ON THE HECKE FIELDS OF GALOIS REPRESENTATIONS

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1 ON THE HECKE FIELDS OF GALOIS REPRESENTATIONS Version DOHOON CHOI AND YUICHIRO TAGUCHI Abstract. It is known that, in many cases, the field (known as the Hecke field) generated by all Hecke eigenvalues of an elliptic modular newform f is in fact generated by the pth Hecke eigenvalue a p of f for a single prime number p. In this paper, we study the density of such primes in a more general situation in terms of Galois representations. This is applied to the study of the fields of rationality of certain automorphic representations. 1. Introduction Let f(z) be an elliptic modular newform of weight k 1 and level N. If f = a n q n is the Fourier expansion of f, where q = e 2π 1z, then the field E f generated over the rational number field Q by a n for all n 1 with (n, N) = 1 is of finite degree over Q and is called the Hecke field of f. In many cases, it is known that E f is in fact generated by a p for a single prime number p. For instance, in Corollary 1.1 of [8], it is proved that, under a mild condition, E f is generated by a p for a single p in a set of primes of density 1. (Such studies are motivated, at least partly, by Maeda s conjecture [10]). Their proof depends on the analysis of Galois representations associated with f. In fact, such a phenomenon is prevalent in, and seems better understood in terms of, Galois representations. In this paper, we give some results in this vein. A typical case of our main results is as follows: Let K be a finite extension of Q and G K = Gal( K/K) its absolute Galois group, where K is a fixed algebraic closure of K. Let l be prime number and E l := E Q Q l, where E is a finite extension of Q and Q l is the l-adic number field. Then E l is a finite product λ l E λ of finite extensions E λ of Q l. Consider a continuous E l -linear representation ρ : G K GL n (E l ) of G K of finite degree n which is unramified outside a finite set S of places of K. For each finite place p of K outside S, put a p := Trρ(Frob p ), where Frob p denotes the Frobenius conjugacy class for p. Assume that a p E for all finite places p S and E is generated by them, and that the Zariski-closure of Im(ρ) in GL n (E l ) is connected. Then the set of finite places p S of K such that Q(a p ) = E has density 1 (Crorollary 2.2). We also give a characteristic polynomial (rather than trace ) version of this result (Corollary 2.6). In our theorems, we cannot replace ρ by one of its components ρ λ : G K GL n (E λ ). This is because, in our proof in Section 3, we may not have the key inequality (3.1) unless we use the whole E l Mathematics Subject Classification. Primary: 11F80, Secondary: 11F30. Key words and phrases. Hecke field, Galois representation, automorphic representation. 1

2 2 DOHOON CHOI AND YUICHIRO TAGUCHI These results are formulated in a more general form in Section 2 and proved in Section 3. In Section 4, we apply our theorems to the Galois representations associated with certain automorphic representations π to deduce a result (Corollary 4.2) on the positivity of the density of finite places p such that the field of rationality Q(π p ) of the p-th factor π p of π coincides with the one Q(π) of π. Acknowledgments. This work has an origin in a series of discussions the authors have had at the Korea Institute for Advanced Study (KIAS) in 2012, where the second author was staying as a visitor. The authors thank KIAS and Youn-Seo Choi for their hospitality. The second author is partially supported by JSPS KAKENHI Statements of the results Let K be a finite extension of Q and G K = Gal( K/K) its absolute Galois group. Let l be a prime number and let Q be a subfield of Q l. Let E be a finite extension field of Q and set E l := E Q Q l. Two practical cases are the following: (i) ( Local Hecke field case) Q = Q l and E = E l is a finite extension field of Q l ; (ii) ( Global Hecke field case) Q = Q and E is a finite extension field of Q. Let F l be a finite Q l -algebra which contains E l as a Q l -subalgebra (so E is identified with a Q-subalgebra of F l ). Consider a continuous F l -linear representation ρ : G K GL n (F l ) of G K of finite degree n unramified outside a finite set S of places of K. In this section, we assume that S contains all finite places of K lying above l. We regard GL n (F l ) as an algebraic group GL n,fl /Q l over Q l (= the Weil restriction Res Fl /Q l (GL n )), and at times we regard it also as the l-adic Lie group GL n,fl /Q l (Q l ). Let G = Im(ρ) be the image of ρ, which is a compact l-adic Lie subgroup