Hecke Eigenvalues for Real Quadratic Fields

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Hecke Eigenvalues for Real Quadratic Fields

Kaoru Okada

CONTENTS 1. Introduction

2. The Trace Formula for Totally Real Number Fields 3. Computation for Real Quadratic Fields

4. Numerical Examples forQ(√

257)andQ(√ 401) Acknowledgments

References

2000 AMS Subject Classification:Primary 11F41; Secondary 11F60, 11F72, 11R42

Keywords: Hilbert cusp form, Hecke operator, eigenvalue, trace formulaL-value

We describe an algorithm to compute the trace of Hecke operators acting on the space of Hilbert cusp forms defined relative to a real quadratic field with class number greater than one. Using this algorithm, we obtain numerical data for eigenvalues and characteristic polynomials of the Hecke operators.

Within the limit of our computation, the conductors of the orders spanned by the Hecke eigenvalue for any principal split prime ideal contain a nontrivial common factor, which is equal to a HeckeL-value.

1. INTRODUCTION

LetF be a totally real algebraic numberfield with nontrivial class group. We shall study the spaceSk(c,ψ) of Hilbert cusp forms (relative toF) and the Hecke opera- torsT(a) acting on it. We shall describe our result using the frameworkfirst introduced in [Shimura 78]. Fol- lowing Shimura’s work, the trace formula (whose origin goes back to fundamental work of Eichler, Selberg, and Shimizu) was made more explicit in [Saito 84]. Saito’s formula gives us a method for computing Hecke eigenvalues as long as the dimension of the space remains reason- ably small. It is then natural to expect Hecke eigenvalues for prime idealspin a given ideal class to have a new feature specific to the ideal class. Such a new feature can be detected only by computing Hecke eigenvalues for the

basefield with nonprincipal ideal classes. The purpose of

this paper is to compute examples of such Hecke eigenvalues for real quadraticfields with class number greater than 1 and to present a new phenomenon that we have discovered through our numerical examples.

We summarize our observations for the data of Hecke eigenvalues when the weight is parallel (k₁, k₁), the level cis the maximal order o_F of F, and the Hecke character ψ is the identity 1. Let f be a primitive form con- tained in S(k1,k1)(o_F,1) that is orthogonal to any base change lift fromQ(that is, f is a primitive form in the

“F-proper” subspace ofS^(k1,k1)(o_F,1) as defined in [Doi et al. 98]). We denote by Cf(p) the eigenvalue ofT(p) satisfying f|T(p) = C_f(p)f, by K_f⁺ the subfield of the

c A K Peters, Ltd.

1058-6458/2001$0.50 per page Experimental Mathematics11:3, page 407

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HeckefieldKf off generated byCf((p)F) for allrational primes p, and by o_K⁺

f the maximal order of K_f⁺. For split prime idealsp, we computed the discriminant of the order Λ_f(p) spanned by the eigenvalue of T(p) (that is, Λf(p) =o_K+

f +Cf(p)N(p)^(k¹⁻^2)/2o_K+

f ) to see whether it has extra factors outside the discriminant of the maximal order. Extra factors show up as the conductor of the order (for the definition of the conductor, see just above Lemma 2.3); so, we writec(Λ_f(p)) for the conductor. Surprisingly enough, as long as the prime ideals p are principal and split, the conductorsc(Λ_f(p))contain a nontrivial common factor F_f, at least within the limit of our computations. (see Sections 4.1 and 4.2).

In Section 2, we recall the space of Hilbert cusp forms for totally real number fields and Hecke operators. We then reformulate Saito’s formula into a more computable one. The notion of the conductor of an order plays an important role in this process. In Section 3, we give an algorithm to compute the trace of Hecke operators for a real quadratic field F. Key points of the computation are the determination of the relative discriminantD_K/F, the character ^K_p , and the Hecke L-value LF(0,χ_K/F) for any totally imaginary quadratic extensionK overF.

In particular, the computation of the Hecke L-value by Shintani’s method [Shintani 76] has been reduced to that of Hilbert symbols (cf. [Okazaki 91]). In Section 4, we give examples of eigenvalues and characteristic polynomials of Hecke operators restricted to the case where the weight is (2,2) and F is Q(√

257) or Q(√

401), and we describe our analysis of the data to convince the reader of the conclusion we have already described.

While I was preparing the revision of this paper following the request of the referee to provide a more detailed study of F_f, Professor Haruzo Hida provided the following crucial suggestion:

(1) Within the limits of the computations carried out, check that F_fo_K_f is divisible by the common factor of 1 +N(p)−C_f(p) for the principal primesp.

(2) As is well known, several outstanding mathemati- cians have worked out the congruence primes between a primitive cusp form and an Eisenstein series, which are essentially given by the value at the weight of a HeckeL-function of the base field. No- tably, A. Wiles studied in depth such an Eisenstein congruence, which is a key step in his proof of the Iwasawa conjecture for totally realfields. Therefore, if (1) is aﬃrmative, his result presumably implies that F_f is divisible by the congruence primes. Here

the congruence prime can be found in the prime factors of the numerator of the algebraic part of the Hecke L-valueLF(2,χ) associated with a nontrivial class character χ.

