Examples of 2-dimensional, odd Galois representations of A_{5}-type over ℚ satisfying the Artin conjecture

Ian Kiming and Xiangdong Wang

0. Let us retain the notation of the introduction of I: We are considering irreducible representations:

ρ : Gal (\overline{ℚ}/ℚ) → GL_{2}(ℂ),

and we say that ρ is of dihedral-, A_{4}-, S_{4}-, or A_{5}-type, if the image of the projective representation associated with ρ is a dihedral group or isomorphic to A_{4}, S_{4}, A_{5}, respectively. Also, for such a representation ρ we denote by N and ε the Artin conductor and determinant character of ρ, respectively.

The purpose of this lecture is to provide new examples of odd representations of A_{5}-type which satisfy, together with all their twists by 1-dimensional characters, the Artin conjecture, i.e.: odd representations whose Artin L-series are L-series of cusp forms of weight 1 on some Γ_{0}(N). For a general description of how such examples may be obtained, we refer to the introduction of I.

Concerning the choice of the examples, we remark the following. First we have to start with a Galois extension K/ℚ with Galois group isomorphic to A_{5}. Such a field gives us a projective representation:

\overline{ρ}_{0} : Gal (\overline{ℚ}/ℚ) → P GL_{2}(ℂ)

by choosing 1 of the 2 embeddings A_{5} ↪ P GL_{2}(ℂ). Now, \overline{ρ}_{0} has a lifting ρ_{0} with some Artin conductor N and determinant character ε. We must of course demand that ε be odd. If we want to verify the Artin conjecture for ρ_{0} we must be able to enumerate all Galois representations ρ with the same conductor and determinant. This forces us to require first that:

A. The field K should occur in table 1 (which was discussed in II). This means that the discriminant of a quintic subfield of K should be bounded by 2083^{2}.

Secondly, we must compute the dimension of the space S_{1}(N, ε), and in order to keep the complexity of this computation within reasonable bounds, this imposes certain restrictions on N and ε: We require that dim S_{1}(N, ε) should be computable by the second method presented in I, which means that:

B. N should have form 2^{α} · n, where n is odd and squarefree.

Further, we have found it necessary to require:

C. ε^{2} = 1 and l.c.m. (N, 4N) ≤ 10000, again for the purpose of keeping the complexity managable. The reason for the requirement l.c.m. (N, 4N) ≤ 10^{4} instead of just N ≤ 10^{4} is that we may have to compute:

dim S_{1}(4N, ε) instead of just dim S_{1}(N, ε)

if N is odd. Summarizing, we have to look for A_{5}-fields K in table 1 such that a projective representation in P GL_{2}(ℂ) associated with K (and hence any such) has a lifting with Artin conductor N and odd determinant character ε, where (N, ε) satisfies the requirements B and C. An inspection of table 1 (using table 2 and the theory of I) shows that there are exactly 7 such fields: These are the fields with numbers 27, 69, 128, 135, 181, 191, and 238 in table 1. (Had we just required ε odd with ε^{2} = 1 and N ≤ 10000, we would have found 6 additional fields, namely the fields with numbers 1, 16, 44, 46, 73, and 88 in table 1). As was shown in section 4 of I, the involved pairs (N, ε) are (not respectively):

N = p, ε = χ_{−p} for p = 2083,
N = 4p, ε = χ_{−p} for p = 487, 751, 887, 919,

and

N = 2^{5}p, ε = χ_{−p} for p = 73, 193,

where χ_{−p} denotes the character of ℚ(\sqrt{−p})/ℚ. Note that the numbers 73, 193, 487, 751, 887, 919, 2083 are prime numbers. In this connection, we also found in section 4 of I the number of inequivalent irreducible representations ρ with conductor N and determinant ε for certain pairs (N, ε). Let us recall in the following table what these numbers are:

Type:  Dihedral  A_{4}  S_{4}  A_{5}
N  ε
487 χ_{−487}  3  0  0  0
751 χ_{−751}  7  0  2  0
887 χ_{−887}  14  0  0  0
919 χ_{−919}  9  0  0  0
2083 χ_{−2083}  3  0  0  4
2^{2}· 487 χ_{−487}  0  0  0  4
2^{2}· 751 χ_{−751}  0  0  0  4
2^{2}· 887 χ_{−887}  0  0  0  4
2^{2}· 919 χ_{−919}  0  0  0  4
2^{5}· 73 χ_{−73}  0  0  0  8
2^{5}· 193 χ_{−193}  0  0  0  8

Let us also recall that for each of the prime numbers p = 73 and p = 193, there is at least 1 representation ρ of dihedral type with N = 2^{2}p and ε = χ_{−p}, cf. section 4.3 in I.

