Modular Curves of Given Level and Genus

The intrinsics described in this section are designed to construct curves \(C\) over Q of genus at least 2 which are images of \(X_1(N)\) (or \(X_0(N)\)) under a morphism \(\pi: X_1(N) \rightarrow C\) defined over Q. Such curves will be referred to as modular curves. The code uses some of the ideas and methods from the papers [Gonzalez-Jimenez and Gonzalez, 2003], [Baker et al., 2005] and [Gonzalez-Jimenez and Oyono, 2010].

The morphism \(\pi\) induces an isogeny \(\pi^*\) from the Jacobian of \(C\), \(Jac(C)\) onto a Q-rational abelian subvariety \(B\) of \(J_1(N)\) (or \(J_0(N)\)), i.e. a Q-rational modular abelian variety of dimension \(g\), the genus of \(C\). The space of holomorphic differentials of \(Jac(C)\) identified with those of \(C\) pull back under \(\pi^*\) to the holomorphic differentials of \(B\) which can be identified with the space of weight 2 cusp forms associated to \(B\).

If \(f_1,\ldots,f_g\) is a Q-basis of this space, then these forms will satisfy the canonical relations for some canonical embedding of \(C\) in the non-hyperelliptic case. Conversely, if a basis of the space of cusp forms for a Q-rational modular abelian variety \(B\) of dimension \(g\) satisfies the polynomial relations defining a canonical curve \(C\) of genus \(g\), then \(C\) is a modular curve, but the pullback of the space of holomorphic differentials under the obvious map \(\pi: z\mapsto [f_1(z):f_2(z):\ldots : f_g(z)]\) from \(X_1(N)\) to \(C\) may NOT give the space of differentials \(\langle f_1,\ldots,f_g\rangle\) and \(Jac(C)\) may be isogenous to a different modular abelian subvariety of \(J_1(N)\) to \(B\).

In the hyperelliptic case, criteria for the pullback of the differentials of the curve \(C\) to give precisely the space of forms of a particular modular abelian variety \(B\) are given in [Gonzalez-Jimenez and Gonzalez, 2003] and [Baker et al., 2005], provided that \(B\) lies in the new part of \(J_1(N)\). In the genus 3 non-hyperelliptic case, a simple explicit criterion for \(Jac(C)\) to pull back to \(B\) is given in [Gonzalez-Jimenez and Oyono, 2010].

Modular curves \(C\) such that the pullback \(\pi^*(Jac(C))\) lies in the new part of \(J_1(N)\) (or \(J_0(N)\)) are referred to as new modular curves.

Intrinsics are provided that determine all of the new modular hyperelliptic curves or all of the new modular genus 3 non-hyperelliptic curves of \(X_1(N)\) (or \(X_0(N)\)) for a given level \(N\). There are also intrinsics that determine whether a given modular abelian subvariety \(B\) corresponds to a hyperelliptic or genus 3 non-hyperelliptic curve \(C\) and give the equations of \(C\) (as a Weierstrass model or canonical model, respectively) in the affirmative case.

In future releases it is planned to add intrinsics to deal with non-hyperelliptic modular curves of genus greater than 3 and to add more functionality to cover non-new cases.

SetVerbose("ModularCurve", v): MonStgElt, RngIntElt

Set the printing level for verbose output for the following intrinsics. Currently the legal values for \(v\) are true, false, \(0\), \(1\), \(2\) (false is the same as \(0\), and true is the same as \(1\)).

NewModularHyperellipticCurves(N, g): RngIntElt, RngIntElt -> SeqEnum

check: BoolElt                      Default: false
prec : RngIntElt                    Default: 100
gamma: RngIntElt                    Default: 1

For level \(N\), this function returns a list of all hyperelliptic curves of genus \(g \ge 2\) which are new modular curves for \(X_1(N)\) if parameter gamma equals \(1\) (the default) or \(X_0(N)\), if gamma equals \(0\).