(3) Moreover, it is expected that the set of the primes of F_f coincides with that of the congruence primes between the F-proper cusp form f and the Eisen- stein series of weight (k1, k1) with Mellin transform LF(s,χ)LF(s+ 1−k1,χ⁻¹) for a nontrivial class characterχ.

We shall give aﬃrmative numerical evidence for (1) and (2) in Section 4.3. As for (3), we hope to discuss this property in a subsequent paper.

Notation

For an associative ring R with identity element, we denote by R^× the group of invertible elements of R. We writeM2(R) for the ring of 2×2 matrices over R, and 12 for the identity element ofM2(R).

For an algebraic number field F of finite degree, we denote byo_F,d_F, andDF the maximal order ofF, the diﬀerent of F over Q, and the discriminant of F over Q. We writeI(F) for the ideal group ofF, and P(F), Cl(F), andh_F (respectively P⁺(F), Cl⁺(F), andh⁺_F) for the principal ideal group of F, the ideal class group of F, and the class number ofF (respectively those in the narrow sense). Forα∈F^×, we put (α)_F =αo_F.For a prime idealp ofF andm∈I(F), we denote by ordp(m) the order of m at p. For α ∈ F, we set α 0 if α is totally positive. We define o^×_F₊ = {a ∈ o^×_F | a 0}. For integral idealsa,bofF, we writea|bifba⁻¹⊂o_F; for elementsα(= 0),β of o_F, we writeα | β if βα⁻¹ ∈ oF. Forα1, . . . ,αr∈F, we write [α1, . . . ,αr] for theZ- submodule ofF generated byα₁, . . . ,α_r. We denote by ζF the Dedekind zeta function ofF.

For an extension K of F of finite degree, we denote byD_K/F the relative discriminant of K overF. For an elementα ofK, we denote byD_K/F(α), N_K/F(α), and TrK/F(α) the relative discriminant, the norm, and the trace of α in K over F. We denote by N(a) the norm of an ideala of F. (We also use the symbols N_K/F(α), Tr_K/F(α), andN(a) whenK andF are localfields.)

For a∈R, we denote by [a]the greatest integer not greater thana. Let ^a_p be the Legendre symbol fora∈ Z and a prime number p. For a set X, we denote the cardinality ofX by|X| and also by X. For a subgroup Hof a groupG, we write [G:H] =|G/H|. For a subfield

Fof afieldK, the symbol [K:F] means the degree ofK

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over F. The disjoint union of setsY1, . . . , Ys is denoted by ^s_i=1Y_i.

2. THE TRACE FORMULA FOR TOTALLY REAL NUMBER FIELDS

In this section, wefirst recall the definition of Hecke operators acting on the space of Hilbert cusp forms as given in [Shimura 78,§2]. (Cf. also [Shimura 91].)

2.1 Hilbert Cusp Forms and Hecke Operators

LetF be a totally real algebraic numberfield of degree g, and denote byaandhthe sets of archimedean primes and nonarchimedean primes of F. For p ∈ h, we also denote bypthe corresponding prime ideal ofF. For any set X, we write X^a for the set of all indexed elements (x_v)_v_∈_a with x_v ∈ X. Let F_A be the ring of adeles of F, andF_A^× the group of ideles of F. For v ∈a∪hand x∈FA, letFv be the v-completion of F, and xv its v- component. We write F_a and F_h for the archimedean and nonarchimedean factors ofFA, and identifyFawith R^a. For a ∈ I(F) and p ∈ h, we denote by ap the topological closure of a inFp. We abbreviate (o_F)p and (dF)p byopanddp, for short. We then setoh= _p_∈_hop

and d_h = _p_∈_hd_p. Fora ∈F_A^×, we denote byao_F the fractional ideal of F such that (aoF)p =apop for every p ∈ h (i.e., ao_F = F ∩F_a _p_∈_ha_po_p). For a ∈ F_A^×, we set ordp(a) = ordp(aoF). We denote by πp a prime element ofFp. By aHecke character ofF, we understand a character ofF_A^× with values in T={z∈C| |z|= 1} that is trivial onF^×.

LetG= GL₂(F). We setG_v= GL₂(F_v) for everyv∈ a∪h. We consider the adelizationGAofG, and denote by G_aandG_h its archimedean and nonarchimedean factors.

We set Ga+ = {x ∈ Ga | det(x) 0} and G+ = G∩ G_a+G_h. For an element x of G_A, we denote by x_a its a-component. For x∈ GA, we set x^ι = det(x)x⁻¹ and x⁻^ι = (x^ι)⁻¹. We take an element δh of Fh such that δho_h =d_h, define subsetsYh andWh of Gh by

Yh = 1 0

0 δh M2(oh) 1 0 0 δh

−1

∩Gh,

Wh = 1 0

0 δh GL2(o_h) 1 0 0 δh

−1

, and set

Y =Ga+Yh, W =Ga+Wh.