It follows that in order to verify by computational means the Artin conjecture for the above representations of A_{5}-type, we have to show computationally that:

dim S_{1}(2083, χ_{−2083}) = 7 ,
dim S_{1}^{new}(2^{2} · p, χ_{−p}) = 4 , for p = 487, 751, 887, 919,

and

dim S_{1}^{new}(2^{5} · p, χ_{−p}) = 8 , for p = 73, 193.

1. Computations

1.1 Before describing the concrete computations, we want to start with a few general remarks.

Let k be a non-negative integer, N a natural number, ε a Dirichlet character mod N with ε(−1) = (−1)^{k}, and let Γ = SL_{2}(ℤ). We have an exact sequence of complex vectorspaces:

0 ⟶ S_{k+2}(Γ_{0}(N), ε)⟶^{ι}H^{1}(Γ, U_{k,ε})_{+}⟶^{r_{+}}H^{1}(Γ_{∞}, U_{k,ε})_{+} ⟶ 0,

where U_{k,ε} is a certain ℂΓ-module which was described in V, Γ_{∞} is the infinite cyclic group generated by:

( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ,

“+” means “fixed points under the action of ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} )”, and r_{+} is the restriction

map. Let:

X_{k}(N, ε)_{+} = Ker (r_{+}) .

For any n ∈ ℕ we have the Hecke operator on S_{k+2}(Γ_{0}(N), ε):

T_{n} = ∑_{l,m} T (l, m) ,

where the summation is over natural numbers l and m with lm = n, l|m and (l, N) = 1, and:

T (l, m) = Γ_{0}(N) ( \begin{matrix}l & 0 \\ 0 & m \end{matrix} ) Γ_{0}(N) ,

cf. [3], chap. 4. We learned in IV and V how to obtain a concrete model for the space H^{1}(Γ, U_{k,ε})_{+} and how to define Hecke operators T̃_{n} on this space which are compatible with ι. For convenience we shall drop the “˜ ” and denote the cohomological Hecke operators also by T_{n}.

Suppose now that t is a divisor of N such that (t, N/t) = 1. We can then write:

ε = ε_{1}ε_{2} ,

where ε_{1} and ε_{2} are Dirichlet characters mod t and mod N/t , respectively. Choosing a, b, c, d ∈ ℤ such that:

adt − bc \frac{N}{t} = 1 ,

we have an isomorphism:

W_{N,t} : S_{k+2}(Γ_{0}(N), ε_{1}ε_{2}) ⟶^{}˜ S_{k+2}(Γ_{0}(N), \overline{ε}_{1}ε_{2}) ,

where the “Artin-Lehner operator” W_{N,t} (cf. [1]) is given by:

W_{N,t} : f ↦ f|W_{N,t} = f|_{k+2} ( \begin{matrix}at & b \\ Nc & dt \end{matrix} ) ,

for f ∈ S_{k+2}(Γ_{0}(N), ε_{1}ε_{2}). This action does not depend on the choice of a, b, c, d. We learned in IV how to define W_{N,t} on H^{1}(Γ, U_{k,ε})_{+} in a way which is compatible with ι.

Now, the following simple lemma is important to us. We formulate it for the sake of simplicity only in the case of trivial nebentype; in case of a non-trivial nebentype, one must identify (in an obvious way) the complex vectorspaces X_{k}(N, ε_{1}ε_{2})_{+} and X_{k}(N, \overline{ε}_{1}ε_{2})_{+} .

