An Algorithm for Massive Realizations of Projective Linear Groups as Galois Groups

over ℚ via Modular Representations

Luis Dieulefait^{∗} and Nuria Vila^{†} Dept. d’Algebra i Geometria, Universitat de Barcelona Gran Via de les Corts Catalanes 585 08007 - Barcelona Spain

August 23, 1999

1 Introduction

The images of the ℓ-adic Galois representations attached to a newform f = ∑_{n=1}^{∞} a_{n}q^{n} of weight 2 on Γ_{0}(N) are known to be “as large as possible” for almost every ℓ, with the sole condition that f doesn’t have complex multiplication (CM), thanks to the work of Momose and Ribet (cf. [Mo 81] , [Ri 85] ). Actually, if f has any inner twist, this is only true for the restriction of these representations to a certain open subgroup of G_{ℚ} = Gal(\overline{ℚ}/ℚ) defined in terms of the twisting characters. Even in this case, thanks to a theorem of Papier, something can be said about the image of G_{ℚ} (see [Ri 85] and [D]).

In this paper we will restrict ourselves to newforms without inner twists and without CM. We will give an algorithm for an explicit determination of the finitely many primes ℓ (seeming to be) exceptional, i.e., such that the image of the corresponding representation is not “as large as possible” . As a consequence, we will be able to realize many families of projective linear groups over finite fields as Galois groups over ℚ.

∗supported by TMR - Marie Curie Fellowship ERB4001GT974451

†partially supported by DGES grant PB96-0970-C02-01

We have concentrated our efforts on newforms f such that ℚ_{f}, the number field generated by its coefficients a_{n}, is an abelian extension of ℚ, to get a simple characterisation of primes with a prescribed residue class degree. The examples are designed to realize, for high density sets of primes, the following groups: P SL(2, 𝔽_{ℓ^{2}}), P GL(2, 𝔽_{ℓ^{3}}) and P SL(2, 𝔽_{ℓ^{4}}) as Galois groups over ℚ. In particular, the groups in the exponent 2 case will be realized as Galois groups over ℚ for every 3 < ℓ < 5000000, and the ones in the exponent 3 case for every 3 < ℓ < 500000.

2 The Algorithm

Let f ∈ S_{2}(N) = S_{2}(Γ_{0}(N)) be a newform, i.e., a normalized eigenform for the whole Hecke algebra. We assume from now on that f does not have CM nor inner twists. Let ℚ_{f} be the number field generated by its coefficients a_{n} and O its ring of integers. For every prime ℓ put: O_{ℓ} = O ⊗_{ℤ} ℤ_{ℓ} and ℚ_{f,ℓ} = ℚ_{f} ⊗_{ℚ} ℚ_{ℓ}

By Deligne’s theorem, there exists a continuous Galois representation:

ρ_{ℓ} : Gal(\overline{ℚ}/ℚ) → GL(2, O_{ℓ}) ⊆ GL(2, ℚ_{f,ℓ})

unramified outside ℓN satisfying, for every prime p ∤ ℓN :

trace ρ_{ℓ}(F rob p) = a_{p}, det ρ_{ℓ}(F rob p) = p

Let G_{ℚ} = Gal(\overline{ℚ}/ℚ) and G_{ℓ} = ρ_{ℓ}(G_{ℚ}), closed subgroup of GL(2, O_{ℓ}). From the decompositions: ℚ_{f,ℓ} = ∏_{λ|ℓ} ℚ_{f,λ} and O_{ℓ} = ∏_{λ|ℓ} O_{λ} we get a decomposition of ρ_{ℓ} as direct sum of representations :

ρ_{λ} : G_{ℚ} → GL(2, O_{λ}) ⊆ GL(2, ℚ_{f,λ})

Let λ be a prime in ℚ_{f}. Consider the reduction \overline{ρ}_{λ} of ρ_{λ} , obtained by composing ρ_{λ} with the reduction map: GL(2, O_{λ}) → GL(2, 𝔽_{λ}), where 𝔽_{λ} is the residue field of λ. Let ℓ = ℓ(λ) be the rational prime such that λ | ℓ and let \overline{G}_{λ} be the image of \overline{ρ}_{λ}. In order to compute the images of the representations \overline{ρ}_{λ} we will use the following result of Ribet [Ri 85]:

Theorem 2.1 : For almost every λ we have:

a) The representation \overline{ρ}_{λ} is an irreducible 2-dimensional representation over 𝔽_{λ}

b) The order of the group \overline{G}_{λ} = \overline{ρ}_{λ}(G_{ℚ}) is divisible by ℓ

