Relative Hecke's integral formula for an arbitrary extension of number fields
aa r X i v : . [ m a t h . N T ] F e b Relative Hecke’s integral formula for an arbitrary extensionof number fields
Hohto Bekki
Abstract
In this article, we present a generalized Hecke’s integral formula for an arbitraryextension
E/F of number fields. As an application, we present relative versionsof the residue formula and Kronecker’s limit formula for the “relative” partial zetafunction of
E/F . This gives a simultaneous generalization of two different knownresults given by Hecke himself and Yamamoto.
Let k be a number field (of finite degree), and O k be the ring of integers of k . Let A be an ideal class of k , and let a ∈ A . Then the partial zeta function ζ k ( A , s ) = ζ k ( a , s )associated to the ideal class A (or to the ideal a ) is defined as, ζ k ( A , s ) := X b ∈ A b ⊂O k N b s (Re( s ) > . (1.1)In the case where k is real quadratic, the classical Hecke’s integral formula expressesthis partial zeta function as the integral of the real analytic Eisenstein series along theclosed geodesic on the modular curve SL ( Z ) \ h associated to the ideal class A , where h := { z ∈ C | Im( z ) > } is the Poincar´e upper half plane.To be precise, let k be real quadratic, and fix an embedding k ֒ → R . Suppose a ∈ A is taken to be of the form a = Z + Z α ⊂ k , where α ∈ k is a real quadratic irrational.Let ¯ α be the conjugate of α over Q , and let ̟ be the geodesic on h connecting α and ¯ α : ̟ : R > → h ; t αti + ¯ αt − ti + t − . (1.2)Then it is known that ̟ projected to SL ( Z ) \ h becomes periodic. More precisely, ̟ induces a closed geodesic ̟ : R > /ε Z → SL ( Z ) \ h , (1.3)where ε > O k (cf. [2], [15]).Now, let E ( z, s ) be the real analytic Eisenstein series defined by E ( z, s ) := 12 X ( c,d ) ∈ Z ( c,d )=1 Im( z ) s | cz + d | s , for z ∈ h and s ∈ C , Re( s ) > , (1.4)1hich is well-defined on SL ( Z ) \ h . Then we have the following Hecke’s integral formula. Theorem 1.0.1 ([10]) . We have Z R > /ε Z E ( ̟ ( t ) , s ) dtt = 12 d s/ k Γ( s/ Γ( s ) ζ k ( a − , s ) ζ Q (2 s ) , (1.5) where d k is the discriminant of k . Many authors including Hecke himself have studied generalizations of this formula.Hecke [10] generalizes the formula to the case of an arbitrary number field, and Hiroe-Oda [11] extend Hecke’s result to L -functions twisted by Grossencharacters. Anothergeneralization is obtained by Yamamoto [18], who generalizes the formula to the case ofan arbitrary quadratic extension E/F of number fields. In the following we refer to anygeneralization of Theorem 1.0.1 as Hecke’s integral formula.In a previous paper [2], motivated by the analogy between the above periodicity ofthe geodesic ̟ and the classical Lagrange’s theorem in the theory of continued fractions,we have considered generalization of closed geodesics in the symmetric space for GL n .As a result, we have established some new geodesic multi-dimensional continued fractionalgorithms, and have proved generalizations of Lagrange’s theorem. In this paper, usingthe same idea as in [2], we present Hecke’s integral formula for an arbitrary extension E/F of number fields (Theorem 3.3.3). Hecke’s result corresponds to the case where F = Q , and Yamamoto’s result corresponds to the case where [ E : F ] = 2. In our argu-ment, we are naturally led to introduce a “relative” partial zeta function ζ E/F, A ( A, s )( A ∈ Cl F , A ∈ Cl E ) (Definition 3.3.2), which gives a “decomposition of the partial zetafunction ζ E ( A, s ) along Cl F ”: ζ E ( A, s ) = X A ∈ Cl F ζ E/F, A ( A, s ) . (1.6)As an application of our Hecke’s integral formula, we obtain the residue formula andKronecker’s limit formula for this relative partial zeta function ζ E/F, A ( A, s ), that is,formulas for the residue and the constant term of ζ E/F, A ( A, s ) at s = 1 (Theorem 5.3.1).The author thinks it is interesting that both the special value of ζ E ( A, s ) at s = 1 andthe special value of ζ F ( A − , s ) at s = n appear simultaneously in the residue formula(5.13) of the relative partial zeta function ζ E/F, A ( A, s ). As far as the author is aware,such a phenomenon has not been observed in the previous works.There are also many preceding works on Kronecker’s limit formula for the zeta func-tions of number fields. Hecke remarks in [10] that one can deduce Kronecker’s limit for-mula for general number fields from the result of Epstein [7, p. 644]. Liu and Masri [14]use this formula for totally real fields to obtain an analogue of Kronecker’s solution ofPell’s equation. Bump and Goldfeld [5] give a different proof in the case of totally realcubic fields. The case of relative quadratic extensions of number fields is obtained byYamamoto [18], and our result generalizes all of these results.2 utline of this paper
In Section 2, we set up generalizations of the Poincar´e upperhalf plane h , the above geodesic ̟ on h , and the real analytic Eisenstein series E ( z, s ).We use the symmetric space for Res F/ Q GL n as a generalization of h where F is anumber field (cf. Borel [3]), and define a certain totally geodesic submanifold called theHeegner object in the symmetric space which plays a part of the above geodesic ̟ . Thenwe consider the Eisenstein series for Res F/ Q GL n following the general construction ofLanglands [13]. In Section 3, we prove our first main theorem: Theorem 3.3.3.Sections 4 and 5 are devoted to prove the residue formula and Kronecker’s limitformula for the relative partial zeta function ζ E/F, A ( A, s ). In Section 4, we computethe Fourier expansion of the Eisenstein series as a preparation. In Section 5, we firstprove the residue formula and Kronecker’s limit formula for the Eisenstein series usingthe Fourier expansion. Then using Theorem 3.3.3, we obtain our second main theorem:Theorem 5.3.1.
Some remarks on Eisenstein series
In this paper, we construct Eisenstein series bygeneralizing the argument of Goldfeld [8]. This construction fits into the general theoryof Eisenstein series established by Langlands [13]. Therefore some of the results in thispaper concerning the Eisenstein series (e.g., the convergence, analytic continuation, andsome part of Fourier coefficients) might be obtained directly from the general theory.However, as far as the author is aware, our Eisenstein series E L, A ( z, s ) ( A ∈ Cl F )(Definition 2.2.7) has not been studied well, while the sum E L ( z, s ) = P A ∈ Cl F E L, A ( z, s )is a generalization of those traditional Eisenstein series studied by Asai [1], Jorgenson-Lang [12], Yoshida [19], and Yamamoto [18] in the case where n = 2. In Sections 4and 5, we give an explicit Fourier expansion formula, residue formula, and Kronecker’slimit formula for our Eisenstein series E L, A ( z, s ), which gives a generalization of thecorresponding results proved in [1], [12], [18] and [14]. Let Λ be any index set, and let { X λ } λ ∈ Λ be any family of sets indexed by Λ. For x ∈ Q λ ∈ Λ X λ , we often denote by x λ ∈ X λ the λ -component of x without specifying.In this paper, a number field is always assumed to be of finite degree over the field Q of rational numbers. For a number field k , we denote by S k the set of archimedean placesof k . For σ ∈ S k , we denote by k σ the completion of k at σ , and denote by n σ := [ k σ : R ]the local extension degree. We also denote by σ : k ֒ → k σ the completion map. Wedenote by k ∞ := k ⊗ Q R the infinite adele of k . Then we have k ∞ ≃ Q σ ∈ S k k σ . Thenumber of archimedean (resp. real, complex) places is denoted by r k (resp. r ( k ) , r ( k )).As usual, we denote by O k the ring of integers, O × k its group of units, Cl k the ideal classgroup, and h k the class number. We denote by d k ∈ Z the absolute discriminant and by d k the different ideal. For a fractional O k -ideal a ⊂ k , we denote by [ a ] ∈ Cl k the idealclass of a . 3e equip C (resp. C n ) with the following normalized absolute value | | (resp. || || ), | | : C → R ≥ ; x + iy ( x + y ) / ( x, y ∈ R ) , (1.7) || || : C n → R ≥ ; ( x , . . . , x n ) ( | x | + · · · + | x n | ) / . (1.8)For σ ∈ S k , let us choose an embedding k σ ֒ → C . Then the above absolute value inducesan absolute value | | (resp. || || ) on k σ (resp. k nσ ). This is clearly independent of thechosen embedding. For r ≥
1, we define the pairing h , i : k r ∞ × k r ∞ → R by h ( x , . . . , x r ) , ( y , . . . , y r ) i := r X i =1 T r k/ Q ( x i y i ) , ( x i , y i ∈ k ∞ ) . (1.9)Here T r k/ Q : k ∞ → R is the field trace map naturally extended to k ∞ .Suppose now k ′ /k is an extension of number fields. For σ ∈ S k , we denote by S k ′ ,σ the set of places of k ′ above σ . Then for σ ∈ S k and τ ∈ S k ′ ,σ , we fix an embedding k σ ⊂ k ′ τ , and denote by n τ | σ := [ k ′ τ : k σ ] the degree of the local filed extension.In the following, we basically fix an extension E/F of number fields in our argument.However, when we introduce general notation, we use k or k ′ /k to represent an arbitrarynumber field or extension of number fields. F In order to generalize closed geodesics on the modular curve SL ( Z ) \ h , first observe thatthe geodesic ̟ (in Section 1) on h connecting conjugate real quadratic irrationals α and¯ α is obtained by the left translation of the imaginary axis I := i R > ⊂ h by the matrix W = (cid:18) α ¯ α (cid:19) ∈ GL ( R ). See Figure 1 below. ̟ = WI α ¯ α ¯ α h ¯ α h W = ! α ¯ α " ! ∞ i ∞ W I Γ = SL ( Z ) !! Figure 1: The geodesic ̟ In the following, we generalize each of these objects h , W , I , ̟ , and SL ( Z ).Let F be a number field of degree d . We fix an embedding F σ ⊂ C for each σ ∈ S F . Let us consider an algebraic group G := Res F/ Q GL n , where Res F/ Q is the Weil4estriction. Then we take a standard maximal compact subgroup K = Q σ ∈ S F K σ of G ( R ) ≃ Q σ ∈ S F GL n ( F σ ) as follows: K σ := O ( n ) ⊂ GL n ( F σ ) = GL n ( R ) , if σ is real, (2.1) K σ := U ( n ) ⊂ GL n ( F σ ) = GL n ( C ) , if σ is complex . (2.2) Definition 2.0.1 (Generalized upper half space over F , Cf. [3], [8]) . Set h nσ := GL n ( F σ ) /F × σ K σ , for σ ∈ S F . (2.3)Then we define the generalized upper half space for G to be the symmetric space h nF := G ( R ) /F ×∞ K ≃ Y σ ∈ S F h nσ . (2.4)In the following, for g ∈ G ( R ) = Q σ ∈ S F GL n ( F σ ), we always denote by g σ the σ -component of g , and denote by [ g ] ∈ h nF the class represented by g . Remark .