Lemma: Consider the above situation with k even and ε = 1. Suppose that x ∈ X_{k}(N, 1)_{+}, and that φ ∈ Hom(X_{k}(N, 1)_{+}, ℂ). Then, as we saw in IV, the function f given by:

f(z) = ∑_{n=1}^{∞} φ(T_{n}x)q^{n} , q = e^{2πiz}, Im(z) > 0 ,

is the Fourier expansion at ∞ of an element in S_{k+2}(Γ_{0}(N)). We have:

(f|W_{N,t})(z) = ∑_{n=1}^{∞} φ(T_{n}W_{N,t}x)q^{n}.

Proof: Let us write W = W_{N,t} for convenience and note that W commutes with all Hecke operators T_{n}. Let:

a_{1} : S_{k+2}(Γ_{0}(N)) ⟶ ℂ

be the linear map “coefficient of e^{2πiz} in the Fourier expansion at ∞”; we then have:

g(z) = ∑_{n=1}^{∞} a_{1}(g|T_{n})q^{n} , for g ∈ S_{k+2}(Γ_{0}(N)).

Define:

A_{1} : X_{k}(N, 1)_{+} ⟶ ℂ ,

by A_{1} = a_{1} ∘ ι^{−1}. Then we have for g ∈ S_{k+2}(Γ_{0}(N)) with y = ιg:

g = ∑ a_{1}(g|T_{n})q^{n}
= ∑ a_{1}(ι^{−1}y|T_{n})q^{n}
= ∑ a_{1}(ι^{−1}(T_{n}y))q^{n}
= ∑ A_{1}(T_{n}y)q^{n} .

Now, there is h ∈ S_{k+2}(Γ_{0}(N)) such that:

x = ιh ;

then:

W x = ι(h|W) ,
h = ∑ A_{1}(T_{n}x)q^{n} ,

and:

h|W = ∑ A_{1}(T_{n}W x)q^{n}.

Further, if m ∈ ℕ we have:

\begin{aligned}h|T_{m} & = ∑_{n} A_{1} ( ∑_{{d|(m,n) \atop (d,N)=1}} T_{mn/d^{2}}x) q^{n} \\ & = ∑_{n} A_{1}(T_{n}T_{m}x)q^{n}. \end{aligned}

There is an element T of the Hecke algebra 𝕋 = ⟨T_{n}⟩ such that:

φ(y) = A_{1}(T y) , for y ∈ X_{k}(N, 1)_{+} ;

this follows from the analogous result on S_{k+2}(Γ_{0}(N)). Then:

f = ∑ φ(T_{n}x)q^{n}
= ∑ A_{1}(T T_{n}x)q^{n}
= ∑ A_{1}(T_{n}T x)q^{n}
= h|T ,

whence:

f|W = (h|T)|W = (h|W)|T
= (∑ A_{1}(T_{n}W x)q^{n})|T
= ∑ A_{1}(T_{n}W T x)q^{n}
= ∑ A_{1}(T (T_{n}W x))q^{n}
= ∑ φ(T_{n}W x)q^{n}.

Q.E.D.

1.2. We want to compute dim S_{1}(Γ_{0}(2^{s+1}p), χ_{−p}) , where p is an odd prime number and:

s ≥ \{ \begin{matrix}2, & if p ≡ 1 (4) \\ 1, & if p ≡ 3 (4), \end{matrix}

by the last-mentioned method in section 0 of I. In other words, we consider the theta-function:

θ_{2}(z) = ∑_{{n=−∞ \atop n≡1 (2)}}^{∞} e^{πin^{2}z/4} , Im(z) > 0 ,

and

g_{s,p}(z) = θ_{2}(2^{s}z) θ_{2}(2^{s}pz) , Im(z) > 0 ,

and we have an embedding:

0 ⟶ S_{1}(Γ_{0}(2^{s+1}p), χ_{−p}) ⟶ S_{2}(Γ_{0}(2^{s+1}p))

given by multiplication with g_{s,p}; the image consists of forms f ∈ S_{2}(Γ_{0}(2^{s+1}p)) such that the first 2^{s−3}(p+1) Fouriercoefficients at ∞ (counting from the coefficient of e^{2πiz}) of both f and f|W_{s,p} all vanish, where:

W_{s,p} = ( \begin{matrix}p & b \\ −2^{s+1}p & dp \end{matrix} ) ,

where b and d are integers with pd + 2^{s+1}b = 1.

