Icosahedral Galois Representations


A short introduction

For each newform f on Gamma_0(N) of weight 1 there exists, by a theorem of [D-S], a finite Galois extension K of Q and a two-dimensional representation r:Gal(K)--->GL(2,C) such that the Artin L-series of r and the L-series of f coincide. The representation r is odd, i.e. r maps the restriction of the complex conjugation to Gal(K) onto a matrix with determinant equal to -1.

It is conjectured that the converse holds true, i.e. for every such representation r there is a newform f of weight 1 on Gamma_0(N) whose L-series equals the one of r. If the conjecture holds true for r, then r is called modular. The conjecture is proven for those representations r such that the image of r'=pr is a cyclic or dihedral group, or is isomorphic to the tetrahedral or octahedral group, i.e. isomorphic to A_4 or S_4, respectively. Here p denotes the canonical map p:GL(2,C)--->PGL(2,C) and r'=pr the composite map. The cyclic and dihedral cases follow from the fact that here the representations are reducible respectively monomial, and hence their Artin L-series are entire, which in turn follows from results of [H]. The tetrahedral and certain octahedral cases follow from [L], the general octahedral case was dealt with in [T].

By a result of F. Klein (cf. [K]) every finite subgroup of PGL(2,C) is cyclic, dihedral, tetrahedral, octahedral or icosahedral (i.e. isomorphic to A_5=PGL(2,F_5) in the latter case). A representation r and the associated newform are called cyclic, dihedral according to the isomorphism type of the image of r'. The above conjecture is still open for general icosahedral representations, though it is known to hold true for various families among these (cf. [???]).

For an explicitly given icosahedral representation it is possible to check its modularity, at least theoretically, by identifying the (conjecturally) associated newform by sufficiently (but finitely) many of its Hecke eigenvalues, by constructing the representation associated to the newform and by finally comparing the two representations.

The Modi data base contains all icosahedral newforms of quadratic nebentypus on Gamma_0(N) for N ≤ 4027. A result of Jehanne and Müller states:

Theorem ([J-M]) All complex Galois representations of degree 2, of icosahedral type with quadratic determinant, and with conductor less or equal to 4027 are modular.

Tables of icosahedral newforms with quadratic nebentypus

Table 1: Icosahedral newforms in the Modi Data Base
NDdim.fixed field of projective repr.ref.

The Modi data base can be queried for all icosahedral newforms of quadratic nebentypus on Gamma_0(n) for N ≤ 4027. In particular, it contains 5000 Fourier coefficients of each form. The precomputed data are courtesy of Arnaud Jehanne.

The first two columns of Table 1 list all levels N and all all d such that S_1(Gamma_0(N),chi_D) contains an icosahedral newform. Here chi_D denotes the character of Q(sqrt(D)). The third column shows the number of icosahedral newforms in S_1(Gamma_0(N),chi_D), and the last one provides a reference to a place in the literature where, to our knowledge, the modularity of the associated representation was first proved.

The quintic polynomials describe the number fields which are fixed by the kernel of the projective representations associated to the icosahedral forms. The Galois group of their field of decomposition is thus A_5. This fixed fields do, for a given level and discriminant (and within the range of our data) not depend on the particular newforms. This is due to the fact, that, first of all, the icosahedral forms come in quadruples whose members are pairwise conjugate under Galois action (see the next section for the explanation). Secondly, if there are two quadruples of icosahedral forms in a given space (and within the range of our tables), then either one can be obtained from the other by twisting its forms by the quadratic character mod 4 (or,likewise, by multiplying the Galois representations of its forms by the corresponding quadratic character).

There are three pairs of spaces whose icosahedral forms have the same projective kernel (i.e. the listed quintic polynomials coincide). If two Galois representations, say, of the absolute Galois group, induce the same projective representation, then they coincide up to multiplication by a 1-dimensional representation. Indeed, for each of the mentioned pairs of spaces, the forms in one space are obtained by twisting the forms in the other space by a character mod 5 of order 4.

Polynomials describing the fixed fields of the Galois representations associated to the quadruples can be obtained by submitting the form below. These fields are of degree 240, their Galois group is UA_5 ( see next section), they can be described as splitting fields of polynomials of degree 24.

Icosahedral newforms on Gamma0(N) with quadratic nebentypus
associated Galois representation
Fourier coefficients
Fourier Development

Remarks on 2-dimensional icosahedral representations

The icosahedral group A_5 is a perfect group. Hence A_5 possesses a universal central extension UA_5. If one views A_5 as the group PGL(2,F_5) then UA_5 can be described as the subgroup of matrices in GL(2,F_5) with determinant +1 or -1. In other words, UA_5 equals SL(2,F_5).<x>, where x is a scalar matrix such that x^2=-1. The extension map sequence is then

1 --> <x> --> UA_5 = SL(2,F_5).<x> --> A_5 = PGL(2,F_5) --> 1,

where the map onto PGL(2,F_5) is the canonical map. To every homomorphism h:<x>-->C into an Abelian group C we can associate the central extension

1 --> C --> UA_5 x C/{y x h(y): y in <x>}  -->  A_5 x C -->  1

(with the obvious maps 1->1 x c and a x c -> a). Every central extension of the icosahedral group is obtained in this way (up to isomorphism).

Table 2: Characters of the 2-Dimensional Irreducible Representations of UA_5

The group UA_5 has exactly four irreducible characters of order 2. They are all conjugate under Galois action. The characters of these representations are given in Table 2. (The table can easily be derived from the well-known character table for SL(2,F_5)). The first column runs through the conjugacy classes nl and x.nl of UA_5, where n denotes the order of the elements of the respective conjugacy class, where x is a central element of order 4, and where x.nl is obtained from nl by multiplying the the elements of nl by x. The letters A and A' stands for the roots of x^2+x-1 (the golden ratio), and I is a root of -1.

Assume that G is a subgroup of GL(2,C) such that its image in PGL(2,C) is isomorphic to A_5. Since A_5 admits no irreducible representation of dimension 2 G must be a nontrivial extension of A_5. Hence G is isomorphic to a quotient of, UA_5 x C with a suitable Abelian group C. Since G cannot be Abelian, it must be irreducible. Since UA_5 possesses exactly 4 irreducible representations, which are all faithful, we deduce that G = q(UA_5).<c>, where a is one of these representations and c a scalar matrix.

We deduce from this: If r:Gal(K)-->GL(2,C) is an icosahedral representation such that det(r) is a character of order 2, then the image of r equals the image of UA_5 under one of its 4 irreducibel 2-dimensional representations (since, if the image of r equals, in the above notation, q(UA_5).<y>, then det(y)^2=y^4=+1 implies y in q(UA_5)). The fixed field of r is in particular of order 240, and 24 is the smallest degree d such that the fixed field can be characterized as splitting field of a polynomial of degree d (since 24 is the smallest d such that UA_5 can be realized as transitive subgroup of the symmetric group of d elements).