In the case where F = Q , n = 2, we have an isometry h Q ∼ → h ; [ g ] gi , wherethe action of g on i ∈ h is the usual linear fractional transformation. In this section, we define a certain totally geodesic submanifold of h nF called the Heegnerobject. We closely follow the construction in [2, Section 2], but have slightly modifiedthe argument in order to deal with the ideal class group of F .Let F be the same as above, and let E/F be a field extension of degree n . For σ ∈ S F and τ ∈ S E,σ , we fix an embedding E τ ⊂ C so that F σ ⊂ E τ ⊂ C . Let us fixa basis w , . . . , w n ∈ E of E over F . Set w := t ( w · · · w n ) ∈ E n . First, we define twoisomorphisms E ∞ ≃ F n ∞ of F ∞ -modules using local and global data on E ∞ . local : For σ ∈ S F and τ ∈ S E,σ , we fix an isomorphism E τ ≃ F n τ | σ σ of F σ -vectorspaces as follows: If n τ | σ = 1, then simply E τ = F σ . Otherwise, σ is real and τ is complex,and n τ | σ = 2. Therefore using C ≃ R ; x + iy ( y, x ), we obtain E τ = C ≃ R = F σ .This induces E σ := E ⊗ F F σ ≃ Y τ | σ E τ ≃ Y τ | σ F n τ | σ σ ≃ F nσ , (2.5)and by taking the product over σ ∈ S F , we get ι : E ∞ ∼ → F n ∞ . (2.6) global : Since w , . . . , w n is a basis of E over F , we have an isomorphism w : F n ∼ → E ; x x · w , of F -vector spaces. Here we regard x as a row vector, and x · w is thescalar product. Thus by tensoring R over Q , we obtain w : F n ∞ ∼ → E ∞ . (2.7)5hen we define W ∈ G ( R ) so that W = ι ◦ w in End( F n ∞ ), that is, ι ◦ w : F n ∞ ∼ → E ∞ ∼ → F n ∞ ; x xW. (2.8)Next, we generalize I . For a number field k , we set T k := Q σ ∈ S k R > . Then T k actsnaturally on k ∞ ≃ Q σ ∈ S k k σ by the component-wise multiplication. For an extension k ′ /k of number fields, the field norm map N k ′ /k : k ′ → k induces a homomorphism, N k ′ /k : T k ′ → T k ; ( t τ ) τ ∈ S k ′ (cid:16) Y τ | σ t n τ | σ τ (cid:17) σ ∈ S k . (2.9)We denote by T k ′ /k := ker( T k ′ N k ′ /k → T k ) the kernel of this norm homomorphism.Then the action of T E/F on F n ∞ via E ∞ ι ≃ F n ∞ , which is clearly as F ∞ -module,induces a group homomorphism I = I E/F : T E/F → G ( R ) = GL n ( F ∞ ) . (2.10) Definition 2.1.1 (Cf. [2]) . We define the Heegner object associated to the basis w of E over F by ̟ = ̟ w : T E/F → h nF ; t [ W I ( t )] , (2.11)where [ W I ( t )] denotes the class of W I ( t ) in h nF as remarked before. Remark . If F = Q and E is imaginary quadratic, then the image of ̟ w is just a Heegnerpoint on h = h Q . Arithmetic subgroup and periodicity
We define an arithmetic subgroup Γ of G ( R )and discuss the periodicity of the Heegner object ̟ with respect to Γ.Let L ⊂ F n be an O F -lattice, that is, an O F -submodule such that L ⊗ O F F = F n .Consider the natural right action of SL n ( F ) on the space F n of row vectors. DefineΓ L := Stab SL n ( F ) ( L ) = { γ ∈ SL n ( F ) | Lγ = L } , (2.12)where “Stab” is the stabilizer subgroup. Then Γ L acts properly discontinuously on h nF from the left.Now let A ⊂ E be a fractional O E -ideal. We take L ⊂ F n so that L corresponds to A under the isomorphism (2.7) (i.e., w : L ∼ → A ), and setΓ = Γ A := Γ L . (2.13)For a number field k , we denote by U k the image of the unit group O × k under thefollowing “multiplicative” regulator map:reg × k : O × k → T k ; u ( | σ ( u ) | ) σ ∈ S k . (2.14)Then, by Dirichlet’s unit theorem, U k is a lattice in T k/ Q ⊂ T k , that is, a discretecocompact subgroup of T k/ Q . 6ow, for an extension k ′ /k of number fields, let O × k ′ /k := ker( O × k ′ N k ′ /k → O × k ) be therelative unit group of k ′ /k . We denote by U k ′ /k the image of O × k ′ /k under the followingrelative regulator map reg × k ′ /k :reg × k ′ /k := reg × k ′ | O × k ′ /k : O × k ′ /k → T k ′ /k , (2.15)which is just the restriction of reg × k ′ . Then we easily see that U k ′ /k is a lattice in T k ′ /k .Let E/F be as before, and let π : h nF → Γ \ h nF be the natural projection. Proposition 2.1.2.
The map T E/F ̟ → h nF π → Γ \ h nF factors through ̟ : T E/F /U E/F → Γ \ h nF . (2.16)In order to prove this proposition, we prepare a lemma. Let ̺ w : E × → Aut F ( E ) ≃ GL n ( F ) ⊂ G ( R ) (2.17)be the regular representation of E over F with respect to the basis w of E over F , thatis, for any row vector x ∈ F n , we have x̺ w ( α ) w = αxw in E . Lemma 2.1.3.
For u ∈ O × E/F , we have W − ̺ w ( u ) W ≡ I (reg × E/F ( u )) mod K .Proof. Set g := W − ̺ w ( u ) W and ρ := reg × E/F ( u ). By the definition of W and ̺ w , theleft hand side g = W − ̺ w ( u ) W represents the multiplication by u on E ∞ via E ∞ ι ≃ F n ∞ .Now, for τ ∈ S E , the multiplication by τ ( u ) on E τ ⊂ C decomposes into the scaling by | τ ( u ) | and a rotation. Therefore g decomposes as g = I ( ρ ) R for some R ∈ K . Proof of Proposition 2.1.2.