A hyperelliptic curve is returned as a univariate polynomial \(f(x)\) such that \(y^2=f(x)\) is a Weierstrass equation of the actual curve.

The parameter prec (default 100) is the precision to which modular forms are expanded in the computations. If parameter check (default false) is true, the forms are actually computed to a precision slightly in excess of the precision \(prec_1\) needed to guarantee that polynomial relations on them (of the degrees that are used in the computations) vanish if they vanish to precision \(prec_1\).

NewModularHyperellipticCurve(B): [ModSym] -> BoolElt, RngUPol

check: BoolElt                      Default: false
prec : RngIntElt                    Default: 100
gamma: RngIntElt                    Default: 1

Given a sequence \(B\) of distinct modular abelian subvarieties of \(J_1(N)^{new}\), presented as subspaces of modular symbols, this function true if the modular abelian variety \(M\) which is the direct sum of the \(B\) corresponds exactly to a hyperelliptic curve \(C\) as described in the introduction (\(M\) is isogenous to \(Jac(C)\)). If so, a univariate polynomial \(f(x)\) is also returned such that \(y^2=f(x)\) is a Weierstrass equation for \(C\).

The parameters check and prec have the same meaning as in the previous intrinsic. The parameter gamma is only relevant if check is true, is \(1\) by default, and should only be set to \(0\) if all of the \(B\) are abelian subvarieties of \(J_0(N)\). It is then used to get sharper bounds for the precision required for the \(q\)-expansions.

NewModularHyperellipticCurve(F): [RngSerPowElt] -> BoolElt, RngUPol

This is a variant on the intrinsic directly above where, instead of the sequence of modular abelian subvarieties as argument, the sequence of \(q\)-expansions of the basis of weight 2 forms for the modular abelian subvariety \(M\) is given instead.

ModularHyperellipticCurve(B): [ModSym] -> BoolElt, RngUPol

prec: RngIntElt                    Default: 100

Given a sequence \(B\) of distinct modular abelian subvarieties of \(J_1(N)^{new}\), where the direct sum modular abelian variety \(M\) that \(B\) defines does not have to lie in the new part \(J_1(N)^{new}\), the function determines whether the basis of the differentials (forms) of \(M\) satisfy the correct relations so as to arise from a hyperelliptic curve \(C\) of genus \(g\). If so, the intrinsic also returns a univariate polynomial \(f(x)\), such that \(y^2=f(x)\) is a Weierstrass equation for \(C\). The difference between this and the new case is that, although \(C\) is a modular curve, it is not guaranteed that \(Jac(C)\) is isogenous to \(M\): it may be isogenous to a different modular abelian subvariety of \(J_1(N)\).

The parameter prec (default 100) is the precision to which modular forms are expanded in the computations.

ModularHyperellipticCurve(F): [RngSerPowElt] -> BoolElt, RngUPol

This is a variant on the intrinsic immediately above where, instead of a sequence of modular abelian varieties \(B\) with direct sum \(M\) passed as the argument, a sequence of \(q\)-expansions of the basis of weight 2 cusp forms of such an \(M\) is given instead.

NewModularNonHyperellipticCurvesGenus3(N): RngIntElt -> SeqEnum

check: BoolElt                      Default: false
prec : RngIntElt                    Default: 100
gamma: RngIntElt                    Default: 1

Given an integer \(N\), this function returns a list of all non-hyperelliptic curves of genus \(3\) which are new modular curves, for \(X_1(N)\) if the parameter gamma equals 1 (the default), or for \(X_0(N)\) if gamma equals 0.

The parameter prec (default 100) is the precision to which modular forms are computed to in the computations. If parameter check (default false) is true, the forms are actually computed to a little over the precision \(prec_1\) needed to check that polynomial relations on them (of the degrees that are used in the computations) are definitely guaranteed to vanish if they vanish to precision \(prec_1\).

