REPRESENTATIONS OF SOME SMALL QUADRATIC FIELDS
HYUNSUK MOON AND YUICHIRO TAGUCHI
Abstract. For a few quadratic fields, the non-existence is proved of continuous irreducible mod 2 Galois representations of degree 2 unramified outside{2,∞}.
1. Introduction
In this paper, we prove the following theorem, which settles some special cases of versions (cf. Conj. 1.1 of [3]; Conj. 1 of [11]; and Question 1 of [5]) of Serre’s modularity conjecture ([14], [16]) for a few quadratic fields:
Theorem. Let F be one of the following quadratic fields:
Q(√
−1), Q(√
±2), Q(√
±3), Q(√
±5), Q(√
±6).
Then there exist no continuous irreducible representations ρ:GF →GL2(F2) unram- ified outside {2,∞}.
Here, GF denotes the absolute Galois group Gal(F /F) ofF, and F2 is an algebraic closure of the finite field F2 of two elements.
The proof is based on the method of discriminant bound as in [18], [15], [1], [7], [8], [2]. However, we need to improve the known upper bounds at the prime 2. This is done in Section 2. The proof of the Theorem is given in Section 3.
It is desirable to have such a theorem for mod p representations for other primes p, but this seems almost impossible at least by our method.
The first version of this paper did not include the cases of Q(√
±6). After it was circulated, M. H. S¸eng¨un communicated to us that, combining his lower bound in [12]
and our upper bound in Section 2, he could prove the non-existence in these cases.
Then we examined the cases and found that our proof actually covered them as well.
Thus we are grateful to him very much for his communication.
Acknowledgments. The writing of this paper was greatly encouraged by “Ecole d’´et´e sur la conjecture de modularit´e de Serre”, Luminy, 9–20, July, 2007. We thank its
2000Mathematics Subject Classification. 11R39, 11F80, 11R32.
The first author was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-531-C00004). The main part of this work was done while the second author was visiting Pohang University of Science and Technology; he thanks YoungJu Choie and POSTECH for their hospitality, and KOFST for financial support.
1
organizers, especially J.-M. Fontaine, and speakers for their big efforts. We thank also A. Ash, F. Diamond, S. Hattori, C. Khare, K. Klosin, T. Saito, J.-P. Serre and R.
Taylor for their information on the literature on various versions of Serre’s modularity conjecture and/or comments on the first version of this paper.
Convention. For a finite extensionE/F of non-Archimedean local fields, we denote by DE/F the different ideal of E/F. The 2-adic valuationv2 is normalized by v2(2) = 1, and is used to measure the order of ideals (such as DE/F) in algebraic extensions of the 2-adic field Q2. We denote by (∗ ∗∗) and (1 ∗1) respectively the subgroups {(a0 bd)}
and {(10 b1)} of GL2(F2).
2. Local lemmas
LetF be a finite extension ofQ2,D=GF its absolute Galois group, andI its inertia subgroup. In this section, we consider mod 2 representations ρ:D→GL2(F2) ofD.
LetE/F be the extension cut out byρ. We shall estimate the differentDE/F of E/F. LetE0 (resp.E1) be the maximal unramified (resp. tamely ramified) subextension of E/F, and let e1 = [E1 : E0] be the tame ramification index of E/F. Then we have DE/F = DE/E1DE1/E0, and v2(DE1/E0) = (1−1/e1)/eF, where eF is the ramification index of F/Q2. Thus it remains for us to calculateDE/E1. We assume E/F is wildly ramified, with wild ramification index 2m. Then the wild inertia subgroup G1 of G := Im(ρ) is a non-trivial 2-group and, after conjugation, we may assume it is contained in (1 ∗1). Since G1 is normal in G and the normalizer of G1 in GL2(F2) is (∗ ∗∗), we may assume that ρ is of the form
(2.1) ρ =
µψ1 ∗ ψ2
¶ ,
where ψi : D → F×2 are characters of D. Note that the ψi’s have odd order, so that they are at most tamely ramified.
Lemma 1. Let the notation be as above. Assume further thatF/Q2 has ramification index 2. If E/F has ramification index 2m (i.e. if e1 = 1), then there exists a non- negative integer m2 ≤m such that
v2(DE/F) = (9
4 − 2m22m+1 and m2 ≤m−1; or 2− 2m22m+1.