In the next section we will give an algorithm to compute a set of primes containing those that do not verify these two conditions. The following result of Ribet [Ri 85] completes the determination of the images:

Theorem 2.2 : Let A_{ℓ} = {x ∈ GL(2, O_{ℓ}) / det(x) ∈ ℤ_{ℓ}^{∗}} The equality G_{ℓ} = A_{ℓ} holds for almost every prime. In fact, this equality holds whenever the following conditions are satisfied :

0) ℓ does not ramify in ℚ_{f}/ℚ

1) The determinant map G_{ℓ} → ℤ_{ℓ}^{∗} is surjective

2) ℓ ≥ 5

3) G_{ℓ} contains an element x_{ℓ} such that: (trace x_{ℓ})^{2} generates O_{ℓ} as a ℤ_{ℓ}- algebra

4) for each λ | ℓ, the group \overline{ρ}_{λ}(G_{ℚ}) is an irreducible subgroup of GL(2, 𝔽_{λ}) whose order is divisible by ℓ.

To see that these conditions hold for almost every prime, theorem 2.1 is applied (to deal with condition (4)) together with the following facts: -there exists an integer v such that v is relatively prime to N and a_{v}^{2} generates ℚ_{f} over ℚ (condition (3)) -the determinant of ρ_{ℓ} coincides on G_{ℚ} with the ℓ-adic cyclotomic character χ_{ℓ} (condition (1)).

This result allows us to realize projective linear groups over finite fields as Galois groups over ℚ in the following way: Let ℓ be a prime verifying the conditions of theorem 2.2 and λ a prime in O over ℓ. Then \overline{G}_{λ} ⊆ GL(2, 𝔽_{λ}) and P (\overline{G}_{λ}), its image in P GL(2, 𝔽_{λ}), is (cf. [Re-Vi 95]): P SL(2, 𝔽_{λ}) if [𝔽_{λ} : 𝔽_{ℓ}] is even, P GL(2, 𝔽_{λ}) otherwise.

2.1 The Exceptional Primes for Theorem 2.1

2.1.1 Reducible Representations

Let us suppose that for a prime λ ∈ O dividing ℓ, \overline{ρ}_{λ}(G_{ℚ}) is a reducible subgroup of GL(2, 𝔽_{λ}). Then

\overline{ρ}_{λ} ≅ ( \begin{matrix}ϕ_{1} & ∗ \\ 0 & ϕ_{2} \end{matrix} )

with ϕ_{i} = ϵ_{i}\overline{χ}_{ℓ}^{m_{i}}, for i = 1, 2, ϵ_{1}, ϵ_{2} Dirichlet characters unramified outside N with image in 𝔽_{λ} and \overline{χ}_{ℓ} the cyclotomic character mod ℓ. By Deligne’s theorem, for every prime p ∤ ℓN we have:

a_{p} = trace ρ_{λ}(F rob p) ≡^{mod λ} ϕ_{1}(F rob p)+ϕ_{2}(F rob p) = ϵ_{1}(p) p^{m_{1}}+ϵ_{2}(p) p^{m_{2}}

p = det ρ_{λ}(F rob p) ≡^{mod λ} ϕ_{1}(F rob p) ϕ_{2}(F rob p) = ϵ_{1}(p) ϵ_{2}(p) p^{m_{1}+m_{2}}

¿From the last formula we see that: m_{1}+m_{2} ≡ 1 (mod ℓ−1) and ϵ_{1}(p)ϵ_{2}(p) = 1 for every p ∤ N, so that we can choose 0 ≤ m_{1} < m_{2} < ℓ − 1. Generalizing results of [Sw 73] to the case of general level, Faltings and Jordan proved in [Fa-Jo 95] the following result, that we will state for general weight, i.e., when the representation comes from a newform f ∈ S_{k}(N):

Theorem 2.3 : Suppose that the representation \overline{ρ}_{λ} is reducible. Then if ℓ > k, ℓ ∤ N, \overline{ρ}_{λ} = ϵ_{1} ⊕ ϵ_{2}\overline{χ}_{ℓ}, with the characters ϵ_{i} unramified outside N.

Carayol and Liv ([Ca 89] , [Li 89]) have given bounds for the conductors of modular Galois representations. Using these results, together with the ones of Faltings and Jordan, we have (see [Fa-Jo 95]) :

Corollary 2.4 : Let f be a newform of weight 2 and level N and λ | ℓ a prime in O such that \overline{ρ}_{λ} is reducible. Then if ℓ > 2, ℓ ∤ N, we have, for every p ∤ ℓN:

a_{p} ≡ ϵ(p) + p ϵ^{−1}(p) (mod λ)

with ϵ a character unramified outside N whose conductor c verifies: c^{2} | N.