Let us put N = 2^{s+1}p and retain the notation of subsection 1.1. We have the following concrete model for the complex vectorspace H^{1}(Γ, U_{0,1})_{+} : It is generated by elements:

(u, v) ∈ (ℤ/ℤN)^{2} of order N ,

with the following relations:

(λu, λv) − (u, v) = 0 , for λ ∈ (ℤ/ℤN)^{x} ,
(u, v) + (u, −v) = 0 ,
(u, v) + (−v, u) = 0 ,

and

(u, v) + (u − v, u) + (−v, u − v) = 0.

Let us write \overline{(u, v)} for the class in H^{1}(Γ, U_{0,1})_{+} of (u, v) . The action of W_{s,p} on H^{1}(Γ, U_{0,1})_{+} can now be described in explicit terms as follows (cf. IV and [2]):

Suppose that x = \overline{(u, v)}, where g.c.d. (u, v) = 1. Let α, β ∈ ℤ be such that αv−βu = 1. For an element r ∈ ℙ^{1}(ℚ) define ψ(r) ∈ H^{1}(Γ, U_{0,1})_{+} : If r = ∞ , put ψ(r) = 0. Otherwise, consider the continued fractions expansion of r :

r = [c_{0}, …, c_{n}]

determined by the requirements:

rc_{0} ≥ 0 , rc_{i} ≥ 1 for i ≥ 1 , and |c_{n}| ≥ 2 , if n ≥ 1.

Let ρ_{i}/σ_{i} for i ≥ 1 be the convergents with the sign conventions:

sgn(σ_{2i+1}) = (−1)^{i} , sgn(σ_{2i}) = (−1)^{i}sgn(r).

Then:

ψ(r) = ∑_{k=1}^{n} \overline{(σ_{k−1}, σ_{k})}.

Writing:

W_{s,p} · \frac{α}{u} = η , W_{s,p} · \frac{β}{v} = ζ ,

we then have:

W_{s,p} · x = ψ(η) − ψ(ζ) ,

(up to a non-zero constant depending only on p).

Now the computation of dim S_{1}(N, χ_{−p}) is performed in the following steps.

1. Construct a ℂ-basis {b_{i}} for H^{1}(Γ, U_{0,1})_{+} where each b_{i} is a ℤ-linearcombination of the above symbols \overline{(u, v)}. Let φ_{i} be the corresponding projections, i.e.:

x = ∑_{i} φ_{i}(x)b_{i} , for x ∈ H^{1}(Γ, U_{0,1})_{+}.

2. Choose a large prime number l (≠ 2, 3) and an element x ∈ X_{2}(N, 1)_{+} , which is a ℤ-linearcombination of the symbols \overline{(u, v)}. If dim S_{2}(Γ_{0}(N)) = dim H^{1}(Γ, U_{0,1})_{+} we may choose any x in H^{1}(Γ, U_{0,1})_{+}. In any case, it can always be determined whether a given x belongs to X_{2}(N, 1)_{+} = Ker(r_{+}) , since the map r_{+} can be determined explicitely.

Consider the matrix:

C(x) = ( \begin{matrix}φ_{1}(T_{1}x) & … & φ_{1}(T_{n}x) \\ ⋮ & & ⋮ \\ φ_{d_{1}}(T_{1}x) & … & φ_{d_{1}}(T_{n}x) \end{matrix} ) ,

where d_{1} = dim H^{1}(Γ, U_{0,1})_{+} and n = \frac{1}{6}μ(N) with μ(N) = [Γ : Γ_{0}(N)]. The numbers φ_{i}(T_{j}(x)) are all in ℤ[\frac{1}{6}]. We can thus reduce C(x) mod l and compute its rank mod l. If this rank is not d_{0} = dim S_{2}(Γ_{0}(N)), choose another x_{1} ∈ X_{2}(N, 1)_{+} , and compute the rank of:

(\frac{C(x)}{C(x_{1})}) .