Let ρ ∈ U E/F . We have to show that there exists γ ∈ Γ suchthat ̟ ( ρt ) = γ̟ ( t ) holds for all t ∈ T E/F . Take any u ∈ O × E/F such that reg × E/F ( u ) = ρ .Let γ := ̺ w ( u ). Since N E/F ( u ) = 1 and the multiplication by u preserves the ideal A , wehave γ ∈ Γ. Then, by Lemma 2.1.3, we obtain ̟ ( ρt ) = [ W I ( ρt )] = [ γW I ( t )] = γ̟ ( t )for all t ∈ T E/F . Res F/ Q GL n In this section, we set up basic definitions of Eisenstein series on h nF . We apply the generalconstruction of the so-called Langlands Eisenstein series to our case G = Res F/ Q GL n .We basically follow the argument in Goldfeld [8]. However, since [8] deals only with thecase where F = Q , we need some additional consideration.In order to define the Langalnds Eisenstein series, we have to choose a parabolicsubgroup of Γ L . Let us denote by a ֒ → L a data consisting of– L ⊂ F n , an O F -latiice (not necessarily defined from the fractional ideal A ⊂ E ),– a ⊂ F , a fractional O F -ideal such that 1 ∈ a (we call such a an anti-integral ideal ),7 a ֒ → L , a split injective O F -homomorphism, that is, the exact sequence 0 → a ֒ → L → L/ a e → e ∈ L the image of 1 ∈ a .In the following, we refer to such a data a ֒ → L a parabolic data . Set Γ := Γ L (cf. (2.12)).Then we define the parabolic subgroup P a ֒ → L associated to the data a ֒ → L ; 1 e as P = P a ֒ → L := Stab Γ L ( a ֒ → L ) = { γ ∈ Γ L | a eγ = a e in L } . (2.18)Next, we define a certain left P -invariant function on h nF . Definition 2.2.1.
We define a function Det = Det a ֒ → L : h nF → R ≥ byDet([ g ]) := Y σ ∈ S F | det g σ | n σ || eg σ || nn σ , for g = ( g σ ) σ ∈ G ( R ) , (2.19)where || || is the normalized absolute value defined as (1.8). Here by an abuse of notation,we denote also by e the image of e ∈ L ⊂ F n under the completion map σ : F n ֒ → F nσ ,and eg σ is an element of F nσ . It is clear that this function is well-defined on h nF . Lemma 2.2.2.
The function
Det( z ) is left P -invariant, that is, Det( γz ) = Det( z ) forall z ∈ h nF and γ ∈ P .Proof. Since γ ∈ Γ L , we have det γ = 1. Now, since γ ∈ P preserves a e ⊂ L , γ actson e by the multiplication by some u ∈ O × F . Then, Q σ || eγg σ || n σ = Q σ || ueg σ || n σ = Q σ | σ ( u ) | n σ || eg σ || n σ = Q σ || eg σ || n σ . This shows the lemma. Definition 2.2.3.
We define the Eisenstein series associated to the data a ֒ → L as E a ֒ → L ( z, s ) := X γ ∈ P \ Γ Det( γz ) s , for z ∈ h nF and s ∈ C , Re( s ) > . (2.20)We prove the absolute convergence of E a ֒ → L ( z, s ) in Section 3.2. More explicit form
In order to study E a ֒ → L ( z, s ), we rewrite the sum more explicitly.For an O F -lattice L ⊂ F n and a fractional O F -ideal a ⊂ F , we define L a := { x ∈ L − { } | F x ∩ L = a x } (2.21)= { x ∈ L | a → L ; α αx, is a split injective O F -homomorphism } . (2.22) Lemma 2.2.4. (1)
We have a decomposition L − { } = ` a :anti-int. L a , where the union istaken over the anti-integral ideals. (2) For a parabolic data a ֒ → L ; 1 e , we have the following bijection: P a ֒ → L \ Γ L ∼ −→ O × F \ L a ; γ eγ. (2.23)We need the following structure theorem for finitely generated projective modulesover Dedekind domains. See [4, Chapter 7, §
4, Proposition 24] for example.8 roposition 2.2.5.
Let A be a Dedekind domain. Then any non-zero finitely generatedprojective module M over A is isomorphic to A r − ⊕ a for some r ∈ Z ≥ and a fractional A -ideal a . Moreover r and the ideal class of a in this presentation are unique.Proof of Lemma 2.2.4. (1) is clear. To see (2), it suffices to show that the right (matrix)action of Γ L on O × F \ L a is transitive. Take any x ∈ L a . Then by (2.22), there existisomorphisms ϕ : a ⊕ L/ a e ∼ → L which sends (1 ,
0) to e , and ϕ : a ⊕ L/ a x ∼ → L whichsends (1 ,
0) to x . Then by Proposition 2.2.5, we see that there exists an O F -isomorphism ϕ : L/ a e ∼ → L/ a x . Then for any u ∈ O × F , we obtain an automorphism L ϕ − ∼ −→ a ⊕ L/ a e × u ⊕ ϕ ∼ −→ a ⊕ L/ a x ϕ ∼ −→ L (2.24)of L which sends e to x . This extends to γ ∈ GL n ( F ) such that det γ ∈ O × F . By replacing u with u (det γ ) − , we obtain γ ∈ Γ L such that eγ ≡ x mod O × F . Corollary 2.2.6.
We can rewrite the sum (2.20) as E a ֒ → L ([ g ] , s ) = X x ∈O × F \ L a Y σ ∈ S F | det g σ | n σ s || xg σ || nn σ s , for g ∈ G ( R ) . (2.25) In particular, E a ֒ → L ( z, s ) depends only on the lattice L and the anti-integral ideal a . We define some variants of E a ֒ → L ( z, s ). Definition 2.2.7.
Using the expression (2.25), for A ∈ Cl F , we define E L, A ( z, s ) := X a ∈ A a :anti-int. E a ֒ → L ( z, s ) , (2.26)where the sum is taken over the anti-integral ideals in A , and define E L ( z, s ) := X A ∈ Cl F E L, A ( z, s ) = X x ∈O × F \ L −{ } Y σ ∈ S F | det g σ | n σ s || xg σ || nn σ s . (2.27)Here the last equality follows from Lemma 2.2.4 (1) and Corollary 2.2.6. Example 2.2.8.
In the case where F = Q , n = 2, L = Z ⊕ Z , and a = Z , we have L a = { ( c, d ) ∈ Z | ( c, d ) = 1 } , and E Z ֒ → Z ( z, s ) is nothing but the usual real analyticEisenstein series (1.4). In this section, we first show the convergence of Eisenstein series defined in Section 2.2.Then we prove the relative Hecke’s integral formula for
E/F .9 .1 Haar measures First, we fix the normalization of the Haar measures. Let k be an arbitrary numberfield. For σ ∈ S k , we normalize the Haar measure dx σ (resp. d × t σ ) on k σ (resp. R > ) as dx σ = dx, d × t σ = dt/t, if σ is real , (3.1) dx σ = idxd ¯ x, d × t σ = dt /t , if σ is complex . (3.2)Then we define the Haar measure dx k (resp. d × t k ) on k ∞ (resp. T k ) to be the productmeasure dx k := Q σ ∈ S k dx σ (resp. d × t k := Q σ ∈ S k d × t σ ).Now, let k ′ /k be an arbitrary extension of number fields. We normalize the Haarmeasure d × t k ′ /k on T k ′ /k so that the induced Haar measure on the quotient group T k ≃ T k ′ /T k ′ /k coincides with d × t k , that is, for any integrable function φ on T k ′ , we have Z T k ′ φ d × t k ′ = Z T k (cid:16) Z T k ′ /k φ d × t k ′ /k (cid:17) d × t k . (3.3)Next, we fix the notion of the relative gamma factor for k ′ /k . Lemma 3.1.1.
Let r be a positive integer, and let also n , . . . , n r be positive integers.Set T := { t = ( t i ) i ∈ R r> | Q ri =1 t n i = 1 } . Take any i ∈ { , . . . , r } , and let p i : R r> → R r − > : ( t i ) ≤ i ≤ r ( t i ) ≤ i ≤ r,i = i (3.4) be the natural projection, which induces an isomorphism p i : T ∼ → R r − > . We define theHaar measure d × t on T to be the pull-back measure d × t := ( p i ) ∗ Q i = i dt n i /t n i . (1) For any integrable function φ on R r> , the measure d × t satisfies Z R r> φ ( t ) r Y i =1 dt n i i t n i i = Z u = t n ··· t nrr ∈ R > (cid:16) Z T φ ( t ) d × t (cid:17) duu . (3.5) In particular, the Haar measure d × t is independent of the choice of i . (2) Set n := P ri =1 n i , and let Γ( s ) be the gamma function. Then n Γ( ns ) Z T t + · · · + t r ) ns d × t = r Y i =1 n i Γ( n i s ) . (3.6) Proof.
This can be proved by a straightforward computation, and we omit the proof.Let k ′ /k be an arbitrary extension of number fields of degree n . We define the relativegamma factor Γ k ′ /k ( s ) ( s ∈ C , Re( s ) >
1) asΓ k ′ /k ( s ) := Z T k ′ /k Y σ ∈ S k P τ ∈ S k ′ ,σ t τ ) nn σ s/ d × t k ′ /k . (3.7)10 emma 3.1.2. We have Γ k ′ /k ( s ) = Q τ ∈ S k ′ n τ Γ( n τ s ) Q σ ∈ S k nn σ Γ( nn σ s ) , for Re( s ) > . (3.8) Proof.