If ρ is non-abelian, then the former case does not occur.
Here, we sayρ is (non-)abelian if the group Im(ρ) is (non-)abelian.
Proof. By assumption, we have E1 = E0 and the characters ψi are unramified. By local class field theory, the Galois group G1 = Gal(E/E1), which is an elementary 2-group, is identified with a quotient of the group (1 +πA)/(1 +πA)2, where A is the ringOE1 of integers of E1,π is a uniformizer of A, and (1 +πA)2 is the subgroup
of the square elements in the multiplicative group (1 +πA). The character group X = Hom(G1,C×) ofG1 is identified with a subgroup of Hom((1+πA)/(1+πA)2,C×).
The subgroup Xi of X consisting of the characters with conductor dividing πi is identified with a subgroup of Hom((1 +πA)/(1 +πiA)(1 +πA)2,C×). It is easy to see that
{1} = X1 ⊂ X2 = X3 ⊂ X4 ⊂ X5 = X.
Indeed, the equalityX2 =X3follows from the fact that (1+π2A) = (1+π3A)(1+πA)2, and the equality X5 = X follows from the fact that (1 +π5A) ⊂ (1 +πA)2; cf. the proof Lemma 2.1 of [7]. Just as in [18], we can show that the index (X5 : X4) is 1 or 2, since the image of (1 +π4A) in (1 +πA)/(1 +πA)2 has order 2. To see this, consider the equation
(2.2) 1 +aπ4 = (1 +xπ2)2
for a givena ∈A× and unknown x∈A×. If 2 =cπ2 with c∈A×, then the equation (2.2) has a solution x if and only if the congruence
cx+x2 ≡ a (mod π)
has a solution. Since the F2-linear map℘ :A/πA→A/πAgiven by x7→cx+x2 has dimF2Coker(℘) = 1, the equation (2.2) has a solution for “half” of the a’s.
By assumption, X5 has order 2m. Suppose X2 has order 2m2. Then the 2-adic order of the different DE/F = DE/E1 can be calculated as follows by using the F¨uhrerdiskriminantenproduktformel ([13], Chap. VI, §3):
(1-i) If (X5 :X4) = 2, then v2(DE/F) = 1
2 · 1 2m
¡(2m−2m−1)×5 + (2m−1−2m2)×4 + (2m2 −1)×2¢
= 9
4 −2m2 + 1 2m . (1-ii) If (X5 :X4) = 1, then
v2(DE/F) = 1 2· 1
2m((2m−2m2)×4 + (2m2 −1)×2) = 2− 2m2 + 1 2m .
Letψi be the characters in (2.1). Ifρ is non-abelian (or equivalently, if ψ1 6=ψ2 as characters on D), then Gal(E0/F) = G/G1 acts on G1 (identified with a subgroup of (1 ∗1)) via ψ1ψ2−1 (cf. [10], Proof of Prop. 2.3). This induces a similar action on X which respects the filtration Xi. Each orbit by this action has odd cardinality
|Im(ψ1ψ−12 )|, while X5rX4 has 2-power cardinality if it is non-empty. Thus we must have X5 =X4, and we are in the case (1-ii) above. ¤ Specializing theF/Q2, we calculate the value ofv2(DE/F) more precisely as follows:
Lemma 2. Assume F/Q2 is a totally ramified quadratic extension. Then the exten- sion E/F has ramification index 2m. If ρ is non-abelian, then there exists a non- negative integer m2 ≤m such that
v2(DE/F) = 2−2m2 + 1 2m .
If ρ is abelian, then we have m ≤ 3 and v2(DE/F) ≤ 15/8. In fact, more precisely, we have:
v2(DE/F) =
15/8 if m = 3,
7/4, 3/2 or 5/4 if m = 2, 5/4, 1 or 1/2 if m = 1.
Proof. F/Q2 being totally ramified, any abelian extension of F has no non-trivial tame ramification since O×F is a pro-2 group. Thus the characters ψi in (2.1) are unramified, and E/F has ramification index 2m.