NewModularNonHyperellipticCurveGenus3(B): [ModSym] -> BoolElt, RngMPolElt

check: BoolElt                      Default: false
prec : RngIntElt                    Default: 100
gamma: RngIntElt                    Default: 1

Given a sequence \(B\) of distinct modular abelian subvarieties of \(J_1(N)^{new}\), where the direct sum modular abelian variety \(M\) that \(B\) defines does not have to lie in the new part \(J_1(N)^{new}\), the function determines whether the basis of the differentials (forms) of \(M\) satisfy the correct relations so as to arise from a non-hyperelliptic curve \(C\) of genus \(3\). If so, the intrinsic also returns a a defining polynomial for the canonical image of \(C\). The parameters check and prec have the same meaning as for the intrinsic NewModularHyperellipticCurves. The parameter gamma, which is only relevant if check is true, and is \(1\) by default, may be set to 0 if \(M\) is an abelian subvariety of \(J_0(N)\) when sharper bounds for the required precision of \(q\)-expansions will be used. This intrinsic is similar to NewModularHyperellipticCurve(B).

NewModularNonHyperellipticCurveGenus3(F): [RngSerPowElt] -> BoolElt, RngMPolElt

This is a variant on the intrinsic NewModularNonHyperellipticCurveGenus3(B) above where, instead of the sequence of modular abelian varieties \(B\) with direct sum \(M\) passed as the argument, a sequence \(F\) of \(q\)-expansions of the basis of weight 2 cusp forms of such an \(M\) is given instead.

ModularNonHyperellipticCurveGenus3(F): [RngSerPowElt] -> BoolElt, RngMPolElt

This is the same as the intrinsic NewModularNonHyperellipticCurveGenus3(F) immediately preceding except that it is not required that the modular abelian variety \(M\) which corresponds to \(F\) lies in the new part \(J_1(N)^{new}\). This may be used to search for non-new non-hyperelliptic genus 3 curves.

Example: Jim Ex Mod Crvs Hyp (ex-dc59de)

We give some examples of the use of these intrinsics.

The modular curve \(X_1(13)\) is of genus \(2\), therefore hyperelliptic, and the space of modular forms of weight \(2\) for \(\Gamma_1(13)\) is generated by an unique newform \(f\) of nebentypus a Dirichlet character of order \(6\), such that the modular abelian variety attached to \(f\), \(A_f\), is \(\bf Q\)-isogenous to \(J_1(13)\):

> chi := DirichletGroup(13,CyclotomicField(6)).1; //order 6 character mod 13
> A13:=Af(chi)[1];
> NewModularHyperellipticCurve([A13]);
true x^6 + 4*x^5 + 6*x^4 + 2*x^3 + x^2 + 2*x + 1
> f13:=qIntegralBasis(A13,100);
> NewModularHyperellipticCurve(f13);
true x^6 + 4*x^5 + 6*x^4 + 2*x^3 + x^2 + 2*x + 1
> SetVerbose("ModularCurve",1);
> NewModularHyperellipticCurve([A13]:check:=true);
Checking ...
          ... bound =113
true x^6 + 4*x^5 + 6*x^4 + 2*x^3 + x^2 + 2*x + 1

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Let us compute all the new modular hyperelliptic curves parameterized by \(X_0(80)\):

> NewModularHyperellipticCurves(80,0: gamma:=0);

     Candidates:=3

     All the curves

1 2 3

[
    x^5 + 2*x^4 - 26*x^3 - 132*x^2 - 231*x - 142
]

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Finally, we calculate the new modular hyperelliptic curves parameterized by \(X_1(80)\) (the default):

> SetVerbose("ModularCurve",0);
> NewModularHyperellipticCurves(80,0);
[
    x^5 + 2*x^4 - 26*x^3 - 132*x^2 - 231*x - 142,
    x^5 - 2*x^4 - 2*x^3 + 20*x^2 - 47*x + 30,
    x^7 - 4*x^6 - 4*x^5 + 39*x^4 - 64*x^3 + 40*x^2 - 8*x,
    x^7 + 2*x^5 + 7*x^4 - 4*x^3 - 20*x^2 - 16*x - 4
]

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Example: Jim Ex Mod Crvs Jeon (ex-c93378)

In this exercise, an interesting example from [Jeon and Kim, 2007] is presented.