If we do not obtain the rank d_{0} after a reasonable amount of effort, choose another prime number l_{1} and repeat the process with l_{1}.

This process must eventually terminate, and when it does, we have a prime number l ≠ 2, 3 , and a number of elements x_{1}, x_{2}, … ∈ X_{2}(N, 1)_{+} such that the matrix:

C = ( \underline{C(x_{1})} ; \underline{C(x_{2})} ; ⋮ )

has rank d_{0} mod l.

Experimentally, it has been found that if one chooses l large enough (for example l ∼ 3 · 10^{4}) then one finds in most cases 1 or 2 elements x_{i} ∈ X_{2}(N, 1)_{+} such that this matrix C has rank d_{0} mod l.

3. Compute the matrix:

D = ( \underline{D(x_{1})} ; \underline{D(x_{2})} ; ⋮ ) ,

where:

D(x_{i}) = ( \begin{matrix}φ_{1}(T_{1}W_{s,p}x_{i}) & … & φ_{1}(T_{n}W_{s,p}x_{i}) \\ ⋮ & & ⋮ \\ φ_{d_{1}}(T_{1}W_{s,p}x_{i}) & … & φ_{d_{1}}(T_{n}W_{s,p}x_{i}) \end{matrix} ) .

Then D has coefficients in ℤ[\frac{1}{6}].

4. Put δ_{s,p} = 2^{s−3}(p + 1). Use Gauss-elimination on the matrix (C∥D) to bring it in the form:

(+)  ( A ; …; 0  D̃  ) ,

\overbrace{δ_{s,p}}  \overbrace{n}

where A has “echelon form”:

( \begin{matrix}∗0 … 0 \\ & ∗0 … 0 \\ & & . & . & . \\ & 0 & & & & ∗0 … 0 \end{matrix} )

with non-zero *’s. Then use Gauss-elimination on D̃ to bring it in the form:

(++)  ( \begin{matrix}A′ \\ 0 & D′ \end{matrix}  ) ,

\overbrace{δ_{s,p}}

again with A′ in echelon form. The dimension of S_{1}(Γ_{0}(2^{s+1}p), χ_{−p}) is then the rank of D′.

Let us observe the following simple fact: That if l ≠ 2, 3 is a prime number such that the matrix C in step 2 has rank d_{0} mod l and such that the *’s in (+) , and hence in (++) , are all invertible mod l , then dim S_{1}(Γ_{0}(2^{s+1}p), χ_{−p}) is the rank of D′ mod l. These conditions can be checked by computations modulo l if the upper left parts of (+) and (++) appear in upper triangular form:

( \begin{matrix}∗ \\ & . & . & . \\ 0 & & & & ∗ \end{matrix} ) ,

with non-zero *’s mod l; this turned out to be the case in all the cases considered below for the following choice of l:

l = 32749.

This circumstance of course reduced the computational complexity of step 4 significantly.

Recall our notation:

d_{0} = dim S_{2}(Γ_{0}(2^{s+1}p)) ,
d_{1} = dim H^{1}(Γ, U_{0,1})_{+} ,
x_{1}, x_{2}, …: elements of X_{2}(2^{s+1}p, 1)_{+} giving the matrix C in step 2,

and put:

t_{1} = rank (D̃) ,
t_{2} = rank (D′) = dim S_{1}(Γ_{0}(2^{s+1}p), χ_{−p}).

The results of the computations are then given by the following table:

s  p  d_{0}  d_{1}  x_{1}, x_{2}, …  t_{1}  t_{2}
1  487  242  242  \overline{(1, 19)}  120  13

1  751  374  374  \overline{(1, 14)}  186  31

1  887  442  442  \overline{(1, 14)}  220  46

1  919  458  458  \overline{(1, 9)}  228  31

1  2083  1040  1040  \overline{(1, 10)}  519  28

2  73  71  71  \overline{(1, 17)}  34  2

3  73  143  145  \overline{(1, 89)}  69  5

4  73  289  293  \overline{(1, 89)}  141  16

2  193  191  191  \overline{(1, 21)}  94  2

3  193  383  385  \overline{(1, 25)}  189  5

4  193  769  773  \overline{(1, 3)}  381  16

\overline{(1, 11)}

The computing time was in no case greater than for the case (s, p) = (1, 2083), which required about 40 hours of computation.