We denote by H the complex upper half-plane. For α= (α_v)_v_∈_a = ^a_c^v ^b^v

v dv v∈a∈G_a+, z= (z_v)_v_∈_a∈ H^a,

k= (kv)v∈a∈Z^a, and aC-valued functionf onH^a, we set

α(z) = (a_vz_v+b_v)/(c_vz_v+d_v) _v

∈a, Jk(α, z) =

v∈a

det(αv)⁻^k^v^/2(cvzv+dv)^k^v , (f kα)(z) =Jk(α, z)⁻¹f(α(z)),

and denote by ˜S^k the space of all holomorphic functions f onH^asatisfying the following two conditions:

(ia) There exists 0< N ∈Z such that f kγ=f for all γ∈SL₂(o_F)∩(1₂+N·M₂(o_F)).

(i_b) For everyα∈G₊, one has (f _kα)(z) =

0 ξ∈Lα

c_α(ξ)e_F(ξz)

with a lattice Lα of F and cα(ξ) ∈ C, where e_F(ξz) = exp(2π√

−1 _v_∈_aξ_vz_v).

Let ψ be a Hecke character of F offinite order such thatthe nonarchimedean part of its conductor is equal to o_F (i.e.ψ(o^×_h) ={1}). We denote byS^k(o_F,ψ) the space of allC-valued functionsf onGAsatisfying the following two conditions:

(ii_a) f(sαxw) = ψ(s)f(x) for s ∈F_A^×, α∈ G, and w∈ W_h (x∈G_A).

(iib) For every x∈Gh, there exists an element fx of ˜Sk

such that f(x⁻^ιu) = (fx ku)(i) for all u ∈ Ga+, where i= (√

−1, . . . ,√

−1)∈H^a.

The elements ofS^k(o_F,ψ) are called (adelic)Hilbert cusp forms of weight k, level o_F, and character ψ. We note that if S^k(o_F,ψ) = {0}, then ψv(−1) = (−1)^k^v for all v ∈ a; moreover, kv > 0 for allv ∈a (cf. [Shimura 78, Proposition 1.1]).

Let RC(W, Y) be the free C-module generated by the double cosets W\Y /W. ForW yW, W zW, W wW ∈ W\Y /W, we take coset decompositions W yW =

m

i=1W y_i andW zW = ⁿ_j=1W z_j, and set

m(W yW, W zW;W wW) = {(i, j)|W yizj =W w}. We then define the product (W yW)(W zW) by

(W yW)(W zW)

=

W wW∈W\Y /W

m(W yW, W zW;W wW)W wW.

Note that the above sum isfinite. We extend this product C-linearly onRC(W, Y). ThenRC(W, Y) becomes aC- algebra, which is called theHecke algebra for W andY.

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For every y ∈ Y, we may assume that W yW =

m

i=1W y_i and (y_i)_a = 1 (∈ G_a). For f ∈ Sk(o_F,ψ), we define a functionf|W yW onGA by

(f|W yW)(x) =

m

i=1

f(xy^ι_i) (x∈GA).

Then, for s ∈ F_A^×, α ∈ G, w ∈ Wh, and x ∈ GA, we have (f|W yW)(sαxw) = ψ(s) ^m_i=1f(x(yiw^ι)^ι) = ψ(s)(f|W yW)(x); moreover, for x∈G_h, we have

(f|W yW)(x⁻^ιu) =

m

i=1

f((xy_i⁻¹)⁻^ιu)

= (

m

i=1

(f_xy−1

i ) _ku)(i) for all u ∈ G_a+, where f_xy−1

i is as in (ii_b). Thus f|W yW ∈S^k(o_F,ψ). Extending this actionC-linearly to the whole of R_C(W, Y), we have a ring homomorphism φof RC(W, Y) into theC-linear endomorphism algebra End_C Sk(o_F,ψ) . We call an element ofφ(R_C(W, Y)) a Hecke operator.