Ifρis non-abelian, thenv2(DE/F) has the second value in Lemma 1. If ρis abelian (or equivalently, ifψ1 =ψ2as characters onD), thenG1is identified with a quotient of OF×/(OF×)2. The groupO×F/(O×F)2 has order 8. The different is the largest in the case whereG1 ' O×F/(O×F)2, in which casem = 3, (X5 :X4) = (X4 :X3) = (X2 :X1) = 2, and
v2(DE/F) = 1 2 · 1
8(4×5 + 2×4 + 1×2) = 15 8 .
Other cases can be calculated similarly. Note that (Xi+1 : Xi) = 1 or 2 since the
residue field of OF is F2. ¤
Recall that e1 = [E1 :E0] denotes the tame ramification index of E/F.
Lemma 3. If F/Q2 is the unramified quadratic extension, then we have e1 = 1 or 3.
If ρ is non-abelian, there exist non-negative integers m2 ≤m4 ≤m such that
v2(DE/F) = (
2−2m−11 if e1 = 1,
8
3 − 2m3·24+2m−1m2+1 if e1 = 3.
If ρ is abelian, then m ≤3 and v2(DE/F)≤35/12. In fact, more precisely, we have:
v2(DE/F) =
35/12 if m= 3, 8/3 or 13/6 if m= 2, 13/6 or 5/3 if m= 1,
if e1 = 3. If e1 = 1, then the values of v2(DE/F) are the above values minus 2/3.
Proof. By local class field theory, the characters ψi in (2.1) are identified with char- acters of F×/(1 + 2OF)×. Since OF×/(1 + 2OF)× 'F×4, the tamely ramified extension E1/E0 has degree either 1 or 3.
As in the proof of Lemma 1, identify the Galois group G1 = Gal(E/E1) (resp. the character group X = Hom(G1,C×)) with a quotient of (1 +πA)/(1 +πA)2 (resp. a subgroup of Hom((1 +πA)/(1 +πA)2,C×)), where A =OE1 and π is a uniformizer of A. Let Xi be the subgroup ofX consisting of the characters of G1 with conductor dividing πi.
If e1 = 1, then the value of v2(DE/F) can be calculated as in Proposition 2.3 of [10]; we have {1} = X1 ⊂ X2 ⊂ X3 = X and (X3 : X2) ≤ 2. If ρ is abelian (i.e.
ψ1 = ψ2), then X is in fact identified with a subgroup of the character group of (1 + 2OF)/(1 + 2OF)2, and one has (X2 :X1)≤4 since F has residue field F4. Thus
|X3| ≤8, and
(2.3) v2(DE/F) =
1
8(4×3 + 3×2) = 94 if m= 3,
1
4(2×3 + 1×2) = 2 or
1
4(3×2) = 32 if m= 2,
3
2 or 22 = 1 if m= 1.
If ρ is non-abelian (i.e. ψ1 6= ψ2), then as in the last part of the proof of Lemma 1, we have X3 =X2, and hence
v2(DE/F) = 1
2m ((2m−1)×2) = 2− 1 2m−1. Assume e1 = 3. Then as in the proof of Lemma 1, one can show that
{1} = X1 ⊂ X2 = X3 ⊂ X4 = X5 ⊂ X6 ⊂ X7 = X,
with (X7 :X6) = 1 or 2. By assumption, X7 has order 2m. Suppose |X2| = 2m2 and
|X4|= 2m4. Ifρis abelian, thenX is identified with a subgroup of the character group of (1 + 2OF)/(1 + 2OF)2 (so X1 = X2 and X5 = X6), and v2(DE/E1) is calculated to have the same values as in (2.3). Adding the tame part v2(DE1/E0) = 2/3, we see that v2(DE/F) has the values as in the statement of the lemma. If ρ is non-abelian, then as in the former case, we have X7 =X6, and hence
v2(DE/E1) = 1 3 · 1
2m ((2m−2m4)×6 + (2m4 −2m2)×4 + (2m2 −1)×2)
= 2− 2m4 + 2m2 + 1 3·2m−1 . Adding the tame part, we obtain
v2(DE/F) = 8
3 −2m4 + 2m2 + 1 3·2m−1 .