This follows from Lemma 3.1.1.Let again k (resp. k ′ /k ) be a number field (resp. an extension of number fields).We define the regulator R k (resp. relative regulator R k ′ /k ) to be the volume of T k/ Q /U k (resp. T k ′ /k /U k ′ /k ) with respect to the Haar measure d × t k/ Q (resp. d × t k ′ /k ). We define µ k (resp. µ k ′ /k ) to be the subgroup of torsion elements in O × k (resp. O × k ′ /k ), and set w k := µ k (resp. w k ′ /k := µ k ′ /k ). Lemma 3.1.3.
Let k ′ /k be an extension of number fileds. Then we have R k ′ /k = [ U k ′ : U k U k ′ /k ][ k ′ : k ] r k − R k ′ R k . (3.9) Proof.
This follows easily from the exact sequence 1 → U k ′ /k → U k U k ′ /k → N k ′ /k U k → Now we prove the convergence of Eisenstein series defined in Section 2.2. Let F be thesame as in Section 2. Note that we can identify T F/ Q as a subgroup of G ( R ) = GL n ( F ∞ )lying in the center F ×∞ ⊂ GL n ( F ∞ ). Lemma 3.2.1.
The infinite series E ([ g ] , s ) := X x ∈ L −{ } P σ ∈ S F || xg σ || ) dns/ , (3.10) converges absolutely and compactly for Re( s ) > and g = ( g σ ) σ ∈ G ( R ) . Moreover, forany g ∈ G ( R ) and ρ ∈ U F , we have E L ([ gρ ] , s ) = E L ([ g ] , s ) .Proof. This is the well-known convergence of the Epstein zeta function. See [17, p.47]for example.
Proposition 3.2.2.
We have Z T F/ Q /U F E ([ gt ] , s ) d × t F/ Q = w F Γ F/ Q ( ns ) Q σ ∈ S F | det g σ | n σ s E L ( z, s ) , (3.11) for Re( s ) > and g = ( g σ ) σ ∈ G ( R ) . In particular, E L ( z, s ) converges absolutely andcompactly for Re( s ) > and g = ( g σ ) σ ∈ G ( R ) . roof. We see this by the classical argument as follows.
LHS = Z T F/ Q /U F X x ∈ L −{ } P σ ∈ S F || xg σ || t σ ) dns/ d × t F/ Q (3.12)= w F Z T F/ Q X x ∈O × F \ L −{ } P σ ∈ S F || xg σ || t σ ) dns/ d × t F/ Q (3.13)= w F X x ∈O × F \ L −{ } Z T F/ Q P σ ∈ S F || xg σ || t σ ) dns/ d × t F/ Q . (3.14)Now, since ρ := || xg σ || Q σ ′∈ SF || xg σ ′ || nσ ′ /d ! σ ∈ T F/ Q , the Haar measure d × t F/ Q is invariantunder the change of variables t ρ − t . Then we have LHS = w F X x ∈O × F \ L −{ } Q σ ′ ∈ S F || xg σ ′ || nn σ ′ s Z T F/ Q P σ ∈ S F t σ ) dns/ d × t F/ Q . (3.15)The proposition now follows from (3.7). Proposition 3.2.3.
Let a ֒ → L be a parabolic data, and let A ∈ Cl F . (1) Both E a ֒ → L ( z, s ) (Definition 2.2.3, Corollary 2.2.6) and E L, A ( z, s ) (Definition 2.2.7)converge absolutely and compactly for Re( s ) > and z ∈ h nF . (2) The Eisenstein series E a ֒ → L ( z, s ) is an automorphic function on h nF with respect to Γ L , that is, we have E a ֒ → L ( γz, s ) = E a ֒ → L ( z, s ) , for all γ ∈ Γ L . (3.16)(3) We have E L, [ a ] ( z, s ) = ζ F ( a − , ns )( N a ) ns E a ֒ → L ( z, s ) . (3.17) Proof. (1) follows directly from Proposition 3.2.2, and (2) follows directly from thedefinition of E a ֒ → L ( z, s ) (Definition 2.2.3). To prove (3), it suffices to see E α a ֒ → L ( z, s ) = | N F/ Q ( α ) | ns E a ֒ → L ( z, s ) , for α ∈ F × . (3.18)This follows from the identity L α a = α L a and Corollary 2.2.6.12 .3 Relative Hecke’s integral formula for E/F
Now we prove our first main theorem. Let the notations be the same as in Section2.1. That is,
E/F is an extension of number fields of degree n , ̟ is the Heegnerobject associated to a basis w of E over F , L ⊂ F n is the O F -lattice correspondingto a fractional O E -ideal A with respect to the basis w , and Γ = Γ L is the arithmeticsubgroup of SL n ( F ) associated to L . Let a be an anti-integral O F -ideal and let a ֒ → L be a parabolic data. We consider the Eisenstein series E a ֒ → L ( z, s ) associated to thisdata. In this section, we assume Re( s ) > Theorem 3.3.1.
Put ∆ w := N F/ Q ( d w ) , where d w := (det W ) ∈ F is the “discriminant”of the basis w . Then we have Z T E/F /U E/F E a ֒ → L ( ̟ ( t ) , s ) d × t E/F = | ∆ w | s R E/F w E R − E Γ E/ Q ( s ) w F R − F Γ F/ Q ( ns ) X x ∈O × E \ A a | N E/ Q ( x ) | s , (3.19) for Re( s ) > , where A a is the image of L a under the isomorphism w (2.7), or equiva-lently, the subset of A defined in the same way as (2.22).Proof. The proof is similar to that of Proposition 3.2.2. Let W = ( W σ ) σ ∈ G ( R ), and I ( t ) = ( I ( t ) σ ) σ ∈ G ( R ) ( t ∈ T E/F ) be as in Section 2.1 so that ̟ ( t ) = [ W I ( t )]. Let usfix a maximal torsion free subgroup ˜ U E (resp. ˜ U F , ˜ U E/F ) of O × E (resp. O × F , O × E/F ) sothat ˜ U F , ˜ U E/F ⊂ ˜ U E . Note that we have reg × E/F : ˜ U E/F ∼ → U E/F , etc. Using Corollary2.2.6, we obtain
LHS = 1 w F Z T E/F /U E/F X x ∈ ˜ U F \ L a Y σ | det W σ | n σ s || xW σ I ( t ) σ || nn σ s d × t E/F (3.20)= 1 w F Z T E/F /U E/F X u ∈ ˜ U E/F X x ∈ ˜ U F ˜ U E/F \ L a Y σ | det W σ | n σ s || x̺ w ( u ) W σ I ( t ) σ || nn σ s d × t E/F (3.21)= 1 w F Z T E/F /U E/F X u ∈ U E/F X x ∈ ˜ U F ˜ U E/F \ L a Y σ | det W σ | n σ s || xW σ I ( ut ) σ || nn σ s d × t E/F (3.22)= 1 w F Z T E/F X x ∈ ˜ U F ˜ U E/F \ L a Y σ | det W σ | n σ s || xW σ I ( t ) σ || nn σ s d × t E/F (3.23)Here, the action of ˜ U E/F on L a is defined by the regular representation ̺ w , and theequality (3.22) follows from Lemma 2.1.3. By putting z = xw , we obtain LHS = | ∆ w | s w F X z ∈ ˜ U F ˜ U E/F \ A a Z T E/F Y σ P τ ∈ S E,σ | τ ( z ) | t τ ) nn σ s/ d × t E/F (3.24)13ow, since ρ := (cid:16) | τ ( z ) | Q τ ′ ∈ S E,στ | τ ′ ( z ) | − n τ ′ /nn στ (cid:17) τ ∈ T E/F , where σ τ is the place of F below τ , the Haar measure d × t E/F is invariant under the change of variables t ρ − t .Therefore, we have LHS = | ∆ w | s w F X z ∈ ˜ U F ˜ U E/F \ A a | N E/ Q ( z ) | s Z T E/F Y σ P τ ∈ S E,σ t τ ) nn σ s/ d × t E/F (3.25)= | ∆ w | s [ U E : U F U E/F ]Γ E/F ( s ) w E w F X z ∈O × E \ A a | N E/ Q ( z ) | s (3.26)Now the theorem follows from Lemma 3.1.2 and Lemma 3.1.3. Definition 3.3.2.
Let A be the same as above. For A ∈ Cl F , we define the partial zetafunction ζ E/F, A ( A − , s ) associated to A − relative to A as ζ E/F, A ( A − , s ) := N ( A ) s X a ∈ A anti-int. X x ∈O × E \ A a | N E/ Q ( x ) | s , (3.27)where N ( A ) is the absolute norm of the fractional O E -ideal A .Now, using Proposition 3.2.3 (3), we can rewrite Theorem 3.3.1 as follows. Theorem 3.3.3 (Relative Hecke’s integral formula) . Let [ a ] ∈ Cl F be the ideal class of a . Put c E/F ( s ) := R E/F w E R − E Γ E/ Q ( s ) w F R − F Γ F/ Q ( ns ) . (3.28) Then we obtain Z T E/F /U E/F E a ֒ → L ( ̟ ( t ) , s ) d × t E/F = | ∆ w | s c E/F ( s ) N ( A ) − s N ( a ) − ns ζ E/F, [ a ] ( A − , s ) ζ F ( a − , ns ) . (3.29) Corollary 3.3.4.