We now determine the generators of RC(W, Y). For each integral ideal a ofF, we define elements T(a) and S(a) ofRC(W, Y) by

T(a) =

W yW∈W\Y /W det(y)oF=a

W yW, S(a) =W a 0 0 a W,

where a = (π^ord_p ^p^(a))p∈h ∈ F_h^× (⊂ F_A^×). Now we set T(π_p^l,π_p^l) =W ^π

l p 0

0 π^l_p W for l, l ∈ Z. (Note that πp ∈ F_p^×(⊂F_A^×).) For y, z∈Y, we have

(W yW)(W zW) =W yzW

if gcd(det(y)o_F,det(z)o_F) =o_F. (2—1) Thus we have

T(a) =

p|a

[ordp(a)/2]

lp=0

T(π_p^l^p,π^ord_p ^p^(a)⁻^l^p) , and hence

T(ab) =T(a)T(b) if gcd(a,b) =o_F. (2—2) For any integere≥0, we have

T(1,π_p^e) =

e

f=0 1≤j≤N(p^f) gcd(mf j,π^f_p,π^e−f_p )=1

W πp^e−f mf jδp⁻¹

0 π_p^f ,

where{mf j}^N^(p

f)

j=1 is a complete set of representatives of o_p/π^fpo_p. Moreover, forl, m, n≥0, we have

T(π^l_p,π^l_p)T(π^m_p ,πⁿ_p) =T(π^l+m_p ,π_p^l+n). (2—3) Thus

T(1,πp)T(1,π^e_p)

= T(1,π^e+1_p ) +N(p)T(π_p,π_p)T(1,π_p)^e⁻¹ ife≥2, T(1,π²_p) + (N(p) + 1)T(πp,πp) ife= 1.

(2—4) Therefore, we have

T(p)T(p^e) =T(p^e+1)

+N(p)S(p)T(p^e⁻¹) forp∈hande≥1.

(2—5) From (2—1), (2—3), and (2—4), we see thatR_C(W, Y) is the commutativeC-algebra generated by T(p) andS(p) for all prime idealsp of F. We also denote byT(a) the imageφ(T(a)) in End_C S^k(o_F,ψ) .

An element f of S^k(oF,ψ) is called a primitive form if f is a normalized common eigenfunction of T(p) for all prime ideals p. Here normalized means that the coeﬃcient c(1) of the Fourier expansion f_x(z) =

ξc(ξ)eF(ξz) for x = 12 (∈ Gh) is equal to 1, where f_xis as in (ii_b). (Cf. [Shimura 78, p. 650].)

2.2 The Trace Formula

It is known that the characteristic polynomial of a Hecke operator can be obtained immediately from traces of Hecke operators by using (2—2), (2—5), and Newton’s identities ([Miyake 89, pp. 266—267]). In particular if we take a prime idealp ofF, we can obtain the characteristic polynomialX^r+a₁X^r⁻¹+· · ·+a_r₋₁X+a_r of T(p) as follows:

Letc1, . . . , cr be the eigenvalues ofT(p), and setbl= c^l₁+· · ·+c^l_r= tr (T(p)^l). Then by (2—5), we have tr (T(p)^l) =

[l/2]

i=0

l

i − l

i−1 N(p)ⁱψ(πp)ⁱtrT(p^l⁻²ⁱ) for l = 1, . . . , r, where ^l

−1 = 0. Therefore, we can obtainblfrom trT(p^l⁻²ⁱ) (i= 0, . . . ,[l/2]). By Newton’s formula, we have

bl+bl−1a1+bl−2a2+· · ·+b1al−1+lal= 0 for l = 1, . . . , r. Thus we can obtain a1, . . . , ar from b₁, . . . , b_r.

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Now we describe the trace formula of a Hecke operator T(a) onSk(o_F,ψ) given by [Saito 84, Theorem 2.1]. But first, we introduce the following notation.

Let K be a quadratic extension of F. We denote by O^K/F the set of all orders in K containingo_F. LetΛ∈ OK/F. Since Λ is anF-lattice, we can take x₁, x₂ ∈ K and a₁,a₂ ∈I(F) such that Λ=a₁x₁+a₂x₂. Then we define the integral idealD_K/F(Λ) ofF by

D_K/F(Λ) = (a₁a₂)² x⁽¹⁾₁ x⁽¹⁾₂ x⁽²⁾₁ x⁽²⁾₂

2

,

wherex⁽¹⁾_j , x⁽²⁾_j are the conjugates ofx_j overF. We call D_K/F(Λ) the relative discriminant of Λ with respect to K/F.

Theorem 2.1. Let F (= Q) be a totally real algebraic numberfield of degree g,ψ a Hecke character ofF of finite order such that the nonarchimedean part of its conductor is equal to o_F, and k = (k₁, . . . , k_g) ∈ Z^a such that kj ≥2 andψvj(−1) = (−1)^k^j for each vj ∈a. For every elementbP(F)∈Cl(F), we define a mappingη of Cl(F) into Cl⁺(F) by η(bP(F)) = b²P⁺(F). Then, for any integral ideala ofF, we have

trT(a) =ε(a)δ(a)2ζ_F(2)|D_F|^3/2

(2π)^2g ψ π_p^ord^p^(a)/2 _p

∈h

·

g

j=1

(kj−1) +ε(a)(−1)^g2⁻¹

·

m∈Ma

ψ π_p^ord^p^(m) _p

∈h

−1

·

n∈Nms∈Sn

g

j=1

Φ(s_j, n_j, k_j)

Λ∈Rsn

h(Λ) hF[Λ^× :o^×_F] + (−1)^g⁻¹b(k)

λ∈C(ψ)

λ π^ordp ^p^(a) p∈h b|a b⊂oF

b∈I(F)

N(b).