¤
3. Proof of the Theorem
Suppose there were a continuous irreducible representation ρ : GF → GL2(F2) unramified outside {2,∞}. Let K/F be the extension cut out by ρ and G = Im(ρ) its Galois group. As in [18], we distinguish the two cases where G is solvable and non-solvable.
First we deal with the solvable case. If G is solvable, then it sits in an exact sequence
1→H →G→Z/2Z→1, H ⊂F×2 ×F×2,
as in Theorem 1 in §22 of [17]. Hence K is an abelian extension of odd degree, unramified outside {2,∞}, over the quadratic extension K0/F corresponding to H.
By using class field theory and noticing that Q(√
3) and Q(√
6) have narrow class number 2 (resp. Q(√
−5) and Q(√
−6) have class number 2), we can show that, for each F = Q(√
2), Q(√
3), Q(√
5), Q(√
6) (resp. F = Q(√
−1), Q(√
−2), Q(√
−3), Q(√
−5)), Q(√
−6) there are 7 possibilities (resp. 3 possibilities) for such K0. By examining Jones’ tables [6], we find them as follows:
If F =Q(√
2), then K0 = Q(p
±√
2), Q(p 1±√
2), Q(√ 2,√
−1), Q(p 2 +√
2), Q(p
−2 +√ 2);
If F =Q(√
3), then K0 = Q(p
1±√
3), Q(p
−1±√
3), Q(√ 3,√
−1), Q(√ 3,√
2), Q(√ 3,√
−2);
If F =Q(√
5), then K0 = Q(
q
(1±√
5)/2), Q(p
−1±√
5), Q(√ 5,√
−1), Q(√ 5,√
2), Q(√ 5,√
−2);
If F =Q(√
6), then K0 = Q(p
2±√
6), Q(p
−2±√
6), Q(√ 6,√
−1), Q(√ 6,√
2), Q(√ 6,√
−2);
If F =Q(√
−1), then K0 = Q(√
−1,√
2), Q(p 1±√
−1);
If F =Q(√
−2), then K0 = Q(√
−2,√
−1), Q(p
±√
−2);
If F =Q(√
−3), then K0 = Q(√
−3,√
−1), Q(√
−3,√
2), Q(√
−3,√
−2);
If F =Q(√
−5), then K0 = Q(√
−5,√
−1), Q(√
−5,√
2), Q(√
−5,√
−2);
If F =Q(√
−6), then K0 = Q(√
−6,√
−1), Q(√
−6,√
2), Q(√
−6,√
−2),
All these K0 have class number either 1 or 2. Let OK0,2 = OK0 ⊗Z Z2 denote the 2-adic completion of the integer ring OK0 of K0. Then its multiplicative group O×K0,2
is isomorphic to the direct-product of Z⊕42 and a cyclic group of order dividing 12 (A non-trivial 3-torsion subgroup appears only if K0 contains Q(√
−3) or Q(√ 5)).
Thus there can exist an abelian extensionK/K0 of odd degree at most 3. But in each
case, the 3-torsion subgroup of OK×0,2 is killed (when the reciprocity map is applied) by the global unit ζ3 = (−1 +√
−3)/2 or ε2 = (3 +√
5)/2 (N.B. The latter is totally positive). Thus there is no abelian extension K/K0 of odd degree unramified outside {2,∞}.
Remark. The quadratic fields F in the Theorem do not have abelian extensions of odd degree which are unramified outside {2,∞}. Thus if a representation ρ:GF → GL2(F2) unramified outside {2,∞} has solvable image, then the image is unipotent, and the extension K/F cut out by ρ is contained in the compositum of the above seven (resp. three) quadratic extensions K0 of F.
Next we prove the non-solvable case. This is done by the comparison of the Tate and Odlyzko bounds for discriminants. We denote bydK/Q the discriminant ofK/Q, and d1/nK =|dK/Q|1/n the root discriminant of K, where n = [K :Q]. By the Odlyzko bound [9], we have
d1/nK >
(
17.020 if n ≥120, 20.895 if n ≥1000.
