On the eigenvalues of the Hecke operator T_{2}.

KEVIN BUZZARD*

Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge, CB2 1SB, UK

ABSTRACT: Let K be the splitting field of the characteristic polynomial of the Hecke operator T_{2} acting on the d-dimensional space of cusp forms of weight k and level 1. We show, for various values of k, that the Galois group Gal(K/ℚ) is the full symmetric group on d symbols.

Let k ≥ 2 be a fixed even integer and S_{k} the complex vector space of cusp forms of weight k for the full modular group SL_{2}(ℤ). Say the dimension of S_{k} is d. Let f ∈ S_{k} be an eigenvector for all the Hecke operators T_{p} as p runs through every rational prime. It is well-known that the eigenvalues a_{p} of T_{p} generate a number field L_{f}. Moreover, it is easy to check that if the characteristic polynomial of T_{p} acting on S_{k} is irreducible over ℚ, then its splitting field K is the compositum of the L_{f}s for f running through the d eigenforms in S_{k}, and K is also the Galois closure of any L_{f} over ℚ.

Because the Galois group Gal(K/ℚ) acts faithfully on the d roots of the characteristic polynomial of T_{p}, we can identify Gal(K/ℚ) with a subgroup of the symmetric group Σ_{d} on d symbols. We restrict now to the case where k = 12l for some prime l. Then the dimension of S_{k} is l.

Theorem. If l ∈ {2, 3, 5, 7, 11, 13, 17, 19} and k = 12l then the characteristic polynomial of T_{2} on S_{k} is irreducible and if K is its splitting field over ℚ then Gal(K/ℚ) ≅ Σ_{l}.

Remarks.

i) If l ≤ 7 then the Galois group of the splitting field of an irreducible polynomial of degree l can be evaluated using standard computer algebra packages like PARI-GP or MAPLE. For l > 7 the packages available at present, to my knowledge, are unable to deal with polynomoials of degree l, so one has to use a trick.

ii) One has dim_{ℂ}S_{k} ≤ 7 for 2 ≤ k ≤ 98, k ≠ 96, and in these cases the referee has calculated Gal(K/ℚ), using the cited computer algebra packages, and has observed that it is isomorphic to the full symmetric group.

Corollary. If l ∈ {2, 3, 5, 7, 11, 13, 17, 19} and k = 12l, then for any cusp eigenform f = Σa_{n}q^{n} of weight k for SL_{2}(ℤ) with a_{1} = 1, the field L_{f} generated over ℚ by the a_{n} has degree l over ℚ and the Galois closure K of L_{f} over ℚ satisfies Gal(K/ℚ) ≅ Σ_{l}.

Proof of corollary. This comes from the fact that for these k we have L_{f} = ℚ(a_{2}).

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The proof of the theorem uses the following lemma.

Lemma. Let l ∈ ℤ be a prime, and let P ∈ ℤ[X] be a monic irreducible polynomial of degree l, with splitting field K over ℚ. Say q is a prime such that the mod q reduction \overline{P} ∈ 𝔽_{q}[X] of P satisfies

\overline{P} = ∏_{i=0}^{r} h_{i}

for distinct irreducible polynomials h_{i} in 𝔽_{q}[X] with the following properties:

i) The degree of h_{0} is 2

ii) The degree of h_{i} is odd, for 1 ≤ i ≤ r.

Then Gal(K/ℚ) ≅ Σ_{l}.

Proof of lemma. Let G be Gal(K/ℚ) identified as a subgroup of Σ_{l}. Now P has distinct roots modulo q and hence if ℘ is a prime of K above q, there is a unique element Frob_{℘} ∈ G, the Frobenius element at

* Research supported by the Science and Engineering Research Council.