(2—6) Here

• ε(a) = 1 or 0 depending on whether aP⁺(F) ∈ η(Cl(F))or not;

• δ(a) = 1 or0depending on whethera is a square or not;

• Ma = {m} is the set of all representatives of {mP(F)∈Cl(F)|m²a∈P⁺(F)} such that m⊂oF

andgcd(m,a) =o_F;

• for everym∈Ma, we take an elementnm ofoF such that (n_m)_F = m²a andn_m 0, and we set N_m = nmEF, whereEF is a complete set of representatives of o^×_F₊/(o^×_F)²;

• forn∈Nm, we setSn={s∈m|s²−4n 0};

• let sj, nj be the vj-components of s, n in Fa, and α_j,β_j the roots of X²−s_jX+n_j; then we set

Φ(sj, nj, kj) = α^k_j^j⁻¹−β_j^k^j⁻¹

α_j−β_j n⁻_j^(k^j⁻^2)/2;

• Ksn =F(√

s²−4n), and Rsn is the set of all dis- tinct orders Λ in OKsn/F satisfying D_K_sn_/F(Λ) | (s²−4n)Fm⁻²;

• h(Λ) is the class number of Λ; that is, h(Λ) =

|(Ksn⊗^FFh)^×/K_sn^× _p_∈_hΛ^×_p|, whereΛpis the topological closure ofΛ inK_sn⊗FF_p;

• b(k) = 1 or 0 depending on whether k = (2, . . . ,2) or not;

• C(ψ)is the set of all unramified Hecke characters λ of F such that λ²=ψ.

Note that the second sum of the right-hand side of (2—6) is independent of the choice ofMa, nm, and E_F. We remark also that (2—6) is shortened and corrected from the original formula which appeared in [Saito 84].

2.3 Preliminary Lemmas

We now presentfive lemmas for transforming (2—6) into a more computable form.

Lemma 2.2. LetF be an algebraic number field of finite degree, andKa quadratic extension ofF. For an integral idealcofF, we putρ(c) =oF+coK. Thenρis a bijection of the set of all integral ideals ofF ontoOK/F.

Proof: Thie result follows immediately from [Shimura 71, Proposition 4.11] when F = Q, and we prove our assertion in a similar fashion. It is well known that there existθ∈K anda∈I(F) such thato_K=o_F+θa. Letc be an integral ideal ofF. SincecoK⊂oF+coK⊂oK, we see thato_F+co_K is aQ-lattice inK. Moreover,o_F+co_K is a subring ofKcontainingo_F. Thuso_F+co_K ∈OK/F, and henceρis a mapping. Ifo_F+co_K=o_F+c o_K with integral ideals c, c of F, then o_F +θac = o_F +co_K = oF+c oK =oF +θac. Since{1,θ} is a basis ofK over F, we have c = c. Thus ρ is injective. Let Λ be any order inOK/F. SinceoK is the unique maximal order of K, we have Λ ⊂o_K. Set b= {c ∈ a | θc ∈ Λ}. Since

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oF 'Λ⊂oK=oF+θa, we have{0}'b⊂a. Moreover, bis an o_F-module. Thusb∈ I(F). For anyx∈Λ, we have x=r+θs with r ∈oF and s∈ a, since Λ ⊂oK. Then θs = x−r ∈ Λ, and hence s ∈ b. Therefore, Λ=o_F+θb. Now we setc=ba⁻¹. Thencis an integral ideal of F, and Λ = o_F +θb = o_F +θac = o_F +co_K. Thusρis surjective.

We denote the mappingρ⁻¹byc, and we callc(Λ) the conductor of ΛforΛ∈OK/F.

Lemma 2.3. Let F andK be as in Lemma2.2. Then for Λ∈OK/F, we have

D_K/F(Λ)·D_K/F⁻¹=c(Λ)².

Proof: By Lemma 2.2, we have o_K = o_F +θa and Λ = o_F +θac(Λ) with θ ∈ K and a ∈ I(F). Now let θ⁽¹⁾,θ⁽²⁾ be the conjugates of θ over F. Then we have D_K/F =D_K/F(o_K) = a²(θ⁽²⁾−θ⁽¹⁾)² and D_K/F(Λ) = (ac(Λ))²(θ⁽²⁾−θ⁽¹⁾)².

Let F be an algebraic number field of finite degree, andK a quadratic extension ofF. Forp∈h, we define

K

p =

⎧⎪

⎨

⎪⎩

1 ifpsplits inK,

−1 ifpremains prime inK, 0 ifpramifies inK.

Lemma 2.4. Let F be a totally real algebraic number

field offinite degree, andKa totally imaginary quadratic

extension of F. Then for Λ∈OK/F, we have h(Λ) =hK o^×_K :Λ^× ⁻¹N(c(Λ))

p|c(Λ) p∈h

1− ^Kp N(p)⁻¹ .

