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An ambient space of modular symbols is created by specifying a base ring,
character, weight, and optional sign. Note that spaces of modular symbols
may be defined directly over any base ring (unlike ModularForms, which
can only be base extensions of spaces over the integers).
The most general signature is ModularSymbols(eps, k, sign),
where eps is a Dirichlet character (the level is taken to be the modulus
of eps, and the base ring is taken to be the base ring of eps).
For information on Dirichlet characters, see Section Dirichlet Characters.
Warning: Certain functions, such as DualVectorSpace, may
fail when given as input a space of modular symbols over a field of
positive characteristic, because the Hecke operators Tp, with p
prime to the level, need not be semisimple.
The space of modular symbols of level N, weight 2, and
trivial character over the rational numbers.
The space of modular symbols of level N, weight k,
and trivial character over the rational numbers.
The space of modular symbols of level N, weight k, and trivial character
over the field F.
The space of modular symbols of level N, weight k, trivial character,
and given sign (as an integer) over the rational numbers.
The space of modular symbols of level N, weight k, trivial character,
and given sign (as an integer), over the field F.
The space of modular symbols of weight k and character ε
(as an element of a Dirichlet group).
Note that ε determines the level and the base field, so
they do not need to be specified.
The space of modular symbols of weight k and character ε
(as an element of a Dirichlet group).
The level and base field are specified as
part of ε. The third argument
"sign" allows for working in certain quotients.
The possible values are -1, 0, and +1,
which correspond to the -1 quotient, full space, and +1 quotient,
respectively.
The +1 quotient of M is M/( * -1)M, where * is
StarInvolution(M).
We create spaces of modular symbols in several different ways.
> M37 := ModularSymbols(37); M37;
Full modular symbols space for Gamma_0(37) of weight 2 and dimension 5
over Rational Field
> Basis(M37);
[
{-1/29, 0},
{-1/22, 0},
{-1/12, 0},
{-1/18, 0},
{oo, 0}
]
As M37 is a space of modular symbols, it is not incorrect
that its dimension is different than that of the three-dimensional
space of modular forms M 2(Γ 0(37)). We have
dim Mm 2(Γ 0(37)) = 2 x ((dim cusp forms)) +
1 x ((dim Eisenstein series)) = 5.
> MF := ModularForms(Gamma0(37),2);
> 2*Dimension(CuspidalSubspace(MF)) + Dimension(EisensteinSubspace(MF));
5
Next we decompose M37 with respect to
the Hecke operators T2, T3, and T5.
> D := Decomposition(M37,5); D;
[
Modular symbols space for Gamma_0(37) of weight 2 and dimension 1
over Rational Field,
Modular symbols space for Gamma_0(37) of weight 2 and dimension 2
over Rational Field,
Modular symbols space for Gamma_0(37) of weight 2 and dimension 2
over Rational Field
]
The first factor corresponds to the standard Eisenstein series, and
the second corresponds to an elliptic curve:
> E := EllipticCurve(D[2]); E;
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 23*x - 50 over
Rational Field
> Rank(E);
0
We now create the space Mm12(1) of weight 12 modular symbols
of level 1.
> M12 := ModularSymbols(1,12); M12;
Full modular symbols space for Gamma_0(1) of weight 12 and dimension 3
over Rational Field
> Basis(M12);
[
X^10*{0, oo},
X^8*Y^2*{0, oo},
X^9*Y*{0, oo}
]
> DimensionCuspFormsGamma0(1,12);
1
> R<z>:=PowerSeriesRing(Rationals());
> Delta(z)+ O(z^7);
z - 24*z^2 + 252*z^3 - 1472*z^4 + 4830*z^5 - 6048*z^6 + O(z^7)
As a module, cuspidal modular symbols equal
cuspforms with multiplicity two.
> M12 := ModularSymbols(1,12);
> HeckeOperator(CuspidalSubspace(M12),2);
[-24 0]
[ 0 -24]
> qExpansionBasis(CuspidalSubspace(M12),7);
[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 + O(q^7)
]
For efficiency purposes, since one is often interested only
in q-expansion, it is possible to work in the quotient
of the space of modular symbols by all relations
*x = x (or *x= - x), where * is StarInvolution(M). In either
of these quotients (except possibly in characteristic p>0)
the cusp forms appear with multiplicity one instead of two.
> M12plus := ModularSymbols(1,12,+1);
> Basis(M12plus);
[
X^10*{0, oo},
X^8*Y^2*{0, oo}
]
> CuspidalSubspace(M12plus);
Modular symbols space for Gamma_0(1) of weight 12 and dimension 1 over
Rational Field
> qExpansionBasis(CuspidalSubspace(M12),7);
[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 + O(q^7)
]
The following is an example of how to create Dirichlet characters in
Magma, and how to create a space of modular symbols with nontrivial
character. For more details, see Section Dirichlet Characters.