In the algorithm, we will only use the following two cases of this corollary:

a) if p ≠ ℓ, p ≡ 1 (mod c) , then: a_{p} ≡ 1 + p (mod λ).

b) if p ≠ ℓ, p ≡ −1 (mod c) , then: a_{p} ≡ ±(1 + p) (mod λ).

This congruences can’t be equalities, because: |a_{p}| ≤ 2\sqrt{p} . So that only finitely many λ can satisfy them.

In the case of prime level N, a better criterion for reducibility was given by Mazur in [Ma 77]. Here the condition ℓ ∤ N can be removed and the reducible primes are the Eisenstein primes, defined as follows (see [Ma 77]):

Definition 2.5 : Let T be the Hecke ring generated by the Hecke operators acting on J_{0}(N). The Eisenstein ideal ℑ ⊆ T is the one generated by the elements: 1 + l − T_{l} (l ≠ N ), 1 + ω. An Eisenstein prime is a prime ideal β ⊆ T in the support of ℑ. Let n = num(\frac{N−1}{12}). We have a one-to-one correspondence between Eisenstein primes β and prime factors p of n, given by:

Prime factors of n ↔ Eisenstein primes

p ↔ (ℑ,p)

Besides, for every Eisenstein prime β, T/β = F_{p} . In particular, whenever the inertia is nontrivial the involved prime will not be Eisenstein.

Lemma 2.6 Let f be a newform of weight 2 and prime level N. Let λ | ℓ, ℓ ≥ 5, be a prime in O such that \overline{ρ}_{λ} is reducible. Then: ℓ | N − 1

2.1.2 The second condition

We now turn to the second condition in theorem 2.1. Let λ and ℓ be as before and suppose that the order of \overline{G}_{λ} is not divisible by ℓ. The reducible case being already treated, we will also assume that \overline{ρ}_{λ} is irreducible. We consider P (\overline{G}_{λ}), its image in P GL(2, 𝔽_{λ}). The only possibilities for this image are to be:

1) cyclic

2) dihedral or

3) isomorphic to one of the following special groups : A_{4}, S_{4}, A_{5}

In case 1), \overline{ρ}_{λ} would be reducible, contradicting our assumption.

In case 2) there exists a Cartan subgroup C_{λ} of GL(2, 𝔽_{λ}) such that \overline{G}_{λ} is contained in the normaliser N_{λ} of C_{λ}, but not in C_{λ}. Let φ_{λ} : G_{ℚ} → {±1} be the composition:

G_{ℚ} → \overline{G}_{λ} ⊆ N_{λ} ↠ N_{λ}/C_{λ} ≅ \{±1\}

The kernel of φ_{λ} is then an open subgroup of G_{ℚ} of index 2, so its fixed field K_{λ} is a quadratic field unramified outside ℓN. We now consider the restriction of \overline{ρ}_{λ} to I, an inertia subgroup of G_{ℚ} for ℓ. Its semisimplification can be described in terms of fundamental characters as in [Se 87]. Using the fact that the weight of f is 2, it follows (cf. [Ri 97]) that the image of I in P GL(2, 𝔽_{λ}) is cyclic of order ℓ ± 1. Then, this is a cyclic subgroup of the dihedral group P (\overline{G}_{λ}) with order, if ℓ > 3, greater than 2, and therefore must be contained in its center . This implies that the quadratic field K_{λ} does not ramify at ℓ, if ℓ > 3. This is enough to make the dihedral case computable, but there is another observation that will simplify the computations if N has any prime factor q such that q ∥ N. In this case, the ramification subgroup of \overline{G}_{λ} for the prime q is unipotent, if q divides the conductor of \overline{ρ}_{λ}, and trivial otherwise. This, together with the assumption that the order of \overline{G}_{λ} is prime to ℓ, implies that these ramification groups are in fact trivial for all these primes. We conclude that the field K_{λ} can only ramify at those prime factors r of N (if any) such that r^{2} | N. Let N′ be the product of these prime factors. If we call α the quadratic character corresponding to K_{λ} we have proved the following

Lemma 2.7 : Suppose that \overline{ρ}_{λ} falls in case (2). Then if ℓ > 3, we have:

a_{p} ≡ α(p) a_{p} (mod λ)

for every p ∤ ℓN; where α is a quadratic character unramified outside N′. In particular, if N is squarefree, no λ verifies the hypothesis of this lemma.