Proof: This can be proved in exactly the same way as in [Miyake 89, Theorem 6.7.2], which deals with the case F =Q. For any latticeLinKandp∈h, we writeLpfor the topological closure ofLinK⊗FFp. ForΛ∈OK/F, we have

h(Λ) = (K⊗^FFh)^× K^×

p∈h

Λ^×_p

= (K⊗FFh)^× K^×

p∈h

(o_K)^×_p

· K^×

p∈h

(oK)^×_p K^×

p∈h

Λ^×_p .

Generally for an abelian group Gand subgroups H, I, andJ satisfyingI⊃J, the sequence

1→(H∩I)/(H∩J)→I/J →HI/HJ→1

is exact. Thus h(Λ) =h_K·

p∈h

(o_K)^×_p

p∈h

Λ^×_p

· K^×∩

p∈h

(oK)^×_p K^×∩

p∈h

Λ^×_p ⁻¹

=h_K·

p∈h

(o_K)^×_p/Λ^×_p · o^×_K/Λ^× ⁻¹.

Since (o_K)_p=Λ_pif and only ifp|c(Λ), we need to show only that

|(o_K)^×_p/Λ^×_p|=N(c(Λ)p) 1− ^Kp N(p)⁻¹ (2—7) for (o_K)_p =Λ_p. We denote an elementα⊗βofK⊗FF_p simply byαβ. Let p satisfy (o_K)p =Λp. Assume first that p splits in K. Seto_K = o_F +θa with θ ∈K and a ∈ I(F), and take αp ∈ Fp such that αpo_p = a_p. Let f be the minimal polynomial of θ overF, and letθ1,θ2

be the two roots off inFp. Fora, b∈Fp, we setτ(a+ bαpθ) = (a+bαpθ1, a+bαpθ2). Thenτis a topologicalFp- algebra isomorphism ofK⊗FFpontoFp×Fp. (cf. [Weil 67, Chapter III, Theorem 4]). Since (αp(θ2−θ1))²o_p = (D_K/F)p=op, we haveτ((o_K)p) =op×op andτ(Λp) = {(α,β) ∈ o_p×o_p | α−β ∈ c(Λ)p}. Hence τ(Λp)^× = {(α,β)∈o^×_p ×o^×_p |α−β ∈c(Λ)p}. For (α,β)∈o^×_p ×o^×_p, we setρ((α,β)) =αβ⁻¹(1 +c(Λ)_p). Then ρ is a group homomorphism ofo^×_p ×o^×_p onto o^×_p/(1 +c(Λ)p). Since Ker(ρ) = τ(Λ_p)^×, we have τ((o_K)^×_p)/τ(Λ^×_p) = (o^×_p × o^×_p)/τ(Λp)^× ∼=o^×_p/(1 +c(Λ)p). Therefore,

|(oK)^×_p/Λ^×_p|= o^×_p/(1+c(Λ)p) =N(c(Λ)p)(1−N(p)⁻¹).

Thus we obtain (2—7) in this case. Now assume that p remains prime or ramifies inK. ThenK⊗^FFpis afield.

Forβ ∈ o^×_p and γ ∈ (oK)^×_p, we setµ1(β(1 +c(Λ)p)) = β(1 +c(Λ)_p(o_K)_p) and µ₂(γ(1 +c(Λ)_p(o_K)_p)) = γΛ^×_p. Since (1 +c(Λ)p(o_K)p)∩o^×_p = (1 +c(Λ)p+θc(Λ)pa_p)∩ o^×_p = 1 +c(Λ)_p and Λ^×_p = o^×_p +c(Λ)_p(o_K)_p = o^×_p(1 + c(Λ)p(o_K)p), the sequence

1→o^×_p/(1 +c(Λ)p)→^µ¹ (oK)^×_p/(1 +c(Λ)p(oK)p)

µ2

→(o_K)^×_p/Λ^×_p →1 is exact. Therefore,

(oK)^×_p/Λ^×_p = (oK)^×_p/(1 +c(Λ)p(oK)p) · o^×p/(1 +c(Λ)p)⁻¹.

Here we have

(oK)^×_p/(1 +c(Λ)p(oK)p) = ((oK)p/c(Λ)p(oK)p)^×

= N(c(Λ)_p)²(1−N(p)⁻²) ifpremains prime inK, N(c(Λ)p)²(1−N(p)⁻¹) ifpramifies inK,

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and o^×_p/(1 +c(Λ)p) = (op/c(Λ)p)^× = N(c(Λ)p)(1− N(p)⁻¹), which proves (2—7) in this case.

Lemma 2.5. Let F andK be as in Lemma2.4. Let g be the degree ofF overQ. Let χK/F be the ideal character corresponding to the extension K/F (by means of class field theory). Then

L_F(0,χ_K/F) = 2^g⁻¹ hK

h_F o^×_K:o^×_F ,

where LF(s,χ_K/F) is the Hecke L-function associated withχ_K/F.