> G<a,b,c> := DirichletGroup(16*7,CyclotomicField(EulerPhi(16*7)));
> Order(a);
2
> Conductor(a);
4
> Order(b);
4
> Conductor(b);
16
> Order(c);
6
> Conductor(c);
7
> eps := a*b*c;
> M := ModularSymbols(eps,2); M;
Full modular symbols space of level 112, weight 2, character a*b*c,
and dimension 32 over Cyclotomic Field of order 48 and degree 16
> BaseField(M);
Cyclotomic Field of order 48 and degree 16
It is also possible to create many spaces of modular symbols for
Γ0(N) by passing a "descriptive label" as an
argument to ModularSymbols.
The most specific label is a string of the form
[Level]k[Weight][IsogenyClass], where [Level]
is the level, [Weight] is the weight,
[IsogenyClass] is a letter code:
A, B, ..., Z, AA, BB, ..., ZZ,
AAA, ...,
and k is a place holder to separate the level and the weight.
If the label is [Level][IsogenyClass], then the weight k is assumed
equal to 2. If the label is [Level]k[Weight] then the cuspidal
subspace of the full ambient space of modular symbols of
weight k and level N is returned.
The following are valid labels:
11A, 37B, 3k12A, 11k4.
The ordering used on isogenies classes is lt; see the
documentation for SortDecomposition.
Note: There is currently no intrinsic that, given a space of
modular symbols, returns its label.
Warning: For 146 of the levels between 56 and 450, our
ordering of weight 2 rational newforms disagrees with the ordering
used in [Cre97].
Fortunately, it is easy to create a space of modular symbols from one
of Cremona's labels using the associated elliptic curve.
> E := EllipticCurve(CremonaDatabase(),"56A");
> M1 := ModularSymbols(E);
Observe that Cremona's "56A" is different from ours.
> M2 := ModularSymbols("56A");
> M2 eq M1;
false
ModularSymbols(s) : MonStgElt -> ModSym
The space of modular symbols described by a label, the string s, and given
sign.
The cusp form Δ(q) is related to the space
of modular symbols whose label is "1k12A".
> Del := ModularSymbols("1k12A"); Del;
Modular symbols space for Gamma_0(1) of weight 12 and dimension 2 over
Rational Field
> qEigenform(Del,5);
q - 24*q^2 + 252*q^3 - 1472*q^4 + O(q^5)
Next, we create the space corresponding to the first newform on Γ 0(11)
of weight 4.
> M := ModularSymbols("11k4A"); M;
Modular symbols space for Gamma_0(11) of weight 4 and dimension 4 over
Rational Field
> AmbientSpace(M);
Full modular symbols space for Gamma_0(11) of weight 4 and dimension 6
over Rational Field
> qEigenform(M,5);
q + a*q^2 + (-4*a + 3)*q^3 + (2*a - 6)*q^4 + O(q^5)
> Parent($1);
Power series ring in q over Univariate Quotient Polynomial Algebra in
a over Rational Field with modulus a^2 - 2*a - 2
We next create the +1 quotient of the cuspidal subspace
of weight-4 modular symbols of level 37.
> M := ModularSymbols("37k4",+1); M;
Modular symbols space for Gamma_0(37) of weight 4 and dimension 9 over
Rational Field
> AmbientSpace(M);
Full modular symbols space for Gamma_0(37) of weight 4 and dimension
11 over Rational Field
> Factorization(CharacteristicPolynomial(HeckeOperator(M,2)));
[
<x^4 + 6*x^3 - x^2 - 16*x + 6, 1>,
<x^5 - 4*x^4 - 21*x^3 + 74*x^2 + 102*x - 296, 1>
]
Suppose M is a space of weight k modular symbols over a field F.
A modular symbol P(X, Y){α, β} is input as M!<P(X,Y),[alpha,beta]>, where P(X, Y)∈()F[X, Y] is homogeneous of
degree k - 2, and α, β∈P1(Q).
Here is an example:
Any space M of modular symbols is finitely generated. One proof
of this uses that every modular symbol is a linear combination of
Manin symbols.
Let P1(Z/NZ) be the set of pairs
(/line(u), /line(v)) ∈Z/NZ x Z NZ such that
(GCD)(u, v, N)=1, where u and v are lifts of /line(u)
and /line(v) to Z.
A Manin symbols < P(X, Y), (/line(u), /line(v)) >
is a pair consisting of a homogeneous polynomial
P(X, Y)∈F[X, Y] of degree k - 2 and
an element (/line(u), /line(v))∈P1(Z/NZ).
The modular symbol associated to
< P(X, Y), (/line(u), /line(v)) >
is constructed as follows. Choose lifts u, v of
/line(u), /line(v) such that (GCD)(u, v)=1.
Then there is a matrix g=pmatrix(w&z
u&v)
in SL2(Z) whose lower two entries are u and v.
The modular symbol is then
g(P(X, Y){0, ∞}).
The intrinsic ConvertFromManinSymbol computes the modular
symbol attached to a Manin symbol.
Ever modular symbol can be written as a linear combination
of Manin symbols using the intrinsic ManinSymbol.