of GL n (F l ). Let G be the Zariski-closure of G in GL n,e/ql, which becomes automatically a smooth affine algebraic group over Q l. (Note that an algebraic group over a field of characteristic zero is always smooth.) Let G 1,..., G c be the connected components of G, with G 1 the identity component, and put G i := G G i for i = 1,..., c. So we have c = (G : G 1 ) = (G : G 1 ). (Note that, since G is Zariski dense in G, the composite ρ homomorphism G K G(Ql ) (G/G 1 )(Q l ) is surjective, and hence G/G 1 is identified with G/G 1.) Let P K,f be the set of all finite places of K. For i = 1,..., c, we set P ρ,i := {p P K,f S ρ(frob p ) G i is non-empty}, where Frob p denotes the Frobenius conjugacy class for p in G K. Then we have P ρ,i = P ρ,i if and only if G i and G i are conjugate in G. For any subset C of P K,f and a real number x, we set π C (x) := {p C N(p) x},

3 ON THE HECKE FIELDS OF GALOIS REPRESENTATIONS 3 where N(p) denotes the absolute norm of p (= the cardinality of the residue field of p). For each p P K,f S, put a p := Trρ(Frob p ), where Tr : GL n (F l ) F l denotes the trace map. We assume that a p E for all p P K,f S. The subfield E tr := Q(a p p P K,f S) of E generated over Q by a p for all p P K,f S is called the Hecke field of ρ. It follows from Corollary 2.2 below that the Hecke field does not depend on the choice of S. For i = 1,..., c, let E tr i be the field generated over Q by the a p s for all p P ρ,i. By definition, E tr is the composite field of the E tr i s. Set C ρ,i Recall the function Li(x) defined by We set := {p P ρ,i Q(a p ) E tr i }. Li(x) := x 2 dt log t (x > 1). ε(x) := log x(log log x) 2 (log log log x) 1 (x 16), ε R (x) := x 1/2 (log x) 2. We set also 1 α := dim G, where the dim denotes the dimension as an l-adic analytic variety. Our first main theorem is: Theorem 2.1. With the above notation, for each i = 1,..., c, we have π Cρ,i (x) = O(Li(x)/ε(x) α ) as x and, assuming the GRH (:= Generalized Riemann Hypothesis), we have π Cρ,i (x) = O(Li(x)/ε R (x) α ) as x. We apply this theorem to obtain a description of the Hecke field as in (2.3) below. Note first that the sets P ρ,i and C ρ,i depend only on the conjugacy class of i, where the conjugacy of i s is defined by the conjugacy of G i s in G/G 1, or equivalently, by the equality of P ρ,i s. Let c i denote the cardinality of the conjugacy class of i (= the cardinality of the conjugacy class of G i in G/G 1 ). It is also equal to #{i P ρ,i = P ρ,i }. Let h be the number of conjugacy classes of G/G 1. By renumbering the connected components G 1,..., G c, we may assume that the first h of them are not conjugate each other. Then P ρ,1,..., P ρ,h are disjoint from each other and We have also c c h = c. For any subset C of P K,f, we define its density δ(c) by P ρ,1 P ρ,h = P K,f S. (2.1) δ(c) := lim x π C (x) π PK,f (x)

4 4 DOHOON CHOI AND YUICHIRO TAGUCHI if the limit exists. For instance, we have δ(p ρ,i ) = c i /c (2.2) by the Chebotarev Density Theorem. Put T ρ,i := P ρ,i C ρ,i. By the definition of E tr, E tr i and T ρ,i, we have E tr = Q(a p1,..., a ph ) (2.3) for any (p 1,..., p h ) T ρ,1 T ρ,h. Corollary 2.2. We have δ(c ρ,i ) = 0 and δ(t ρ,i ) = c i /c. In particular, we have δ(t ρ,1 T ρ,h ) = 1. Indeed, the first assertion follows from the above theorem and the Prime Number Theorem: π PK,f (x) Li(x) as x. The rest follows from (2.1) and (2.2). To consider how often the Hecke field is generated by a single a p, set T ρ := {p P K,f S Q(a p ) = E tr }. Corollary 2.3. If there exist at least d indices i (1 i c) such that E tr i δ(t ρ ) d/c. = E tr, then we have This applies in particular if G is connected, in which case c = d = 1. Note also that the assumption of the corollary holds if there exist d finite places p 1,..., p d S such that ρ(frob pi ) s are mutually non-conjugate in G and Q(a pi ) = E tr for each i. Remark 2.4. There are representations ρ such that Q(a p ) E tr for any single p P K,f S. Indeed, it is known 1 that the 8th alternating group A 8 has a 66-dimensional semisimple representation ρ : A 8 GL 66 ( Q) which is in fact defined over E tr := Q(Trρ(g) g A 8 ) = Q( 7, 15) = Q(ζ 7 + ζ ζ 4 7, ζ 15 + ζ ζ ζ 8 15), where ζ m denotes a primitive m-th root of unity. Elements of A 8 have order in M := {1, 2, 3, 4, 5, 6, 7, 15}. If g A 8 has order m, then ρ(g) has eigenvalues and trace in Q(ζ m ). For m M, however, we have Q(ζ m ) E tr. Hence there is no g A 8 such that Q(Trρ(g)) = E tr. Thus each A 8 -extension of K gives rise to a 66-dimensional representation ρ : G K GL 66 (E tr ) such that Q(a p ) E tr for any unramified prime p. 1 See e.g.