If G = Gal(K/F) is non-solvable, then n = 2|G| ≥ 120. On the other hand, by Lemmas 2 and 3, we have
d1/nK ≤
2·22 = 8 if F =Q(√
−1), 2√
2·22 < 11.314 if F =Q(√
±2), 2√
3·22 < 13.857 if F =Q(√
√ 3),
3·235/12 < 13.079 if F =Q(√
√ −3),
5·235/12 < 16.885 if F =Q(√ 5), 2√
5·22 < 17.889 if F =Q(√
−5), 2√
6·22 < 19.596 if F =Q(√
±6).
Thus we have a contradiction in all cases but F = Q(√
−5) and Q(√
±6). To deal with these three cases, let 2m be the wild ramification index of K/F at 2. Then the 2-Sylow subgroup of G has order ≥ 2m. If m ≤ 2, then by Lemma 2 applied to a 2-adic completion ofK/F, we have v2(DK/F)≤7/4, and hence
d1/nK ≤ (
2√
5·27/4 < 15.043 if F =Q(√
−5), 2√
6·27/4 < 16.479 if F =Q(√
±6),
which contradicts the Odlyzko bound. If m≥3, then by §§251–253 of [4], the image of G in PGL2(F2) contains a conjugate of PSL2(F8), which has order 504. Hence the Odlyzko bound applies with n = 2|G| > 1000, whence a contradiction in the
remaining cases as well. ¤
References
[1] S. Brueggeman,The nonexistence of certain Galois extensions unramified outside5, J. Number Theory75(1999), 47–52
[2] S. Brueggeman,The nonexistence of certain nonsolvable Galois extensions of number fields of small degree, Int. J. Number Theory1(2005), 155–160
[3] K. Buzzard, F. Diamond and F. Jarvis,On Serre’s conjecrure for mod` Galois representations over totally real fields, preprint, 2005,
http://www.unet.brandeis.edu/~fdiamond/bdj12.pdf [4] L. E. Dickson,Linear Groups, Teubner, 1901, Leibzig
[5] L. M. Figueiredo, Serre’s conjecture for imaginary quadratic fields, Compositio Math. 118 (1999), 103–122
[6] J. Jones,Tables of number fields with prescribed ramification, http://math.asu.edu/~jj/numberfields/
[7] H. Moon, Finiteness results on certain mod p Galois representations, J. Number Theory 84 (2000), 156–165
[8] H. Moon and Y. Taguchi,Refinement of Tate’s discriminant bound and non-existence theorems for modpGalois representations, Kazuya Kato’s fiftieth birthday, Doc. Math. 2003, Extra Vol., 641–654
[9] A. M. Odlyzko,Discriminant bounds, (unpublished manuscript, 1976), available at:
http://www.dtc.umn.edu/~odlyzko/unpublished/index.html
[10] K. Ono and Y. Taguchi, 2-adic properties of certain modular forms and their applications to arithmetic functions, Int. J. Number Theory1(2005), 75–101
[11] M. Schein,Weights in Serre’s conjecture for Hilbert modular forms: the ramified case, preprint, 2006,
http://arxiv.org/abs/math.NT/0610488
[12] M. H. S¸eng¨un, The non-existence of certain representations of the absolute Galois group of quadraticfields, preprint, 2007
[13] J.-P. Serre,Corps Locaux, 3e´ed., 1980, Hermann, Paris
[14] J.P. Serre,Valeurs propres des op´erateurs de Hecke modulo `, Journ´ees Arithm´etiques de Bor- deaux (Conf., Univ. Bordeaux, 1974), pp. 109–117, Ast´erisque24-25, Soc. Math. France, Paris, 1975
[15] J.-P. Serre, Note 229.2on p. 710, Œuvres III, Springer-Verlag, 1986
[16] J.-P. Serre,Sur les repr´esentations modulaires de degr´e2deGal(Q/Q), Duke Math. J.54(1987), 179–230
[17] D. A. Suprunenko,Matrix Groups, A.M.S., Providence, 1976
[18] J. Tate, The non-existence of certain Galois extensions of Qunramified outside2, in: “Arith- metic geometry”, pp. 153–156, Contemp. Math.174Amer. Math. Soc., Providence, RI, 1994 Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 702-701, Korea
E-mail address: [email protected]
Graduate School of Mathematics, Kyushu University 33, Fukuoka 812-8581, Japan E-mail address: [email protected]
