# Icosahedral Galois Representations

### Contents

- A short introduction
- Tables of icosahedral newforms with quadratic nebentypus
- Remarks on 2-dimensional icosahedral 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 | ||||
---|---|---|---|---|

N | D | dim. | fixed field of projective repr. | ref. |

1948 | -487 | 4 | x^5-7x^3-17x^2+18x+73 | [K-W] |

2083 | -2083 | 4 | x^5+8x^3+7x^2+172x+53 | [K-W] |

2336 | -292 | 8 | x^5+2x^3-4x^2-2x+4 | [K-W] |

2707 | -2707 | 4 | x^5-x^4+9x^3-6x^2-32x+93 | [B-D-SB-T] |

2863 | -2863 | 8 | x^5+12x^3+21x^2+22x+7 | [J-M] |

3004 | -751 | 4 | x^5-8x^3+10x^2+160x+128 | [K-W] |

3203 | -3203 | 4 | x^5+8x^3+5x^2-4x+1 | [J-M] |

3547 | -3547 | 4 | x^5-8x^3-2x^2+31x+74 | [B-D-SB-T] |

3548 | -887 | 4 | x^5+10x^3+10x^2+44x+56 | [K-W] |

3587 | -3587 | 8 | x^5+3x^3+24x^2-20x+131 | [J-M] |

3676 | -919 | 4 | x^5-8x^3+28x^2-40x+48 | [K-W] |

3775 | -151 | 8 | x^5-3x^3+x^2+x+3 | [J-M] |

3775 | -755 | 8 | x^5-3x^3+x^2+x+3 | [J-M] |

3875 | -31 | 8 | x^5+10x^3+5x^2+1 | [J-M] |

3875 | -155 | 8 | x^5+10x^3+5x^2+1 | [J-M] |

4000 | -4 | 8 | x^5+20x+16 | [J-M] |

4000 | -20 | 8 | x^5+20x+16 | [J-M] |

4027 | -4027 | 4 | x^5+x^4+9x^3-38x^2+13x+23 | [J-M] |

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`.

### 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, U`A_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 | ||||||||
---|---|---|---|---|---|---|---|---|

1a | 3a | 4a | 5a | 10a | 2a | 6a | 5b | 10b |

2 | -1 | 0 | A | -A | -2 | 1 | A' | -A' |

x.1a | x.3a | x.4a | x.5a | x.10a | x.2a | x.6a | x.5b | x.10b |

2I | -I | 0 | IA | -IA | -2I | I | IA' | -IA' |

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).