Taking the sum over the anti-integral ideals a , we obtain Z T E/F /U E/F E L ( ̟ ( t ) , s ) d × t E/F = | ∆ w | s c E/F ( s ) N ( A ) − s ζ E ( A − , s ) . (3.30) Remark .
In the case where F = Q or n = 2, Corollary 3.3.4 gives the results ofHecke [10, p. 370] and Yamamoto [18, Theorem 3.1.2] respectively. It should be alsoremarked that in the case where F is imaginary quadratic and n = 2, or F = Q and E is totally real, Harder [9] and Sczech [16] respectively obtain results which can be seenas a “cohomological interpretation” of this theorem, and deduce the rationality of thespecial values of zeta functions. 14 The Fourier expansion of Eisenstein series
In this preliminary section, we present a kind of the Fourier expansion formula of theEisenstein series E L, [ a ] ( z, s ) for Re( s ) >
1. This shows the analytic continuation of theEisenstein series, and enables us to compute its residue and the constant term at s = 1.As remarked in Section 1, the Fourier coefficients (Theorem 4.2.1) may be obtained fromthe more general theory. However here we present an explicit formula in terms of idealclasses of F , and give a proof in a self-contained way. Let F/ Q be a number field of degree d , and let a ֒ → L be as before. For another O F -lattice L ′ ⊂ F n which is isomorphic to L , let γ ∈ GL n ( F ) be a matrix such that Lγ = L ′ .Let us consider the parabolic data a ֒ → L γ → L ′ . Then we easily see that E a ֒ → L ( z, s ) = | N F/ Q (det γ ) | s E a ֒ → L ′ ( γ − z, s ) . (4.1)On the other hand, by Proposition 2.2.5 any O F -lattice L ⊂ F n is isomorphic to thelattice of the form a ⊕ · · · ⊕ a n ⊂ F n for some anti-integral ideals a i ⊂ F ( i = 1 , . . . , n ).Therefore, in this section we assume L = a ⊕ · · · ⊕ a n ⊂ F n . We also assume n ≥ a n i n ֒ → L be the n -th inclusion, which is another parabolic data. We consider theFourier expansion of the Eisenstein series E L, [ a ] ( z, s ) at the “cusp” corresponding to theparabolic subgroup P a n ֒ → L = Stab Γ L ( a n ֒ → L ) associated to the data a n ֒ → L . Let N := ξ I ... ξ n − · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ i ∈ a − i a n ⊂ N ( R ) := ξ I ... ξ n − · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ i ∈ F ∞ (4.2)be the nilpotent radical of P a n ֒ → L , and its canonical lifting into G ( R ). In the following,we identify N ( R ) with F n − ∞ via the isomorphism F n − ∞ ∼ → N ( R ); ( ξ , . . . , ξ n − ) ξ I ... ξ n − · · · , (4.3)which clearly induces an isomorphism a − a n ⊕ · · · ⊕ a − n − a n ∼ → N . We also identify N with the O F -lattice a − a n ⊕ · · · ⊕ a − n − a n in F n − ∞ via this isomorphism. We denote by dξ the Haar measure on N ( R ) ≃ F n − ∞ which is the product measure of dx F on F ∞ definedin Section 3.1. We denote by N ∨ := { y ∈ F n − ∞ | h x, y i ∈ Z , ∀ x ∈ N } = a a − n d − F ⊕ · · · ⊕ a n − a − n d − F (4.4)the dual lattice of N with respect to the pairing h , i : F n − ∞ × F n − ∞ → R .15 efinition 4.1.1 (Fourier coefficients) . For ∗ ∈ { a ֒ → L, ( L, [ a ]) } and ν ∈ N ∨ , we definethe ν -th Fourier coefficient of E ∗ ( z, s ) as I ∗ ,ν ( z, s ) := 1 vol ( N \ N ( R )) Z N \ N ( R ) E ∗ ( ξz, s ) e − πi h ν,ξ i dξ. (4.5)Then, by the general theory of the Fourier expansion, we have E ∗ ( z, s ) = X ν ∈ N ∨ I ∗ ,ν ( z, s ) . (4.6) Iwasawa normal form
In order to state the Fourier expansion formula explicitly, for z ∈ h nF , we always take a representative g ∈ G ( R ) = GL n ( F ∞ ) of z (i.e., z = [ g ]) of thefollowing form called the Iwasawa normal form: g = XY with X = x ij . . . , Y = y ′ . . . y ′ n − , (4.7)where x ij ∈ F ∞ , and y ′ i = y i · · · y n − for y i ∈ T F = Q σ R > . Note that the existenceand the uniqueness of the Iwasawa normal form is guaranteed by the Iwasawa decom-position of GL n ( R ) and GL n ( C ). With the above notation, for 2 ≤ j ≤ n , we set x j := t ( x j , . . . , x j − ,j ) ∈ F j − ∞ , which is essentially the j -th column of X . Furthermore,for 1 ≤ j ≤ n −
1, we put X ( j ) := x · · · x ,n − j . . . . . . .... . . x n − j − ,n − j , Y ( j ) := y · · · y n − j y · · · y n − j . . . y n − j , (4.8)and set g ( j ) := X ( j ) Y ( j ) ∈ GL n − j ( F ∞ ). We fix these notations throughout the paper. Some arithmetic functions and Bessel function
We prepare some arithmeticfunctions and Bessel function.Let m ⊂ O F be a non-zero integral ideal. We define– (Euler’s totient function) ϕ ( m ) := O F / m ) × ,– (Divisor sum) σ s ( m , χ ) := P n | m χ ( n ) N n s , ( χ ∈ Hom( Cl F , C × ), s ∈ C ).Furthermore, let b ⊂ F be a fractional O F -ideal such that b ⊂ d − F . Then, since O F isa Dedekind domain, O F / m is a principal ideal ring, and thus bm − / b is isomorphic to O F / m as an O F / m -module. We denote by ( bm − / b ) × the set of generators of bm − / b as an O F / m -module. We define the special case of the Gauss sum as16 (Ramanujan sum) τ ( m , b ) := X x ∈ ( bm − / b ) × e πiT r F/ Q ( x ) ,where the sum is taken over a system of representatives of ( bm − / b ) × . This is well-defined since b ⊂ d − F . Lemma 4.1.2.
Let m ⊂ O F be a non-zero integral ideal, and let b ⊂ F be a fractional O F -ideal such that b ⊂ d − F , then we have X a | m ϕ ( a ) = N m , X a | m τ ( a , b ) = ( N m ( m | bd F )0 ( otherwise ) (4.9) Proof.
This follows directly from the identity bm − / b = ` a | m ( ba − / b ) × . Definition 4.1.3.
Let s = ( s , . . . , s k ) ∈ C k , t = ( t , . . . , t l ) ∈ C l be tuples of complexvariables such that Re( s i ) , Re( t j ) > i, j , and let b j ⊂ d − F ≤ j ≤ l be fractional O F -ideals. For A ∈ Cl F , we define Z A ( s ; t ; ( b j ) j ) := X m ,..., m k ⊂O F n ,..., n l ⊂O F m ··· m k n ··· n l ∈ A k Y i =1 ϕ ( m i ) N m s i i l Y j =1 τ ( n j , b j ) N n t j j , (4.10)if k, l ≥
1. In the case where k = 0 or l = 0, we define Z A ( ∅ , ∅ ) := ( A = [ O F ])0 ( A = [ O F ]) , Z A ( s ; ∅ ) := X m ,..., m k ⊂O F m ··· m k ∈ A k Y i =1 ϕ ( m i ) N m s i i ,Z A ( ∅ ; t ; ( b j ) j ) := X n ,..., n l ⊂O F n ··· n l ∈ A l Y j =1 τ ( n j , b j ) N n t j j . (4.11)Furthermore, for a character χ ∈ Hom( Cl F , C × ), we define Z ( s ; t ; ( b j ) j ; χ ) := X A ∈ Cl F χ ( A ) Z A ( s ; t ; ( b j ) j ) . (4.12) Lemma 4.1.4.
Let the notations be as above. Then Z A ( s ; t ; ( b j ) j ) converges absolutelyand compactly for Re( s i ) , Re( t j ) > . Moreover, for χ ∈ Hom( Cl F , C × ) , we have Z ( s ; t ; ( b j ) j ; χ ) = k Y i =1 L ( s i − , χ ) L ( s i , χ ) l Y j =1 σ − t j ( b j d F , χ ) L ( t j , χ ) , (4.13) where L ( s, χ ) is the Hecke L -function associated to the character χ . roof. This follows from Lemma 4.1.2.On the other hand, we have Z A ( s ; t ; ( b j ) j ) = 1 h F X χ ∈ Hom( Cl F , C × ) χ ( A − ) Z ( s ; t ; ( b j ) j ; χ ) . (4.14)Therefore we get the following: Corollary 4.1.5.