First create the space M = Mm 4(Γ 0(3);F 7).
> F7 := GF(7);
> M := ModularSymbols(3,4,F7);
> R<X,Y> := PolynomialRing(F7,2);
Now we input (X 2 - 2XY){0, 1}.
> M!<X^2-2*X*Y,[Cusps()|0,1]>;
6*Y^2*{oo, 0}
Note that (X 2 - 2XY){0, 1} = 6Y 2{∞, 0} in M.
When k=2, simply
enter M!<1,[alpha,beta]>.
> M := ModularSymbols(11,2);
> M!<1,[Cusps()|0,Infinity()]>;
-1*{oo, 0}
> M![<1,[Cusps()|0,Infinity()]>, <1,[Cusps()|0,1/11]>];
-2*{oo, 0}
In this example, we convert between Manin and modular symbols
representations of a few elements of a space of modular symbols
of weight 4 over F 5.
> F5 := GF(5);
> M := ModularSymbols(6,4,F5);
> R<X,Y> := PolynomialRing(F5,2);
> ConvertFromManinSymbol(M,<X^2+Y^2,[1,4]>);
(3*X^2 + 3*X*Y + 2*Y^2)*{-1/2, 0} + (X^2 + 4*X*Y + 4*Y^2)*{-1/3, 0} +
(X^2 + X*Y + 4*Y^2)*{1/3, 1/2}
> ManinSymbol(M.1-3*M.2);
[
<X^2, (0 1)>,
<2*X^2, (1 2)>
]
Thus the element M.1-3*M.2 of M
corresponds to the sum of Manin symbols
< X 2, (0, 1) > + 2< X 2, (1, 2) >.
The coercion of x into the space of modular symbols M. Here x can be either a modular symbol
that lies in a subspace of M, a 2-tuple that describes a modular
symbol, a sequence of such 2-tuples, or anything that can be coerced
into VectorSpace(M). If x is a valid sequence of such
2-tuples, then M!x is the sum of the coercions into M of
the elements of the sequence x.
The modular symbol associated to the 2-tuple x = < P(X, Y), [u, v] >, where P(X, Y) ∈F[X, Y] is homogeneous of
degree k - 1, F is the base field of the space of modular symbols M, and [u, v] is a sequence
of 2 integers that defines an element of P1(Z/NZ), where N
is the level of M.
An expression for the modular symbol x in terms of Manin symbols, which
are represented as 2-tuples < P(X, Y), [u, v] >.
> M := ModularSymbols(14,2); M;
Full modular symbols space for Gamma_0(14) of weight 2 and dimension 5
over Rational Field
> Basis(M);
[
{oo, 0},
{-1/8, 0},
{-1/10, 0},
{-1/12, 0},
{-1/2, -3/7}
]
> M!<1,[1,0]>;
0
> M!<1,[0,1/11]>;
{-1/10, 0} + -1*{-1/12, 0}
> M![<1,[0,1/2]>, <-1,[0,1/7]>]; // sequences are added
{-1/8, 0} + -1*{-1/12, 0} + -1*{-1/2, -3/7}
> M!<1,[0,1/2]> - M!<1,[0,1/7]>;
{-1/8, 0} + -1*{-1/12, 0} + -1*{-1/2, -3/7}
> M!<1,[Cusps()|Infinity(),0]>; // Infinity() is in Cusps().
{oo, 0}
We can also coerce sequences into the underlying vector space of M.
> VectorSpace(M);
Full Vector space of degree 5 over Rational Field
Mapping from: Full Vector space of degree 5 over Rational Field to
ModSym: M given by a rule [no inverse]
Mapping from: ModSym: M to Full Vector space of degree 5 over Rational
Field given by a rule [no inverse]
> Eltseq(M.3);
[ 0, 0, 1, 0, 0 ]
> M![ 0, 0, 1, 0, 0 ];
{-1/10, 0}
> M.3;
{-1/10, 0}
The "polynomial coefficients" of the modular symbols
are homogeneous polynomials in 2 variables of degree k - 2.
> M := ModularSymbols(1,12);
> Basis(M);
[
X^10*{0, oo},
X^8*Y^2*{0, oo},
X^9*Y*{0, oo}
]
> R<X,Y> := PolynomialRing(Rationals(),2);
> M!<X^9*Y,[Cusps()|0,Infinity()]>;
X^9*Y*{0, oo}
> M!<X^7*Y^3,[Cusps()|0,Infinity()]>;
-25/48*X^9*Y*{0, oo}
> Eltseq(M!<X*Y^9,[1/3,1/2]>);
[ -19171, -58050, -30970 ]
> M![1,2,3];
X^10*{0, oo} + 2*X^8*Y^2*{0, oo} + 3*X^9*Y*{0, oo}
> ManinSymbol(M![1,2,3]);
[
<X^10, (0 1)>,
<2*X^8*Y^2, (0 1)>,
<3*X^9*Y, (0 1)>
]
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