In case 3) it is known (see [Ri 85] ) that for every p ∤ ℓN

a_{p}^{2} ≡ 0, p, 2p or 4p (mod λ), or a_{p}^{4} − 3pa_{p}^{2} + p^{2} ≡ 0 (mod λ) (2.1)

In [Ri 85] it is proved that this can hold only for finitely many ℓ. Actually, we will need to apply (2.1) only to the prime ℓ = 5 because in the case of weight 2 and irreducible \overline{ρ}_{λ}, using again the description of the semisemplification of the restriction of \overline{ρ}_{λ} to an inertia subgroup for the prime ℓ in terms of fundamental characters, it follows that there is an element of order ℓ ± 1 in the projective image of this inertia subgroup, and therefore in P (\overline{G}_{λ}), and this rules out case 3) if ℓ ≥ 7 (cf. [Se 72], [Ri 97]).

It is proved in [Ri 97] that in the squarefree level case, 3) can not hold not even for 5, so that in this case the following result holds:

Theorem 2.8 : Let f be a newform of weight 2 and level N squarefree. Let λ | ℓ, ℓ ≥ 5, be a prime in O such that \overline{ρ}_{λ} is irreducible. Then the image of \overline{ρ}_{λ} has order multiple of ℓ.

2.2 Description of the algorithm

At this point we can give a complete description of the algorithm to compute the exceptional primes in theorem 2.2. Suppose that we have a newform f ∈ S_{2}(N) not having CM nor inner twists, and we consider the representation ρ_{ℓ} for a prime ℓ ≥ 5 . The conditions of theorem 2.2 are treated as follows :

• Step (i): It is enough, in order to verify conditions 0) and 3), to find a coefficient a_{p} with p ∤ ℓN such that: a_{p}^{2} (and a fortriori a_{p}) generates ℚ_{f} over ℚ and ℓ ∤ disc(a_{p}^{2}).

• Step (ii) : Find the maximal c such that c^{2} | N, and a coefficient a_{p} with p ≡ 1 (mod c) such that

p ∤ ℓN and a_{p} − (1 + p) ≢ 0 (mod λ)

for every λ | ℓ. Then, if ℓ ∤ N, the reducible case is excluded.

To verify the congruences, it’s enough to compute the norm of a_{p} − (1 + p) and check that ℓ ∤ N (a_{p} − (1 + p)). Similarly, take a prime p ≡ −1 (mod c) with

p ∤ ℓN and a_{p} ≢ ±(1 + p) (mod λ)

It suffices to verify that ℓ ∤ N (a_{p} ± (1 + p)).

In the prime level case reducible primes ℓ ≥ 5 are those ℓ | N − 1. In particular, the prime N (untractable with the other approach ) is not reducible.

Remark: The remaining conditions have not to be verified in the case of squarefree level. In this case, all exceptional primes ℓ ≥ 5 are detected with steps (i) and (ii) .

• Step (iii) : For every quadratic character α ramifying only at the primes q such that q^{2} | N, find a coefficient a_{p} with:

p ∤ Nℓ, a_{p} ≠ 0, α(p) = −1 and a_{p} ≢ −a_{p} (mod λ)

for every λ | ℓ. Then, the dihedral case is excluded. To verify the congruences, it suffices to check that ℓ ∤ N (a_{p}).

• Step (iv): Take a prime p ∤ 5N and compute the resultants of Q, the minimal polynomial of a_{p} , and the polynomials P_{i} , i = 1, ..., 5 in formula (2.1):

x^{2}, x^{2} − p , x^{2} − 2p , x^{2} − 4p , x^{4} − 3px^{2} + p^{2}

If

5 ∤ Res(P_{i}, Q), i = 1, ..., 5

it follows that none of the P_{i} has a_{p} as a root mod ˆ5, for every ˆ5 | 5. Then, the case of special images is excluded.

We conclude the section with a brief explanation of how, for the newforms in the examples, we have determined the corresponding number fields and checked that the newforms have no CM nor inner twists.

Determination of the number field ℚ_{f}: Let f be a newform in S_{2}(N). If f has a coefficient a_{p}, p ∤ N, which is a simple root of the characteristic polynomial of the Hecke operator T_{p} acting on S_{2}^{new}(N), then it follows that ℚ(a_{p}) = ℚ_{f}.

