.
> Chi28 := FullDirichletGroup(28);
> chi := Chi28.1*Chi28.2;
> Chi28;
Group of Dirichlet characters of modulus 28
over Cyclotomic Field of order 6 and degree 2
> Order(Chi28), Order(chi);
12 6
> M28chi := CuspidalSubspace( ModularSymbols(chi, 2, 1));
> M28chi;
Modular symbols space of level 28, weight 2, character $.1*$.2, and dimension
2 over Cyclotomic Field of order 6 and degree 2
> qEigenform(M28chi);
q + (1/3*(-zeta_6 - 1)*a - zeta_6)*q^2 + a*q^3 + 1/3*(4*zeta_6 - 2)*a*q^4 +
(zeta_6 - 2)*q^5 + (-zeta_6*a + (2*zeta_6 - 1))*q^6 + 1/3*(-zeta_6 - 4)*a*q^7
+ O(q^8)
> Parent(Coefficients(qEigenform(M28chi))[2]);
Univariate Quotient Polynomial Algebra in a
over Cyclotomic Field of order 6 and degree 2
with modulus a^2 + 3*zeta_6
So the q-eigenform is defined over a quadratic extension of Q(ζ6),
In fact, this extension is Q(ζ6, i).
Since the space M28chi has dimension 2 over Q(ζ6),
it is irreducible over Q(ζ6).
The corresponding abelian variety over Q therefore has dimension 4.
> A := ModularAbelianVariety(M28chi); > A; Modular abelian variety of dimension 4 and level 2^2*7 over Q with sign 1 > delta, F := DegreeMap(M28chi); > F; Rational FieldThis means that A is isogenous to B2 for some abelian variety B over Qbar of dimension 2, and that the endomorphism algebra of B is a quaternion algebra over F = Q.
> BrauerClass(M28chi); [ 2, 3 ]This means the endomorphism algebra of B is the quaternion algebra over Q ramified only at 2 and 3. Now we determine the possible fields of definition of B.
> delta; // This came from DegreeMap, above. [ <-7, 3> ]In particular, this tells us that KP = Q(Sqrt( - 7)).
> ObstructionDescentBuildingBlock(M28chi);
[ Place at Prime Ideal
Two element generators:
[2, 0]
[0, 1],
Place at Prime Ideal
Two element generators:
[2, 0]
[3, 1] ]
> Universe($1); // What are these places elements of?
Set of Places of Number Field with defining polynomial x^2 + 7
over the Rational Field
The obstruction is given as a list of places of Q(Sqrt( - 7)).
Recall that B can be defined over any extension
of KP for which the obstruction is trivial. In this case,
any extension of Q(Sqrt( - 7)) which splits the quaternion algebra
over Q(Sqrt( - 7)) ramified at the two primes above 2.