5 ON THE HECKE FIELDS OF GALOIS REPRESENTATIONS 5 Next we give a characteristic polynomial version of these results. We assume that, for each finite place p S, the characteristic polynomial ϕ ρ,p (X) := det(x ρ(frob p )) of ρ(frob p ) is in E[X]. For p P K,f S, let Q(ϕ ρ,p ) denote the subfield of E generated over Q by the coefficients of ϕ ρ,p. More generally, for any subset Φ E[X], we denote by Q(Φ) the subfield of E generated over Q by the coefficients of ϕ for all ϕ Φ. We set E ch := Q(ϕ ρ ) := Q({ϕ ρ,p p P K,f S}). It follows from Corollary 2.6 below that the field E ch does not depend on the choice of S. We have E ch E tr, and they coincide if ρ is semisimple (cf. Remark 2.8 below). For i = 1,..., c, let Ei ch := Q({ϕ ρ,p p P ρ,i }). By definition, E ch is the composite field of the Ei ch s. Set C ch ρ,i := {p P ρ,i Q(ϕ ρ,p ) E ch i }. Theorem 2.5. With the above notation, for each i = 1,..., c, we have and, assuming the GRH, we have Put T ch ρ,i By the definition of E ch, E ch i π C ch ρ,i (x) = O(Li(x)/ε(x) α ) as x π Cρ,i (x) = O(Li(x)/ε R (x) α ) as x. := P ρ,i C ch ρ,i and T ch ρ := {p P K,f S Q(ϕ ρ,p ) = E ch }. and T ch ρ,i, we have E ch = Q(ϕ ρ,p1,..., ϕ ρ,ph ) (2.4) for any of (p 1,..., p h ) Tρ,1 ch Tρ,h ch. As before, the following corollaries follow from the theorem. Corollary 2.6. We have In particular, we have δ(c ch ρ,i) = 0 and δ(t ch ρ,i) = c i /c. δ(t ch ρ,1 T ch ρ,h) = 1. Corollary 2.7. If there exist at least d indices i (1 i c) such that E ch i Remark 2.8. Suppose ρ is defined over E l ; δ(tρ ch ) d/c ρ : G K GL n (E l ). = E ch, then we have Denote by Q(ρ) the smallest subfield E of E such that σ ρ ρ for all σ Gal(Ē/E ), where σ acts on ρ coefficient-wise via the first factor of E l = E Q Q l. We have Q(ρ) E ch E tr. They coincide if ρ is semisimple. Indeed, an argument of Faltings (Proof of Satz 5 in [5]; see

6 6 DOHOON CHOI AND YUICHIRO TAGUCHI also the Lemma in [9], Chap. VIII, Sect. 5) implies that, if ρ is semisimple, then there is a finite set T of finite places of K, which is disjoint from S and l, such that Q(ρ) = Q(a p p T ). 3. Proof of the Theorems We begin by recalling the notion of M-dimension from Section 3 of [12]. Let N be an integer 0 and d a real number 0. A closed subset C of (Z l ) N (= the direct product of N copies of the ring Z l of l-adic integers) is said to be of M-dimension d, and denoted dim M C d, if C n = O(l nd ) as x, where C n is the image of C in (Z l /l n Z l ) N and C n is its cardinality. By using a covering by copies of (Z l ) N, the notion of M-dimension can be defined also for a closed subset of an analytic variety Ω of dimension N over Q l. An analytic subspace of Ω has M-dimension d if it has dimension d as an analytic space over Q l ([12], 3.2, Thm. 8). The following theorem is fundamental to our purpose: Theorem 3.1 ([12], 4, Thm. 10). Let K be a finite extension of Q and L/K a Galois extension which is unramified outside a finite set of places of K and whose Galois group G is an l-adic Lie group of dimension N. Let C be a closed subset of G which is stable by conjugation. Suppose dim M C d for some real number d < N. Put α = (N d)/n. Then we have: π C (x) = O(Li(x)/ε(x) α ) as x and, assuming the GRH, π C (x) = O(Li(x)/ε R (x) α ) as x. Now we prove our theorems. Since the proofs are similar, we give here only the proof of Theorem 2.5. We write E i for Ei ch for the simplicity of notation. Let E i be the (set-theoretic) union of all proper Q-subalgebras E of E i, and E i,l its topological