CM-Inner Twists: It is known that a newform f ∈ S_{2}(N) doesn’t have CM nor inner twists if the level is squarefree. It is also known that if there exists a prime q strictly dividing the level N, i.e., q ∥ N, then in S_{2}(N) there is no newform with CM. In the remaining cases, to avoid CM one has to check, for every character α ramifying only at primes dividing the level N, that there exists a coefficient a_{p} with α(p) = −1, a_{p} ≠ 0. Finally, to avoid inner twists in the non-squarefree level case, one has to find a coefficient a_{p} with ℚ(a_{p}^{2}) = ℚ_{f}, as in step (i) of the algorithm .

3 Examples- ℚ_{f} quadratic or abelian quartic

All the examples of newforms have been computed with an algorithm implemented by W. Stein based on ideas of J. Cremona. The algorithm described in the previous section and all other computations have been done with PARI GP.

3.1 Applying the algorithm

In this section we look for newforms f of weight 2 such that the field ℚ_{f} is quadratic or quartic and abelian. We made a table of all weight 2 newforms with level up to 640 and we found 10 with fields of these types (all different fields). We verified that these newforms have no CM nor inner twists. Then, we applied the algorithm to find the exceptional primes for the representations attached to each one of them. The results are summarised in the following table:

[Level / ℚ_{f}]  [Step (i) : p, a_{p}]
1 23/ x^{2} − 5  2, x^{2} + x − 1
2 29/ x^{2} − 2  2, x^{2} + 2x − 1
3 410/ x^{2} − 3  3, x^{2} − 2x − 2
4 410/ x^{2} − 17  7, x^{2} − 2x − 16 // 11, x^{2} − 2x − 16
5 414/ x^{2} − 7  5, x^{2} + 2x − 6 // 11, x^{2} − 2x − 6
6 496/ x^{2} − 33  5, x^{2} − 3x − 6 // 13, x^{2} − 2x − 32
7 418/ x^{2} − 13  3, x^{2} + 3x − 1
8 546/ x^{2} − 57  5, x^{2} + x − 14 // 11, x^{2} − 3x − 12
9 226/ F_{20} 3, x^{4} − 2x^{3} − 6x^{2} + 12x − 4 // 5, x^{4} − 4x^{3} − 4x^{2} + 16x − 4
10 358/ V_{17}  3, x^{4} + 2x^{3} − 7x^{2} − 8x − 1 // 5, x^{4} + 7x^{3} + 12x^{2} − 3x − 13
[Step (i) : ℓ] [Step (ii):p, a_{p}] [Step (ii) : ℓ] [Step (iii) : p, a_{p} ]
1  5  —-  11  —-
2  empty  —-  7  —-
3  empty  3  5, 41  —-
4  17  3, 2  5, 41  —-
5  7  7, 2  23  5 and 11
6  11  3, −2  31  3 or // 5 and 13
7  13  3  11, 19  —-
8  19  5 // 11  7, 13  —-
9  5  from 3 to 17  113, 19  —-
10  17  3 // 5  179  —-
[Step (iii): ℓ] [Step (iv) : p, a_{p}] [Step (iv) :5?] [Results]
1  —-  —-  —-  5, 11
2  —-  —-  —-  7
3  —-  —-  —-  5, 41
4  —-  —-  —-  5, 41
5  empty  11  NO  7, 23
6  empty  13  NO  11, 31
7  —-  —-  —-  11, 13, 19
8  —-  —-  —-  7, 13, 19
9  —-  —-  —-  5, 19, 113
10  —-  —-  —-  17, 179

Table 1

Explanation of Table 1: In this table we exhibit 10 cusp forms and the result of the algorithm of the previous section applied to them. The first eight are newforms with quadratic number field, and the last two examples correspond to the abelian quartic field case: F_{20} denotes the maximal real subfield of the cyclotomic field of 20-th roots of unity, and V_{17} a totally real subfield of the cyclotomic field of 17-th roots of unity. The minimal polynomials of the coefficients used at each step are included, but the value is not repeated if it has already appeared in a previous column. In every odd column, the exceptional primes found at the corresponding step are listed, and they are put together in the last column. We remark that here exceptional primes are those for which the algorithm was not able to prove that the image is “as big as possible” , but that it may be so for some of them. For instance all primes appearing in the fifth column are primes dividing the level, except for the Eisenstein primes 11 and 7 in the first and second row and the prime 19 appearing in the ninth row. In this last case, having computed the coefficients from a_{3} to a_{17}, the corresponding representation seems to be reducible.