For A ∈ Cl F , we have Z A ( s ; t ; ( b j ) j ) = 1 h F X χ ∈ Hom( Cl F , C × ) χ ( A − ) k Y i =1 L ( s i − , χ ) L ( s i , χ ) l Y j =1 σ − t j ( b j d F , χ ) L ( t j , χ ) . (4.15) In particular Z A ( s ; t ; ( b j ) j ) can be continued meromorphically to whole s i , t j ∈ C . In this section, we use a slightly different gamma factor from those in Section 3. Fora number field k/ Q of degree d , defineΓ k ( s ) := Y σ ∈ S k Γ( n σ s . (4.16)Let K s ( x ) ( s ∈ C , x ∈ R > ) be the K -Bessel function, that is, K s ( x ) = 12 Z ∞ e − x ( u + u ) u s duu . (4.17)Then, for s ∈ C and x = ( x σ ) σ ∈ T k , we define the K -Bessel function K k ( s, x ) over k as K k ( s, x ) := Y σ ∈ S k K nσs ( x σ ) . (4.18) We have the following formulas for the Fourier coefficients I L, [ a ] ,ν ( z, s ). For 1 ≤ j ≤ n −
1, we set L ( j ) := a ⊕ · · · ⊕ a n − j ⊂ F n − j . Theorem 4.2.1. (1) (Constant term, ν = 0 ) For Re( s ) > , we have I L, [ a ] , ( z, s ) ζ F ( a − , ns ) = δ [ a ] , [ a n ] ( N a n ) − ns Y σ | det g (1) σ | n σ s (4.19)+ 2 r ( F ) π d p | d F | N a n Γ F ( ns − F ( ns ) Y σ | det g (1) σ | n σ − sn − X m ⊂O F m =0 ϕ ( m ) N m ns E L (1) , [ am − ] ([ g (1) ] , ns − n − ) ζ F ( a − m , ns − , where δ is the Kronecker delta, g (1) σ is the σ -component of g (1) and we identify [ g (1) ] ∈ h n − F , and E L (1) , [ am − ] ([ g (1) ] , s ) is the Eisenstein series on h n − F . (Non-constant terms, ν = 0 ) For Re( s ) > , we have X ν ∈ N ∨ −{ } I L, [ a ] ,ν ( z, s ) ζ F ( a − , ns ) = 2 d π dns p | d F | ( N a n ) ns X n ⊂O F n =0 N ( nd − F ) ns − X m ∈ [ aa − n nd − F ] τ ( m , nd − F ) N m ns × X ν ∈ ( N ∨ ) n − e πi h ν, x n i K F ( ns − , (2 π || νg (1) σ || ) σ )Γ F ( ns ) Y σ | det g (1) σ | n σ s || νg (1) σ || nσ ( ns − . (4.20)We give a proof of this theorem in Section 4.3. Remark .
We can deduce the corresponding results of Epstein [7], Liu-Masri [14] andYamamoto [18] from this theorem. We go further to get rid of the Eisenstein series inthe constant term. Note that in order to deduce the result in [7] and [14], we use thefunctional equation of the Eisenstein series, and specialize this formula to s = 0 (see alsoTheorem 4.2.3 below). Fourier expansion formula
By using Theorem 4.2.1 recursively, we obtain a kind ofFourier expansion formula for E L, [ a ] ( z, s ). In order to simplify the presentation, we firstdefine some additional notation. For 0 ≤ j ≤ n −
1, we set c j ( z, s ) := r ( F ) π d p | d F | ! j Γ F ( ns − j )Γ F ( ns ) ( N a i ) j − ns N a n · · · N a n − j +1 × n − j − Y k =1 | N F/ Q ( y k ) | ks n − Y k = n − j | N F/ Q ( y k ) | ( n − k )(1 − s ) , (4.21) d j ( z, s ) := 2 d π ds ( n − j ) p | d F | Γ F ( ns − j ) c j ( z, s ) , (4.22)where we define c ( z, s ) := Q n − k =1 | N F/ Q ( y k ) | ks for j = 0. For 1 ≤ j ≤ n −
1, setΛ ( j ) := a a − n − j +1 d − F ⊕ · · · ⊕ a n − j a − n − j +1 d − F . Note that Λ (1) = N ∨ . Definition 4.2.2. (1) For 0 ≤ j ≤ n −
1, Re( s ) >
1, we define Φ j ( z, s ) := c j ( z, s ) Z [ aa − n − j ] ( j − tuple z }| { ns, ns − , . . . , ns − j + 1; ∅ ) , (4.23)where we assume ( ns, ns − . . . , ns − j + 1) is the empty tuple ∅ if j = 0.192) For 0 ≤ j ≤ n −
2, Re( s ) >
1, we define Ψ j ( z, s ) := d j ( z, s ) X n ⊂O F n =0 Z [ aa − n − j nd − F ] ( j − tuple z }| { ns, ns − , . . . , ns − j + 1; ns − j ; nd − F ) × N ( nd − F ) ns − j − X ν ∈ (Λ ( j +1) ) n − e πi h ν, x n − j i K F ( ns − j − , (2 π || νg ( j +1) σ || ) σ ) Q σ || νg ( j +1) σ || nσ ( ns − j − , (4.24)where we assume ( ns, ns − . . . , ns − j + 1) = ∅ if j = 0, as above. Theorem 4.2.3.
The functions Φ j ( z, s ) and Ψ j ( z, s ) can be continued meromorphicallyto whole s ∈ C . Furthermore, Φ j ( z, s ) (resp. Ψ j ( z, s ) ) is holomorphic outside the polesof L ( ns − j,χ ) L ( ns,χ ) (resp. L ( ns, χ ) − ) for χ ∈ Hom( Cl F , C × ) and ≤ j ≤ n − . The proof of this Theorem is given in Section 4.4
Theorem 4.2.4 (Fourier expansion formula) . For
Re( s ) > , we have E L, [ a ] ( z, s ) ζ F ( a − , ns ) = n − X j =0 ( Φ j ( z, s ) + Ψ j ( z, s )) + Φ n − ( z, s ) . (4.25) In particular, E L, [ a ] ( z, s ) can be continued meromorphically to whole s ∈ C Proof.
This follows from (4.6), Theorem 4.2.1, and Theorem 4.2.3.
Example 4.2.5.