Proof: Let W_K (respectively W_F) be the group of the roots of 1 in K (respectively F), and RK (respectively R_F) the regulator ofK(respectivelyF). Setw_K =|W_K| and wF = |WF|. Let ZK(s) = (2π)¹⁻^sΓ(s) ^gζK(s) and ZF(s) = π⁻^s/2Γ(s/2) ^gζF(s). Then we have Res_s=0Z_K(s) = −(2π)^gh_KR_Kw⁻_K¹ and Res_s=0Z_F(s) =

−2^ghFRFw⁻_F¹. By ζK(s) =ζF(s)LF(s,χ_K/F), we have Z_K(s) =π^g(1⁻^s)/2Γ((s+ 1)/2)^gZ_F(s)L_F(s,χ_K/F), and hence Ress=0ZK(s) = π^gRess=0ZF(s) ·LF(0,χ_K/F).

Therefore,

LF(0,χ_K/F) =hKRKw_K⁻¹ h_FR_Fw⁻_F¹. Thus we need to show that

[o^×_K :o^×_F] = 2^g−1w_Kw⁻_F¹R⁻_K¹R_F. (2—8) Let l be the mapping from o^×_K to R^g defined byl(δ) = (log|δ⁽¹⁾|, . . . ,log|δ^(g)|), whereδ⁽¹⁾, . . . ,δ^(g)are the conjugates of δ over F. Then o^×_K/WKo^×_F ∼= l(o^×_K)/l(o^×_F).

Since [l(o^×_K) : l(o^×_F)] = 2^g⁻¹R⁻_K¹RF, we have [o^×_K : W_Ko^×_F] = 2^g⁻¹R⁻_K¹R_F. On the other hand, we have [WKo^×_F :o^×_F] =w⁻_F¹wK. Thus we obtain (2—8).

Lemma 2.6. Let F and K be as in Lemma 2.4. Then, for any integral idealfof F, we have

c|f c⊂oF

c∈I(F)

N(c)

p|c p∈h

1− ^Kp N(p)⁻¹

=

p|f

(^Kp)⁼−1 p∈h

N(p)^ord^p^(f)+1+N(p)^ord^p^(f)−2 N(p)−1

·

p|f

(^Kp)=1 p∈h

N(p)^ord^p^(f)

p|f

(^Kp)=0 p∈h

N(p)^ord^p^(f)+1−1 N(p)−1 .

Proof: For every prime ideal p ofF and 0≤s∈ Z, we set

ϕ(p^s) = 1 ifs= 0,

N(p)^s 1− ^Kp N(p)⁻¹ ifs≥1.

Now letf=p^e₁¹· · ·p^e_r^r be the factorization offinto prime factors. Then

c|f

N(c)

p|c

1− ^Kp N(p)⁻¹

=

e1

s1=0

· · ·

er

sr=0

ϕ(p^s₁¹)· · ·ϕ(p^s_r^r)

=

r

j=1 ej

sj=0

ϕ(p^s_j^j) .

Here we have

e_j

s_j=0

ϕ(p^s_j^j) =

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

N(pj)^e^j if _p^K

j = 1, N(pj)^e^j⁺¹+N(pj)^e^j−2

· N(pj)−1 ⁻¹ if _p^K

j =−1, N(pj)^e^j⁺¹−1 N(pj)−1 ⁻¹ if _p^K

j = 0.

Therefore, we obtain our lemma.

2.4 Formula for Computation

From the above lemmas, we obtain the following result:

Proposition 2.7. With the notation of Theorem2.1, we have trT(a) =ε(a)δ(a)(−1)^g2^1−gζF(−1)ψ π_p^ord^p^(a)/2

p∈h

·

g

j=1

(kj−1) +ε(a)(−1)^g2⁻^g

m∈M_a

ψ π_p^ord^p^(m)

p∈h

−1

·

n∈Nms∈Sn g

j=1

Φ(sj, nj, kj) ·LF(0,χK_sn/F)

·

p|fsn

(^Ksnp )⁼−1 p∈h

N(p)^ord^p^(f^sn⁾⁺¹+N(p)^ord^p^(f^sn⁾−2 N(p)−1

·

p|fsn

(^Ksnp )⁼¹

p∈h

N(p)^ord^p^(f^sn⁾

p|fsn

(^Ksnp )⁼⁰

p∈h

N(p)^ord^p^(f^sn⁾⁺¹−1 N(p)−1

+ (−1)^g⁻¹b(k)

λ∈C(ψ)

λ π^ord_p ^p^(a)

p∈h b|a b⊂o_F b∈I(F)

N(b),

(2—9)