3.2 Galois realizations of groups P SL(2, 𝔽_{ℓ^{2}})

For each of the examples in Table 1, we consider primes with residue class degree 2. The condition for this is easily given in terms of congruences , or using Legendre ’s symbols. Thus , from Table 1 and the discussion following theorem 2.2 we conclude that for the following sets of primes the groups P SL(2, 𝔽_{ℓ^{2}}) are Galois groups over ℚ : ℓ ≥ 5 and ℓ ≡ 2, 3 (mod 5) (already proved in [Me 88]) ℓ ≢ ±1 (mod 8) (\frac{3}{ℓ}) = −1 ℓ ≠ 5, 41 ℓ ≡ ±3, ±5, ±6, ±7 (mod 17) ℓ ≠ 5, 41 (\frac{7}{ℓ}) = −1 ℓ ≠ 23 (\frac{33}{ℓ}) = −1 (\frac{13}{ℓ}) = −1 ℓ ≠ 11, 19 (\frac{57}{ℓ}) = −1 ℓ ≠ 13 ℓ ≡ 9, 11 (mod 20) ℓ ≡ ±2, ±8 (mod 17) ℓ ≠ 179

Observe that the primes listed as exceptional at some of the rows: 5, 41, 23, 11, 19, 13, 179, are covered by some other row. After some other elementary simplifications, we obtain the following result:

Theorem 3.1 : The group P SL(2, 𝔽_{ℓ^{2}}) is a Galois group over ℚ whenever ℓ is a prime greater than 3 satisfying one of the following conditions: ℓ ≢ ±1 (mod 120) ℓ ≢ ±1, ±4 (mod 17) (\frac{7}{ℓ}) = −1 or (\frac{11}{ℓ}) = −1 or (\frac{13}{ℓ}) = −1 or (\frac{19}{ℓ}) = −1

In [Re-Vi 95], using the Galois representations attached to cusp forms of weight k for SL(2, ℤ), other projective linear groups are realized. In particular, for k = 24, 28, 30, 32, 34, 38 , there is a cusp form f with a quadratic ℚ_{f} and it is proved that, except for a few computed exceptions, for every prime ℓ inert in some of these quadratic fields the group P SL(2, 𝔽_{ℓ^{2}}) is a Galois group over ℚ. We computed the primes ℓ ≥ 5 up to 5000000 not covered by this result of [Re-Vi 95] nor by theorem 3.1 and we found only 6 :

620759, 878641, 1782959, 3747241, 3871921, 4490639

To cover these primes, we consider three newforms of prime level:

\begin{matrix}Level & a_{2} \\ 41 & x^{3} + x^{2} − 5x − 1 \\ 59 & x^{5} − 9x^{3} + 2x^{2} + 16x − 8 \\ 79 & x^{5} − 6x^{3} + 8x − 1 \end{matrix}

In each case the coefficient a_{2} generates ℚ_{f}. The 6 primes not yet covered have the property that in some of these three ℚ_{f} there is a prime above them with residue class degree 2. Applying the algorithm, it is easy to see that none of them is exceptional.

Theorem 3.2 : For every prime 5 ≤ ℓ < 5000000, P SL(2, 𝔽_{ℓ^{2}}) is a Galois group over ℚ.

3.3 Galois realizations of groups P SL(2, 𝔽_{ℓ^{4}})

Let us consider the inert primes in the two examples of quartic abelian number field: F_{20} and V_{17}, which are given by the conditions: ℓ ≡ 2, 3 (mod 5) and ℓ ≡ ±3, ±5, ±6, ±7 (mod 17) , respectively. Among the exceptional primes obtained in these two examples, the only one satisfying these conditions is 113 in the first example. But 113 ≡ −6 (mod 17), so that it is covered by the second example.

Theorem 3.3 : For every prime ℓ ≥ 5 satisfying one of the following conditions: ℓ ≡ 2, 3 (mod 5) ℓ ≡ ±3, ±5, ±6, ±7 (mod 17)

P SL(2, 𝔽_{ℓ^{4}}) is a Galois group over ℚ.