closure in E i,l (equivalently, E i,l is the union of E l := E Q Q l for all such E ). We regard F l, E i,l, E i,l, etc. as affine schemes over Q l. Note that dim E i,l < dim E i,l, (3.1) where the dim means the dimension as an Q l -scheme. Let V = (F l ) n (resp. V = (E i,l )n ) be the direct product of n copies of F l (resp. E i,l ). Define a morphism of Q l-schemes Φ : GL n (F l ) V by taking the characteristic polynomial, i.e., by setting Φ(g) := (a 1 (g),..., a n (g)) on the level of Q l -valued points, where, for any g GL n (F l ), the a j (g) are the coefficients of the characteristic polynomial det(x g) = X n a 1 (g)x n ( 1) n a n (g). (For the proof of Theorem 2.1, use the trace map Tr : GL n (F l ) F l instead of Φ.) Since the smooth connected algebraic variety G i is irreducible, so is its image Φ(G i ) by Φ. Hence if Φ(G i )

7 ON THE HECKE FIELDS OF GALOIS REPRESENTATIONS 7 is contained in V (equivalently, if Φ(G i ) V (Q l )), it must be contained in an irreducible component of V, which is of the form E l = E Q Q l with E a proper Q-subalgebra of E i. This implies that E contains the coefficients of ϕ ρ,p (X) for all p P ρ,i. By the definition of E i, we have E = E i, which is a contradiction. Thus we have Φ(G i ) V. (3.2) Let H ( GL n (F l )) be the inverse image by Φ of the point set (E i,l )n, which we identify with the set H(Q l ) of Q l -valued points of the Q l -scheme H := Φ 1 (V ) and regard here as an l-adic analytic subvariety of GL n (F l ). By Theorem 3.1, we are reduced to showing that dim M (G i H) dim(g i ) 1, (3.3) where the dim on the right-hand side means the dimension as an l-adic analytic variety. We claim that no non-empty open subset U of G i is contained in H. Indeed, if U H, then the Zariski closure U of U is contained in H. But since U is non-empty and open in G i, we have G i = U. Thus we have G i = U H, which contradicts (3.2). Now (3.3) follows from: Lemma 3.2. Let G and H be analytic subspaces of an l-adic analytic variety. Suppose no non-empty open subset of G is contained in H. Then we have dim M (G H) dim(g) 1. Proof. Let m = dim(g). For each g G H, there exist an open subset U of G containing g and a bi-analytic map φ : U (Z l ) m (so we identify U with (Z l ) m via φ). Since the problem is local, it is enough to show that dim M (U H) < m. Since U H, the intersection U H is a proper closed analytic subspace of U, and hence has dimension < m. By Theorem 8 in 3.2 of [12], we have dim M (U H) < m. Or, more directly, we can prove this inequality as follows: If we identify U with the Q l -valued points of Sp(Q l x 1,..., x m ), where Q l x 1,..., x m is the ring of power series in x 1,..., x m over Q l convergent on the unit disk (Z l ) m, then U H is defined by a non-unit ideal I of Q l x 1,..., x m. There is no loss of generality if we assume that I is generated by one element f Q l x 1,..., x m. By the Weierstrass preparation theorem (e.g. [2], 5.2.2, Thm. 1), we have (possibly after a change of variables) f = ug, where u is a unit in Q l x 1,..., x m and g Q l x 1,..., x m 1 [x m ] is monic as a polynomial in x m. Thus we are reduced to the case where U H is defined by the polynomial g. Since there are at most deg(g) roots x m of g in Z l for each (x 1,..., x m 1 ) (Z l ) m 1, the projection (Z l ) m (Z l ) m 1 to the first (m 1) factors induces a map U H (Z l ) m 1 whose fiber has cardinality deg(g). Thus, for each n, the image of U H in (Z l /l n Z l ) m has cardinality l n(m 1) deg(g), which proves dim M (U H) < m.