℘. Moreover, if d_{i} is the degree of h_{i}, we see that Frob_{℘} is the product of r + 1 disjoint cycles of lengths d_{0}, d_{1}, …, d_{r}. Hence if t = ∏_{i=1}^{r} d_{i} then (Frob_{℘})^{t} is a transposition in G. So the transitive subgroup G of Σ_{l} contains a transposition. This forces G to be the whole of Σ_{l} as can be seen thus: Put an equivalence relation ∼ on the roots of P by setting a ∼ b if either a = b or the transposition (a, b) is an element of G. Then because the action is transitive on the roots, all equivalence classes have the same size, and because l is a prime there can be either 1 or l of them. But there is some transposition in G, and hence there is only 1 equivalence class and so G contains all transpositions and is thus Σ_{l}.

□

Proof of theorem. Clearly it suffices to show that the characteristic polynomial of T_{2} acting on S_{k} for k = 12l, l ∈ {2, 3, 5, 7, 11, 13, 17, 19} satisfies the conditions of the lemma for some q. As remarked earlier, for l ≤ 7 the characteristic polynomial of T_{2} acting on S_{k} can be easily checked to be irreducible and the Galois group of its splitting field over ℚ can also easily checked to be the full symmetric group. We shall restrict ourselves to the case l > 7. Let χ_{k} be the characteristic polynomial of T_{2} acting on S_{k}. To prove that χ_{k} for k = 12l is irreducible, it suffices to show that it is irreducible mod p for some prime p. So we have reduced the theorem to finding, for all k in question, primes p and q for which χ_{k} is irreducible modulo p and satisfies the conditions of the lemma modulo q.

The first calculations of χ_{k} for k ≤ 158 were done by Maeda in [1]. For larger k we calculated χ_{k} modulo many primes without actually calculating χ_{k} itself. The following table of results finishes the proof.

l  prime  Complete factorisation of χ_{12l} modulo this prime.
11  q = 37  (X^{2} + 30X + 34)(X + 4)(X + 6)(X + 11)(X + 14)
(X + 15)(X + 21)(X + 26)(X + 31)(X + 33)
11 p = 479 X^{11} + 189X^{10} + 343X^{9} + 43X^{8} + 424X^{7} + 323X^{6}+
+58X^{5} + 100X^{4} + 131X^{3} + 307X^{2} + 192X + 133
13 q = 29 (X^{2} + 24X + 26)(X + 9)(X + 11)(X + 13)(X + 14)
(X + 16)(X + 17)(X + 22)(X + 24)(X + 26)(X + 27)(X + 28)
13 p = 353 X^{13} + 287X^{12} + 288X^{11} + 304X^{10} + 252X^{9} + 76X^{8} + 139X^{7}+
+218X^{6} + 62X^{5} + 350X^{4} + 249X^{3} + 299X^{2} + 307X + 73
17 p = 263 X^{17} + 123X^{16} + 97X^{15} + 194X^{14} + 30X^{13} + 60X^{12} + 99X^{11} + 2X^{10}+
+94X^{9} + 209X^{8} + 203X^{7} + 157X^{6} + 46X^{5} + 8X^{4} + 83X^{3} + 209X^{2} + 204X + 4
17 q = 317 (X^{2} + 123X + 261)(X^{15} + 91X^{14} + 66X^{13} + 205X^{12} + 71X^{11} + 191X^{10} + 77X^{9}+
+43X^{8} + 295X^{7} + 28X^{6} + 168X^{5} + 253X^{4} + 18X^{3} + 54X^{2} + 186X + 4)
19 q = 53 (X^{2} + 41X + 6)(X + 13)(X + 17)(X + 21)(X + 23)(X + 37)(X + 45)(X + 46)(X + 49)
(X^{3} + 47X^{2} + 7X + 21)(X^{3} + 35X^{2} + 36X + 52)(X^{3} + 18X^{2} + 35X + 25)
19 p = 251 X^{19} + 186X^{18} + 5X^{17} + 86X^{16} + 71X^{15} + 237X^{14} + 15X^{13} + 145X^{12} + 113X^{11}+
+30X^{10} + 155X^{8} + 162X^{7} + 70X^{6} + 89X^{5} + 241X^{4} + 188X^{3} + 52X^{2} + 217X + 199
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Reference.

1. Y. MAEDA, Table of characteristic polynomials for the Hecke operator T (2) on S_{k}(SL_{2}(ℤ)). Hokkaido University, Sapporo, Japan.