4 Three examples designed to realize P GL(2, 𝔽_{ℓ^{3}}) for many ℓ

In this section we consider three examples. The first two are examples of newforms whose corresponding number field is abelian cubic, namely: ℚ_{f} = F_{7} and ℚ_{f} = F_{9}, where for every n, F_{n} = ℚ(ζ_{n} + ζ_{n}^{−1}), ζ_{n} a primitive n-root of unity. The third example is the only example in all this paper involving relatively big numbers. It consists of a newform whose corresponding number field has degree 27, and is a cubic extension of F_{27}. Using the algorithm, we will compute all exceptional primes in each case, and we will apply the result to the realization of groups P GL(2, 𝔽_{ℓ^{3}}) as Galois groups over ℚ. The following result of Brumer helped us finding these examples (see [Br 96]):

Theorem 4.1 : Let f ∈ S_{2}(N) be a newform not having CM . Suppose that p^{r_{p}} ∥ N . Let s_{p} = ⌈\frac{r_{p}}{2} − 1 − \frac{1}{p−1}⌉ and ζ a primitive p^{s_{p}}-root of unity . Then ℚ_{f} ⊇ ℚ(ζ + ζ^{−1}) if p > 2 (resp. ℚ(ζ^{2} + ζ^{−2}) if p = 2)

So that we looked for newforms without CM with corresponding number field F_{7} and F_{9} in the spaces S_{2}^{new}(343) and S_{2}^{new}(243), respectively, and we found them. We will not use the one of level 343 because there is also a newform f ∈ S_{2}(97) with ℚ_{f} = F_{7} and it is easier to apply the algorithm to this one, because it has prime level.

4.1 Applying the algorithm

• First example: There is a newform f ∈ S_{2}(97) with ℚ_{f} = F_{7}. We will only need the following two coefficients, given by their minimal polynomials:

a_{2} : x^{3} + 4x^{2} + 3x − 1 and a_{17} : x^{3} + 3x^{2} − 4x − 13

Running the algorithm, we find, using both a_{2} and a_{17}, that 7 is the only exceptional prime at step (i). There is no exceptional (Eisenstein) prime at step (ii). The level being prime, we conclude that 7 is the only exceptional prime ℓ ≥ 5 .

• Second example: There is a newform f ∈ S_{2}(243) having not CM nor inner twists, with ℚ_{f} = F_{9}. We apply the algorithm to find the exceptional primes ℓ ≥ 5. At step (i), we use the coefficient : a_{2} : x^{3} − 3x^{2} + 3 to see that there is no exceptional prime. At step (ii), we first observe that we have to take c = 9 because N = 243 = 3^{5}. Then we use the coefficients:

a_{19} : x^{3} + 3x^{2} − 24x + 1 a_{53} : x^{3} − 18x^{2} + 81x − 81

and a_{37}, which is Galois conjugate to a_{19}, to see that there is no exceptional prime. Finally, we use again a_{2} to see that there is no exceptional prime at step (iii) and that 5 is not exceptional at step (iv). So that, in this example, there is no prime ℓ ≥ 5 exceptional.

• Third example: In the space S_{2}^{new}(2187) we haven’t found a newform f with ℚ_{f} = F_{27} but we have found one with ℚ_{f} of degree 27 being a cubic extension of F_{27}. This newform has no CM nor inner twists. The field ℚ_{f} is NOT a Galois field. It is generated by the coefficient a_{2}, with minimal polynomial:

A_{2}(x) = x^{27}+9x^{26}−222x^{24}−459x^{23}+2133x^{22}+7362x^{21}−9045x^{20}−55485x^{19} +4047x^{18}+241677x^{17}+128898x^{16}−643257x^{15}−609714x^{14}+1040283x^{13}

+1377729x^{12}−957987x^{11}−1758753x^{10}+410742x^{9}+1285227x^{8}−5184x^{7}

−520830x^{6} − 53136x^{5} + 106434x^{4} + 14094x^{3} − 8262x^{2} − 972x + 27

To apply step (i) of the algorithm we will use also the coefficient a_{5}, given by the polynomial:

A_{5}(x) = x^{27}+18x^{26}+81x^{25}−399x^{24}−4347x^{23}−4077x^{22}+72270x^{21}+216621x^{20}

−480411x^{19}−2883570x^{18}−56565x^{17}+19123614x^{16}+20821662x^{15}−67078422x^{14}

−137250288x^{13}+98869950x^{12}+423746154x^{11}+82935981x^{10}−662094576x^{9}

−503302086x^{8}+423203940x^{7}+625539429x^{6}+37164501x^{5}−254848680x^{4}

−101328894x^{3} + 21227994x^{2} + 18844893x + 2847447

Instead of factoring the discriminants of a_{2}^{2} and a_{5}^{2}, we compute its greatest common divisor (and check that 5 ∤ disc(a_{2}^{2})). This gives the following exceptional primes: 811 and 7655551041527. We also computed the discriminant of the field ℚ_{f} and we found that these two primes ramify.