Let us then verify the Artin conjecture for the representations of A_{5}-type (as well as for their twists by characters of Gal (\overline{ℚ}/ℚ)) listed in section 0.

1.3. N = 2^{2}p, ε = χ_{−p} with p ∈ {487, 751, 887, 919}: In each of these cases the information in the table of section 0 allows us immediately to write down the dimension of:

S_{1}(Γ_{0}(p), χ_{−p}) ,

since there are no representations of A_{5}-type with conductor p and determinant character χ_{−p} in any of these cases. We find that dim S_{1}(Γ_{0}(p), χ_{−p}) is 3, 9, 14, 9 for p = 487, 751, 887, 919, respectively. Since, as is easily seen,

dim S_{1}^{new}(Γ_{0}(2p), χ_{−p}) = 0 ,

for any prime p, we have:

dim S_{1}^{new}(Γ_{0}(4p), χ_{−p}) = dim S_{1}(Γ_{0}(4p), χ_{−p}) − 3 · dim S_{1}(Γ_{0}(p), χ_{−p}) .

Hence we find:

dim S_{1}^{new}(Γ_{0}(4p), χ_{−p}) = 4 , for p = 487, 751, 887, 919.

1.4. N = 2083 , ε = χ_{−2083}: Here we note that we obtain the Fourier expansions of g_{s,p} · f_{i}, where (f_{i}) is a basis of V = S_{1}(Γ_{0}(4 · 2083), χ_{−2083}), from the matrix D′ at the end of step IV. Dividing these by the Fourier expansion of:

g_{1,2083}(z) = θ_{2}(4z) θ_{2}(4 · 2083z)

at ∞, we obtain the Fourier expansion of the f_{i}’s. This permits the determination of the subspace of V consisting of forms f ∈ V such that f(4z) ∈ V , and this subspace is precisely S_{1}(Γ_{0}(2083), χ_{−2083}). We find that:

dim S_{1}(Γ_{0}(2083), χ_{−2083}) = 7 .

1.5. N = 2^{5} · p , ε = χ_{−p} with p ∈ {73, 193} : If p is one of the prime numbers 73 or 193, we introduce the following notation:

α_{i} = dim S_{1}^{new}(Γ_{0}(2^{i}p), χ_{−p}) for i = 2, 3, 4, 5.

Then from the table of section 1.2, we obtain in each of the cases:

dim S_{1}(Γ_{0}(2^{2}p), χ_{−p}) = α_{2} ,
dim S_{1}(Γ_{0}(2^{3}p), χ_{−p}) = 2α_{2} + α_{3} = 2 ,
dim S_{1}(Γ_{0}(2^{4}p), χ_{−p}) = 3α_{2} + 2α_{3} + α_{4} = 5 ,
dim S_{1}(Γ_{0}(2^{5}p), χ_{−p}) = 4α_{2} + 3α_{3} + 2α_{4} + α_{5} = 16 .

Recall the remark at the end of section 0 that:

dim S_{1}^{new}(Γ_{0}(2^{2}p), χ_{−p}) > 0 for p = 73 and p = 193 .

Thus α_{2} ≥ 1 and we then find successively α_{2} = 1, α_{3} = 0, α_{4} = 2 , and finally:

dim S_{1}^{new}(Γ_{0}(2^{5}p), χ_{−p}) = α_{5} = 8 ,

for both p = 73 and p = 193 .

1.6. The results of sections 1.3, 1.4, 1.5 together with the table of section 0 and the discussion in the introduction of I, give us a verification of the Artin conjecture for all the representations listed in section 0 and for the twists of these by characters of Gal (\overline{ℚ}/ℚ) .

References

[1] A.O.L. Atkin, J. Lehner: Hecke operators on Γ_{0}(m) .

Math. Ann. 185 (1970), pp. 134-160.

[2] L. Merel: Opérations de Hecke pour Γ_{0}(N) et fractions continues.

Ann. Inst. Fourier 41(3), (1991).

[3] T. Miyake: Modular Forms.

Springer Verlag, 1989.