8 8 DOHOON CHOI AND YUICHIRO TAGUCHI 4. Application to automorphic representations of GL n In this section, we apply our results in Section 2 to the Galois representations associated with certain automorphic representations of GL n. Let K be a totally real field 2 and A K the adele ring of K. Suppose that (π, χ) is a polarized, 3 regular, algebraic, cuspidal automorphic representation of GL n (A K ), by which we mean a pair (π, χ) where (1) π is a cuspidal automorphic representation of GL n (A K ), (2) χ = p PK χ p : A K /F C is an algebraic character such that χ p ( 1) is independent of p, (3) π π (χ det). The global representation π and its finite part π f can be written as tensor products π = p P K π p and π f = p P K,f π p of local representations π p of GL n (K p ), where K p is the completion of K at a place p, and P K (resp. P K,f ) denotes the set of places (resp. finite places) of K. For σ Aut(C), let σ π f and σ π p denote respectively the representations obtained from π f and π p by the scalar extension σ 1 : C C of the underlying representation spaces (See Section 3 of [4] for more details). It is known ([4], Thm. 3.13) that σ π f in fact extends to an automorphic representation of GL n (A K ); thus there exists a unique (up to isomorphism) automorphic representation σ π of GL n (A K ) such that ( σ π) f σ π f. Let Q(π) be the smallest subfield E of C such that σ π f π f for all σ Aut(C/E). By Theorem 3.13 of [4], Q(π) is a finite extension of Q. Similarly, for a subset T = {p 1, p 2,...} of P K,f, we define Q(π p1, π p2,...) to be the smallest subfield E of C such that σ π p π p for all σ Aut(C/E) and all p T. It is the composite field of the Q(π p ) s for all p T. With these notations we state the following theorem. Theorem 4.1. Let K and π be as above. (1) There exists a finite number h of disjoint sets T π,1,..., T π,h of finite places of K with total density δ(t π,1... T π,h ) = 1 such that we have for any (p 1,..., p h ) T π,1 T π,h. Q(π) = Q(π p1,..., π ph ) (2) If there exists a finite place p of K such that π p is unramified and Q(π) = Q(π p ), then the set of finite places p of K such that Q(π) = Q(π p ) has density > 0. The number h can in fact be taken to be the number of conjugacy classes of the component group G/G 1 (cf. Section 2) of the Zariski-closure of the image of the Galois representation 2 We assume so for simplicity, but similar results as in this section can be proved, mutatis mutandis, if K is a CM field. See Section 2.1 of [1]. 3 In fact, according to Theorem A of [6] and Theorem I.4 of [11]), our results in this section hold true without the assumption of polarizability, with K either totally real or CM.