Now we turn to step (ii). We have c = 27 because the level is 2187 = 3^{7}. We use the following two coefficients: a_{53} given by:

x^{27}+90x^{26}+3321x^{25}+58986x^{24}+286659x^{23}−8402130x^{22}−173468061x^{21} −997212708x^{20} +8609338368x^{19} +174882898125x^{18} +879497766396x^{17} −3839798343339x^{16}−72574129126800x^{15}−322522524201888x^{14}+507275006736879x^{13} +11881100841555393x^{12}+50880136840412436x^{11}+32267290657623723x^{10} −583118269440460587x^{9}−2803147835818522302x^{8}−6056528831138196135x^{7} −5328052040331553149x^{6}+4040855350703314974x^{5}+14364803283290718444x^{4} +11649146213084963013x^{3}+205986024736020486x^{2}−3863232567933946101x −1195010794014230997

and a_{109}, given by : x^{27} − 1161x^{25} + 1143x^{24} + 552258x^{23} − 943704x^{22} − 141492546x^{21} + 288666909x^{20} +21785693796x^{19}−42415128438x^{18}−2149133334543x^{17}+3249157564620x^{16} +140451320065959x^{15} − 118337820691194x^{14} − 6119920386937629x^{13} +286365050094708x^{12}+174146647974621273x^{11}+137088225897636096x^{10} −3067205587844711523x^{9}−4979500588408020255x^{8}+29948677258338999450x^{7} +76790301210845625861x^{6}−119349417294626396391x^{5}−535760443536103781838x^{4} −167198875027467682545x^{3}+1273594079757375456489x^{2}+1827054356350586351796x +769405689432892210627

We found no exceptional prime at step (ii).

We finally check, using only a_{2}, that there are no exceptional primes at step (iii) and that 5 is not exceptional at step (iv). We conclude that the only exceptional primes are 811 and 7655551041527.

4.2 Galois realizations

We now want to apply these three examples to the realization of groups P GL(2, 𝔽_{ℓ^{3}}) as Galois groups over ℚ. To do this we restrict, in each of the examples, to those primes ℓ ≥ 5 unramified in ℚ_{f} such that there exists a place λ ∈ ℚ_{f} above them having residue class degree 3 . In the first two examples, this is equivalent to say that ℓ is inert, and this gives: ℓ ≢ ±1, 0 (mod 7) and ℓ ≢ ±1 (mod 9). In the third example, these primes can be characterized in terms of the decomposition of the polynomials A_{2} and A_{5}. The fact that [ℚ_{f} : F_{27}] = 3 implies that all these primes ℓ verify: ℓ ≡ ±1 (mod 9), i.e., they are in the complement of the set of primes covered by the second example. We have already shown that in our examples the only exceptional primes are those that ramify in ℚ_{f}. Thus , we have:

Theorem 4.2 : For every prime ℓ ≥ 5 satisfying one of the following conditions: ℓ ≢ ±1 (mod 7) ℓ ≢ ±1 (mod 9) ℓ ∤ disc(A_{2}) and A_{2} has a cubic factor when reduced mod ℓ ℓ ∤ disc(A_{5}) and A_{5} has a cubic factor when reduced mod ℓ

P GL(2, 𝔽_{ℓ^{3}}) is a Galois group over ℚ.

As in the previous section, we want to combine this result with the one obtained in [Re-Vi 95] using cusp forms f for SL(2, ℤ) with cubic ℚ_{f} and weight k = 36, 40, 42, 44, 46 and 50. They computed for these cases the few exceptional inert primes. We computed the primes ℓ ≥ 5 up to 500000 such that the groups P GL(2, 𝔽_{ℓ^{3}}) are not covered by this result of [Re-Vi 95] nor by theorem 4.2 and we found only 12 : 11087, 97649, 176597, 202987, 237691, 297793 358273, 368803, 394631, 407287, 408437 and 496817 To cover these primes, we introduce four newforms of squarefree level :

\begin{matrix}Level & a_{2} \\ 71 & x^{3} + x^{2} − 4x − 3 \\ 87 & x^{3} − 2x^{2} − 4x + 7 \\ 91 & x^{3} − x^{2} − 4x + 2 \\ 97 & x^{4} − 3x^{3} − x^{2} + 6x − 1 \end{matrix}

In each case the coefficient a_{2} generates ℚ_{f}. The 12 primes not yet covered have the property that in some of these four ℚ_{f} there is a prime above them with residue class degree 3. Applying the algorithm, it is easy to see that none of them is exceptional. Then, we have:

Theorem 4.3 : For every prime 5 ≤ ℓ < 500000, P GL(2, 𝔽_{ℓ^{3}}) is a Galois group over ℚ.

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