> S:=CuspidalSubspace(ModularSymbolsH(21,[1,8,13,20],2,+1));
> S;
Modular symbols space of level 21, weight 2, and dimension 3 over Rational Field
(multi-character)
> ModularHyperellipticCurve([S]);
true x^8 - 6*x^6 + 4*x^5 + 11*x^4 - 24*x^3 + 22*x^2 - 8*x + 1

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The above curve is the unique hyperelliptic intermediate modular curve \(X_{\Delta}(N)\) between \(X_1(N)\) and \(X_0(N)\), where \(\Delta\) is a subgroup of \(({\bf Z}/N{\bf Z})^*/\{\pm 1\}\) and \(X_{\Delta}(N)\) is the modular curve associated to some congruence subgroup of \(PSL_2({\bf Z})\) attached to \(\Delta\). In fact, the above hyperelliptic curve is the new modular curve denoted by \(C^A_{21A_{\{0,2\}}}\) in [Baker et al., 2005].

Example: Jim Ex Mod Crvs Jeon (ex-f78f1d)

Given a modular abelian subvariety of \(J_0(97)\), we determine whether it corresponds to a new modular non-hyperelliptic curve of genus \(3\) and level \(97\):

> M:=ModularSymbols(97,2,1);
> NN:=SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M))));
> NN;
[
    Modular symbols space for Gamma_0(97) of weight 2 and dimension 3 over
    Rational Field,
    Modular symbols space for Gamma_0(97) of weight 2 and dimension 4 over
    Rational Field
]
> A97:=NN[1];
> NewModularNonHyperellipticCurveGenus3([A97]);
true -x^2*y^2 + x*y^3 + x^3*z + x*y^2*z - 5*x^2*z^2 + 3*x*y*z^2 - 3*y^2*z^2 +
  6*x*z^3 - y*z^3 - 2*z^4

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Example: Jim Ex Mod Crvs Gen3 (ex-724f38)

We calculate all of the new modular non-hyperelliptic curves of genus \(3\) parameterized by \(X_1(20)\):

> NewModularNonHyperellipticCurvesGenus3(20);
[
    -x^2*y^2 + x*y^3 + x^3*z - 3*x^2*z^2 + 4*x*z^3 - 2*z^4
]

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We now construct an example of a non-new modular non-hyperelliptic curve of genus 3:

> S1:=ModularSymbols(178,2,1);
> N1:=SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(S1))));
> A:=N1[3];A;
Modular symbols space for Gamma_0(178) of weight 2 and dimension 2 over
Rational Field
> S2:=ModularSymbols(89,2,1);
> N2:=SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(S2))));
> B:=N2[1];B;
Modular symbols space for Gamma_0(89) of weight 2 and dimension 1 over
Rational Field
> fA:=qIntegralBasis(A,100);
> fB:=qIntegralBasis(B,100);
> q := Universe(fB).1;
> fB2:=&+[Coefficient(fB[1],k)*q^(2*k) : k in [1..99]];
> g:=fB[1]+2*fB2;
> ModularNonHyperellipticCurveGenus3([g,fA[1],fA[2]]);
true -x^4 + 2*x^2*y^2 - y^4 + 8*x^2*y*z + 8*y^3*z - 6*x^2*z^2 - 38*y^2*z^2 +
    24*y*z^3 + 7*z^4
>

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The above curve is denoted by \(C^{89A}_{178C}\) in [Gonzalez-Jimenez and Oyono, 2010]. Its jacobian is \({\bf Q}\)-isogenous to \(A\times B\).