In the case where F = Q , n = 2, L = Z ⊕ Z , and a = Z , we have Φ ( z, s ) = y s , Φ ( z, s ) = √ πy − s Γ( s − )Γ( s ) ζ Q (2 s − ζ Q (2 s ) , (4.26) Ψ ( z, s ) = 2 π √ y Γ( s ) ζ Q (2 s ) X n ∈ Z , =0 | n | s − σ − s ( n ) K s − (2 π | n | y ) e πinx , (4.27)where σ s ( n ) is the usual divisor sum. In this case, Theorem 4.2.4 is nothing but theclassical Fourier expansion of the real analytic Eisenstein series [8, Theorem 3.1.8]. By Proposition 3.2.3 (3), it suffices to compute the Fourier coefficients I a ֒ → L,ν ( z, s ). Put L := L (1) , g = ( g σ ) σ := g (1) for simplicity.We fix a fundamental domain of N \ N ( R ). Set L a ,i := { x = ( x j ) j ∈ L a ⊂ F n | x j = 0 ( ∀ j ≤ i − , x i = 0 } , ( i ≤ n ) , (4.28) L i := { x = ( x j ) j ∈ L ⊂ F n − | x j = 0 ( ∀ j ≤ i − , x i = 0 } , ( i ≤ n − , (4.29) N i ( R ) := { ξ = ( ξ , . . . , ξ n − ) ∈ N ( R ) | ξ j = 0 ( j = i ) } , ( i ≤ n − , (4.30) N i ( R ) := { ξ = ( ξ , . . . , ξ n − ) ∈ N ( R ) | ξ i = 0 } , ( i ≤ n − , (4.31) N i := N ∩ N i ( R ) , N i := N ∩ N i ( R ) , ( i ≤ n − . (4.32)20ere we identify matrices ξ ∈ N ( R ) with vectors ( ξ , . . . , ξ n − ) ∈ F n − ∞ via the identifi-cation (4.3). We have L a = ` ni =1 L a ,i , N = N i N i , and if i ≤ n − N i acts freely on L a ,i by the matrix action from the right. Step 1. (Decomposition of the integral) For x ∈ F n ∞ , put f ( x ) := Q σ | det g σ | nσs || xg σ || nnσs . Thenwe can decompose the integral I a ֒ → L,ν ( z, s ) as I a ֒ → L,ν ( z, s ) = 1 vol ( N \ N ( R )) n X i =1 X x ∈O × F \ L a ,i Z N \ N ( R ) f ( xξ ) e − πi h ν,ξ i dξ. (4.33)Set I ν,i := P x ∈O × F \ L a ,i R N \ N ( R ) f ( xξ ) e − πi h ν,ξ i dξ . In the case where i = n , we easilysee that I ν,n = 0 unless ν = 0 and [ a ] = [ a n ], in which case we have I ν,n = vol ( N \ N ( R )) (cid:18) N a N a n (cid:19) ns Y σ | det g σ | n σ s . (4.34)In the following, we assume i ≤ n −
1. Then we calculate as I ν,i = X x ∈O × F \ L a ,i /N i X ξ ′ ∈ N i Z N \ N ( R ) f ( xξ ′ ξ ) e − πi h ν,ξ ′ ξ i dξ. (4.35)= X x ∈O × F \ L a ,i /N i Z N i \ N i ( R ) Z N i ( R ) f ( xξ ) e − πi h ν,ξ i dξ i ! dξ i , (4.36)where ξ ′ ξ is the multiplication as matrices, and we use e − πi h ν,ξ ′ ξ i = e − πi h ν,ξ i since h ν, ξ ′ i ∈ Z . Furthermore, dξ i and dξ i are the Haar measures on N i ( R ) and N i ( R )normalized in the same way as dξ . Step 2. (Calculation of the integral)Now, xξ = ( x , . . . , x n − , x ξ + · · · + x n − ξ n − + x n ) for ξ = ( ξ , . . . , ξ n − ). Since x i = 0, we can replace x ξ + · · · + x n − ξ n − + x n with x i ξ i by the change of variables.Then we obtain Z N i \ N i ( R ) Z N i ( R ) f ( xξ ) e − πi h ν,ξ i dξ i ! dξ i (4.37)= e πiT r F/ Q ( νi xi x n ) Z N i \ N i ( R ) e − πiT r F/ Q ( ξ ( ν − νi xi x )+ ··· + ξ n − ( ν n − − νi xi x n − )) dξ i × Z N i ( R ) f ( x , . . . , x n − , x i ξ i ) e − πiT r F/ Q ( ν i ξ i ) dξ i . (4.38)Here the first integral in (4.38) is 0 unless ν = ν i x i ( x , . . . , x n − ), in which case equal to vol ( N i \ N i ( R )). Therefore we assume ν = ν i x i ( x , . . . , x n − ). On the other hand, the21econd integral in (4.38) can be written as Y σ | det g σ | n σ s Z F σ e − πiT r Fσ/ R ( ν i t σ ) ( || ¯ xg σ || + | ¯ x x n,σ + x i t σ | ) nn σ s/ dt σ , (4.39)where ¯ x = ( x , . . . , x n − ) is the first n − x , and x n,σ ∈ F n − σ is the σ -component of x n , and ¯ x x n,σ is the scalar product. By an elementary computation, wehave Z F σ e − πiT r Fσ/ R ( ν i t σ ) ( || ¯ xg σ || + | ¯ x x n,σ + x i t σ | ) nn σ s/ dt σ = n σ π nσ Γ( nn σ s − n σ )Γ( nn σ s ) | σ ( x i ) | − n σ || ¯ xg σ || n σ (1 − ns ) ( ν i = 0)2 n σ π nσns e πiT r Fσ/ R ( ν x n,σ ) K nnσs − nσ (2 π || νg σ || )Γ( nnσs ) | σ ( νi xi ) | nσ ( ns − | σ ( x i ) | nσ || νg σ || nσ (1 − ns ) ( ν i = 0)(4.40) Step 3. (Calculation of the summation)We take a closer look at L a ,i /N i . For x ′ = ( x , . . . , x n − ) ∈ F n − − { } , define b x ′ := { α ∈ F | αx i ∈ a i , ≤ ∀ i ≤ n − } , (4.41) S x ′ := { α ∈ F | ( x , . . . , x n − , α ) ∈ L a } , (4.42)so that we have F x ′ ∩ L = b x ′ x ′ and L a ,i = ` x ′ ∈ L i { x ′ } × S x ′ . Note that b x ′ is afractional O F -ideal since x ′ = 0, and we have, for any integral ideal m ⊂ O F , { x ′ ∈ L − { } | ab − x ′ = m } = L am − , (4.43) Lemma 4.3.1.
We have S x ′ = ∅ unless a ⊂ b x ′ , in which case we have (i) S x ′ = { α ∈ a n a − | v p ( α ) = v p ( a n a − ) ∀ p : prime ideal p | ab − x ′ } , where v p is theusual additive p -adic valuation. In particular, a n b − x ′ acts on S x ′ by addition. (ii) S x ′ / a n b − x ′ = ( a n a − / a n b − x ′ ) × : the set of generators of a n a − / a n b − x ′ as an O F / ab − x ′ -module. (iii) L a ,i /N i = ` x ′ ∈ L i { x ′ } × ( S x ′ /x i a n a − i ) , where x ′ = ( x , . . . , x n − ) .Proof. The first assertion and (i) follows from the fact α ∈ S x ′ ⇔ b x ′ ∩ α a n = a ⇔ O F ∩ α a n b − x ′ = ab − x ′ , ∀ α ∈ F × . (4.44)(ii) follows directly from (i), and (iii) is obvious.22ote that ν i = 0 if and only if ν = 0 by the assumption ν = ν i x i ( x , . . . , x n − ).Therefore, in the case ν = 0, we obtain I ,i = X x ′ ∈O × F \ L i X α ∈ S x ′ /x i a n a − i vol ( N i \ N i ( R )) | N F/ Q ( x i ) | r ( F ) π d Γ F ( ns − F ( ns ) Y σ | det g σ | n σ s || x ′ g σ || n σ ( ns − = vol ( N \ N ( R )) p | d F | r ( F ) π d Γ F ( ns − F ( ns ) (cid:18) N a N a n (cid:19) X x ′ ∈O × F \ L i ab − x ′ ⊂O F ϕ ( ab − x ′ ) N ( ab − x ′ ) Y σ | det g σ | n σ s || x ′ g σ || n σ ( ns − , (4.45)where we use S x ′ /x i a n a − i ) | N F/ Q ( x i ) | vol ( N i \ N i ( R )) = ϕ ( ab − x ′ ) N ( a n b − x ′ ) p | d F | . (4.46)By taking the summation over 1 ≤ i ≤ n −
1, and using (4.43) and Proposition 3.2.3(3), we obtain Theorem 4.2.1 (1).Similarly, in the cases where ν = 0, we obtain I ν,i = vol ( N \ N ( R ))2 d π dns e πi h ν, x n i K F ( ns − , (2 π || νg σ || ) σ )Γ F ( ns ) Y σ | det g σ | n σ s || νg σ || nσ ( ns − × X x ′ ∈O × F \ L i ν = νi xi x ′ | N F/ Q ( ν i x i ) | ns − | N F/ Q ( x i ) | vol ( N i \ N i ( R )) X α ∈ S x ′ /x i a n a − i e πiT r F/ Q ( νi xi α ) . (4.47)Now, suppose ν ∈ ( N ∨ ) n − for an integral ideal n ⊂ O F (and ν i = 0). Then we claimthat the last summation in (4.47) is 0 unless ab − x ′ ⊂ O F , in which case we have X α ∈ S x ′ /x i a n a − i e πiT r F/ Q ( νi xi α ) = a n b − x ′ /x i a n a − i ) τ ( ab − x ′ , nd − F ) . (4.48)Indeed, by (4.43), we have ( N ∨ ) n − = { ν ∈ N ∨ | a n b − ν d F = n } , and therefore, for x ′ ∈ L i such that ν = ν i x i x ′ , we have ν i x i a n b − x ′ = a n b − ν = nd − F ⊂ d − F . (4.49)Now, the claim follows easily from Lemma 4.3.1. Thus, for ν ∈ ( N ∨ ) n − , we obtain I ν,i = 0 unless ν i = 0 and ν j = 0 for all j < i , in which case we have I ν,i = vol ( N \ N ( R )) p | d F | d π dns e πi h ν, x n i K F ( ns − , (2 π || νg σ || ) σ )Γ F ( ns ) Y σ | det g σ | n σ s || νg σ || nσ ( ns − × (cid:18) N a N a n (cid:19) ns N ( nd − F ) ns − X m ∈ [ aa − n nd − F ] m ⊂O F τ ( m , nd − F ) N m ns . (4.50)23ere we use the identity (cid:26) x ′ ∈ O × F \ L i (cid:12)(cid:12)(cid:12)(cid:12) ν = ν i x i x ′ , ab − x ′ = m (cid:27) i -th proj. ∼ −→ (cid:26) x i ∈ O × F \ ( a i − { } ) (cid:12)(cid:12)(cid:12)(cid:12) x i ν i ab − ν = m (cid:27) = (cid:26) x i ∈ O × F \ F × (cid:12)(cid:12)(cid:12)(cid:12) x i ν i ab − ν = m (cid:27) , (4.51)for any integral ideal m ∈ [ aa − n nd − F ]. By taking the summation over 1 ≤ i ≤ n − ν ∈ N ∨ − { } , we obtain Theorem 4.2.1 (2). This completes the proof. The assertion for Φ j ( z, s ) follows directly from Corollary 4.1.5. To prove the assertionfor Ψ j ( z, s ), again by Corollary 4.1.5, we rewrite Ψ j ( z, s ) as Ψ j ( z, s ) = d j ( z, s )( N d − F ) ns − j − h F X χ ∈ Hom( Cl F , C × ) χ ( a − a n − j d F ) L ( ns, χ ) X A ∈ Cl F χ ( A − ) × X n ∈ A n ⊂O F X ν ∈ (Λ ( j +1) ) n − ( N n ) ns − j − σ j − ns ( n , χ ) K F ( ns − j − , (2 π || νg ( j +1) σ || ) σ ) Q σ || νg ( j +1) σ || nσ ( ns − j − e πi h ν, x n − j i . (4.52)Therefore, it suffices to prove that the second row of (4.52) converges absolutely andcompactly for s ∈ C . We use the following asymptotic formulas. Lemma 4.4.1.