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wherefsn=c oF+ ((s+√

s²−4n)/2)oF m⁻¹= (s²− 4n)FDKsn/F−1 1/2

m⁻¹, andχKsn/F is the ideal character corresponding to the extensionKsn/F. (We note that (s²−4n)FD_K_sn_/F⁻¹ is a square by Lemma 2.3.) Proof: In view of Theorem 2.1, we only need to show that

2ζF(2)|DF|^3/2

(2π)^2g = (−1)^g2¹⁻^gζF(−1), (2—10)

Λ∈Rsn

h(Λ)

hF[Λ^×:o^×_F] = 2¹⁻^gLF(0,χKsn/F)

·

p|fsn

(^Ksn_p )⁼−1

N(p)^ord^p^(f^sn⁾⁺¹+N(p)^ord^p^(f^sn⁾−2 N(p)−1

·

p|fsn

(^Ksn_p )⁼¹

N(p)^ord^p^(f^sn⁾

p|fsn

(^Ksn_p )⁼⁰

N(p)^ord^p^(f^sn⁾⁺¹−1 N(p)−1 .

(2—11) By the functional equation of ζF, we have ζF(2) =

|D_F|⁻^3/2(−2π²)^gζ_F(−1), and hence we obtain (2—10).

Next, by Lemma 2.4 and Lemma 2.5, we have

Λ∈Rsn

h(Λ) hF[Λ^×:o^×_F]

=

Λ∈Rsn

hKsn

hF o^×_K_sn:o^×_F N(c(Λ))

p|c(Λ)

1− ^Kp^sn N(p)⁻¹

= 2¹⁻^gLF(0,χK_sn/F)

Λ∈Rsn

N(c(Λ))

·

p|c(Λ)

1− ^K_p^sn N(p)⁻¹ .

Now, from Lemma 2.3, we have

Rsn= Λ∈OKsn/F | D_K_sn_/F(Λ)|(s²−4n)Fm⁻²

= Λ∈OKsn/F | c(Λ)²|(s²−4n)FD_K_sn_/F⁻¹m⁻²

= Λ∈OKsn/F | c(Λ)|f_sn . Thus we have

Λ∈Rsn

N(c(Λ))

p|c(Λ)

1− ^Kp^sn N(p)⁻¹

=

c|fsn

c⊂oF

c∈I(F)

N(c)

p|c

1− ^Kp^sn N(p)⁻¹ ,

since the mappingcis bijective by Lemma 2.2. Therefore we obtain (2—11) by Lemma 2.6.

3. COMPUTATION FOR REAL QUADRATIC FIELDS In this section, we give an algorithm to compute For- mula (2—9) for a real quadraticfieldF. In particular,we assumeF =Q(√m)with a square-free integermsatisfy- ing m≡1 (mod 4Z) exclusively, though the casem≡1 (mod 4Z) can be handled by a similar consideration be- low. Throughout this section, we let ω = (1 +√m)/2, and denote byσthe nontrivial automorphism ofF. We note that for every integral idealaofF there exist ratio- nal integersa >0 andbsuch thata= [a, b+ω]; moreover, it is well known how to check whethera∈P(F) (respectivelya∈P⁺(F)) and to find an explicit generator ofa whena∈P(F) (respectivelya ∈P⁺(F)) by the theory of continued fractions (cf. [Dirichlet 1894], for example).

Forp ∈ h, we callp odd (respectively even) if p z (2)F

(respectivelyp|(2)_F).

3.1 Preliminaries

We note thatζ_F(−1) =ζ(−1)·L −1, ^m = 24⁻¹B_2,(^m), whereB_2,(^m) is the second generalized Bernoulli number associated with ^m . Here ^m is the character corresponding toQ(√m)/Q. It is known that

B_2,(^m) = (6m)⁻¹

m

a=1 m

a (6a²−6am+m²) (cf. [Iwasawa 72,§2]). Hence

ζ_F(−1) = (144m)⁻¹

m

a=1 m

a (6a²−6am+m²).

Thus thefirst and third sums of the right-hand side of (2—9) are easily computable.

Hereinafter, we consider the second sum for the case ε(a) = 1, which implies Ma = ∅. We first explain a method for choosingMa, Nm, and S_n in (2—9). Choose an arbitrary complete set of integral representatives C of Cl(F). Take the set of all elements b1, . . . ,bu of C such that b²_ja ∈ P⁺(F). If gcd(b_j,a) = o_F, then set mj =bj; if gcd(bj,a) =oF, then take a prime idealpj

ofF satisfyingp_j z a and p^σ_jb_j ∈P(F) (i.e., p_jP(F) = bjP(F)), and setmj =pj. Then the setMa is given by

Ma={m₁, . . . ,m_u}.

We fix m ∈ Ma. Then we can take nm satisfying 0 nm ∈oF and (nm)F =m²a. Now we can choose {1} as EF when hF = h⁺_F, and {1,ε} as EF when hF = h⁺_F, where ε is the fundamental unit of F satisfying ε > 1.

Thus we can takeN_m=n_mE_F. (Note that the choice of