9 ON THE HECKE FIELDS OF GALOIS REPRESENTATIONS 9 associated with π. Recall (see for example Theorem of [1]) that, for a π as above, there exist a number field E and a weakly compatible system of semisimple representations ρ λ : G K GL n (E λ ), where λ runs over the finite places of E, such that, for each place p of K not lying above the residue characteristic l of λ, we have ιwd(ρ λ GK p )F-ss rec(π p det (1 n)/2 p ), (4.1) where, for each p P K,f, the local absolute Galois group G Kp is identified with a decomposition group for p, WD(...) denotes the Weil-Deligne representation attached to a representation of G Kp, ι : Ēλ C is a fixed isomorphism, F-ss denotes the Frobenius semisimplification, rec is the unitarily normalized local Langlands correspondence, and... p = N(p) ordp(...) is the normalized absolute value of K p. For a prime number l, set ρ l := λ l ρ λ, which is a representation of G K into GL n (E l ) with E l = E Q Q l = λ l E λ. As an immediate corollary of the theorem, we have: Corollary 4.2. If the Zariski-closure of Im(ρ l ) is connected for some prime number l, then the set of finite places p of K such that Q(π) = Q(π p ) has density 1. Note that the connectedness assumption holds for all primes l once it holds for some l (cf. Théorème in [13] and the remark following it). Proof of Theorem 4.1. Choose a prime number l, and let S be a finite subset of P K,f which contains all places of K lying above l and is such that π p is unramified (and hence so is ρ l ) GK p if p S. Recall the fields Q(ϕ ρl,p) and Q(ϕ ρl ) from Section 2 Q(ϕ ρl,p), for p P K,f S, is the field generated over Q by the coefficients of the characteristic polynomial ϕ ρl,p of ρ l (Frob p ), and Q(ϕ ρl ) is the composite field of the Q(ϕ ρl,p) s for all p P K,f S. The strong multiplicity one theorem for GL n implies that σ π π if and only if σ π p π p for all but finitely many p P K,f (cf. [3], Theorem and the remarks following it). In particular, this implies that Q(π) is equal to the composite field of the Q(π p ) s for all p P K,f S. Now we claim that Q(π p ) = Q(ϕ ρl,p) for each p P K,f S, and hence that Q(π) = Q(ϕ ρl ). Indeed, by Lemma VII of [7], the isomorphism (4.1) is compatible with all field automorphism of C. This implies that Q(π p ) equals the field generated over Q by the coefficients of the characteristic polynomial of WD(ρ λ )(Frob GK p p), which is the same as that of ρ λ (Frob p ). Now the assertion (1) of the theorem follows from Corollary 2.6 applied to ρ l. If Q(π) = Q(π p ) for some p, then we can apply Corollary 2.7 to ρ l to obtain the assertion (2). References [1] T. Barnet-Lamb, T. Gee, D. Geraghty and R. Taylor, Potential automorphy and change of weight, Ann. of Math. 179 (2014), [2] S. Bosch, U. Güntzer and R. Remmert, Non-Archimedean Analysis, Grundlehren der Mathematischen Wissenschaften 261, Springer-Verlag, Berlin, 1984, xii+436 pp.

10 10 DOHOON CHOI AND YUICHIRO TAGUCHI [3] D. Bump, Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics 55. Cambridge University Press, Cambridge, 1997, xiv+574 pp. [4] L. Clozel, Motifs et formes automorphes: applications du principe de fonctorialité, In: Automorphic Forms, Shimura Varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math. 10, pp , Academic Press, Boston, 1990 [5] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), [6] M. Harris, K. Lan, R. Taylor and J. Thorne, On the rigid cohomology of certain Shimura varieties, preprint, arxiv: [7] M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties, Ann. of Math. Studies 151, Princeton Univ. Press, Princeton, 2001 [8] K. T.-L. Koo, W. Stein and G. Wiese, On the generation of the coefficient field of a newform by a single Hecke eigenvalue, J. Théor. Nombres Bordeaux 20 (2008), [9] S. Lang, Algebraic Number Theory, Second ed., Graduate Texts in Math. 110, Springer-Verlag, New York, 1994 [10] Y. Maeda, Maeda s conjecture and related topics, to appear in: RIMS Kôkyûroku Bessatsu [11] P. Scholze, On torsion in the cohomology of locally symmetric varieties, preprint, arxiv: [12] J.-P. Serre, Quelques applications du théorème de densite de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), [13] J.-P. Serre, Lettre à Ken Ribet du 29/1/1981, in: Œuvres, Vol. IV, Springer-Verlag, Berlin, 2000, pp (D.C.) School of Liberal Arts and Sciences, Korea Aerospace University, Goyang, Gyeonggi, , Korea address: choija@kau.ac.kr (Y.T.) Faculty of Mathematics, Kyushu University, Fukuoka, , Japan address: taguchi@math.kyushu-u.ac.jp