Let R be any real number such that R > . (1) For any integral ideal n , any χ ∈ Hom( Cl F , C × ) , and any s ∈ C such that | Re( s ) | ≤ R , we have | σ s ( n , χ ) | ≤ ( N n ) R . (4.53)(2) There exists
C > such that for any x ∈ R > and any s ∈ C such that | Re( s ) | ≤ R ,we have (cid:12)(cid:12) K s ( x ) x − s (cid:12)(cid:12) ≤ Ce − x x − R . (4.54)(3) There exists
C > such that for any x = ( x σ ) σ ∈ T F , we have X u ∈ U F Y σ ∈ S F e − x σ u σ ≤ CN F/ Q ( x ) − . (4.55) See Section 2.1 for the definition of U F ⊂ T F . roof. (1) is elementary and we omit the proof. To prove (2) we use the followingwell-known asymptotic behavior of the K -Bessel function:lim x →∞ √ xK s ( x ) e x = r π , lim x → K s ( x ) x s = 2 s − Γ( s ) , ( s ∈ C , Re( s ) >
0) (4.56)Now, since | K s ( x ) x − s | ≤ K Re( s ) ( x ) x − Re( s ) , we may assume − R ≤ s ≤ R . Then, by theintegral expression (4.17) of K s ( x ), we have K s ( x ) ≤ Z e − x ( u + u ) u − R duu + 12 Z ∞ e − x ( u + u ) u R duu (4.57) ≤ K − R ( x ) + K R ( x ) = 2 K R ( x ) . (4.58)Therefore, we have K s ( x ) x − s e x/ x R ≤ K R ( x ) x R − s e x/ . (4.59)By the asymptotic formulas, the right hand side can be bounded uniformly in s for − R ≤ s ≤ R . This proves (2). To prove (3), first observe that X σ x σ u σ ≥ X σx σ u σ ≥ log( x σ u σ ) ≥ X σx σ u σ ≥ n σ log( x σ u σ ) (4.60)= 14 (log( N F/ Q ( x )) + X σ n σ | log( x σ u σ ) | ) , (4.61)where we use N F/ Q ( u ) = 1. Thus we have X u ∈ U F Y σ e − x σ u σ ≤ N F/ Q ( x ) − X u ∈ U F Y σ e − ( n σ | log( u σ )+log( x σ ) | ) . (4.62)Then, we easily see that there exists C > x such that X u ∈ U F Y σ e − ( n σ | log( u σ )+log( x σ ) | ) ≤ C, (4.63)using the fact that U F is a lattice in T F/ Q . This proves (3).Let R > n ≥
1. Then, by Lemma 4.4.1, there exist C , C > | Re( ns − j − | ≤ R , the second row of (4.52) can be bounded as X n ∈ A X ν ∈ (Λ ( j +1) ) n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( N n ) ns − j − σ j − ns ( n , χ ) Y σ K nσ ( ns − j − (2 π || νg ( j +1) σ || ) || νg ( j +1) σ || nσ ( ns − j − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (4.64) ≤ C X n ∈ A X ν ∈ (Λ ( j +1) ) n − ( N n ) R Y σ || νg ( j +1) σ || − n σ R e − π || νg ( j +1) σ || (4.65) ≤ C X n ∈ A X u ∈O × F X ν ∈O × F \ (Λ ( j +1) ) n − ( N n ) R Y σ || uνg ( j +1) σ || − n σ R e − π || uνg ( j +1) σ || (4.66) ≤ C X n ∈ A X ν ∈O × F \ (Λ ( j +1) ) n − ( N n ) R Y σ || νg ( j +1) σ || − n σ R − n σ . (4.67)25et us fix n ∈ A . Then, (4.67) can be rewritten as C X x ∈O × F \ n − | N F/ Q ( x ) | R − R − X ν ∈O × F \ (Λ ( j +1) ) n − Y σ || νg ( j +1) σ || − n σ (5 R + ) (4.68)= C ( N n ) R + ζ F ( n − , R + 14 ) X ν ∈O × F \ (Λ ( j +1) ) n − Y σ || νg ( j +1) σ || − n σ (5 R + ) , (4.69)where C := C ( N n ) R . By Proposition 3.2.3 (1) applied to the parabolic data n − ֒ → Λ ( j +1) , the last infinite series converges. This completes the proof of Theorem 4.2.3. In this section, we compute the residue and the constant term of our Eisenstein series E L, [ a ] ( z, s ) and the “relative” partial zeta function ζ E/F, [ a ] ( A − , s ) at s = 1.We keep the notations in Section 4. That is, F is a number field of degree d , L ⊂ F n ( n ≥
2) is an O F -lattice of the form L = a ⊕ · · · ⊕ a n , and E L, [ a ] ( z, s ) is the Eisensteinseries associated to a parabolic data a ֒ → L , etc. In the following, for any meromorphicfunction f on C , and a ∈ C , we denote by f ( − ( a ) (resp. f (0) ( a )) the residue (resp.constant term) of f ( s ) at s = a . E L, [ a ] ( z, s ) Recall that by the Fourier expansion formula (Theorem 4.2.4), we have E L, [ a ] ( z, s ) ζ F ( a − , ns ) = n − X j =0 ( Φ j ( z, s ) + Ψ j ( z, s )) + Φ n − ( z, s ) . (5.1)Then, by Theorem 4.2.3, P n − j =0 ( Φ j ( z, s ) + Ψ j ( z, s )) is holomorphic at s = 1. On theother hand, by Corollary 4.1.5, we have Φ n − ( z, s ) = c n − ( z, s ) 1 h F X χ ∈ Hom( Cl F , C × ) χ ([ a a − ]) L ( ns − n + 1 , χ ) L ( ns, χ ) . (5.2)Therefore we get the following. Theorem 5.1.1 (Residue formula for E L, [ a ] ( z, s )) . The Eisenstein series E L, [ a ] ( z, s ) hasa simple pole at s = 1 , and we have E ( − L, [ a ] ( z,
1) = c n − ( z, κ F nh F ζ F ( a − , n ) ζ F ( n )= ( N a · · · N a n ) − r ( F ) π d p | d F | ! n r ( F ) Γ F ( n ) R F nw F ζ F ( a − , n ) ζ F ( n ) , (5.3) where ζ F ( s ) is the Dedekind zeta function of F , and κ F = ζ ( − F (1) is its residue at s = 1 . emark . Note that c n − ( z,
1) and hence E ( − L, [ a ] ( z,
1) are independent of z . We put c n − := c n − ( z,
1) for simple.
Proof.
This follows from the fact that L ( s, χ ) is holomorphic at s = 1 unless χ is trivial,in which case L ( s, χ ) = ζ F ( s ) has a simple pole. E L, [ a ] ( z, s ) First recall the classical case where F = Q , n = 2, L = Z ⊕ Z , and a = Z (cf. Example4.2.5). In this case, we have Φ (0)0 ( z,
1) + Ψ (0)0 ( z,
1) = Φ ( z,
1) + Ψ ( z,
1) = − π log | η ( z ) | , (5.4) Φ (0)1 ( z,
1) = 6 π γ + 12 ψ (cid:18) (cid:19) − ψ (1) − log y − ζ (1) Q (2) ζ Q (2) !! (5.5)= 6 π ( γ − log 2 − log √ y ) − π ζ (1) Q (2) ζ Q (2) , (5.6)where η ( z ) is the Dedekind eta function, γ = ζ (0) Q (1) is Euler’s constant, and ψ ( s ) := Γ ′ ( s )Γ( s ) is the digamma function, that is, the logarithmic derivative of the gamma function. Definition 5.2.1. (1) We define H L, [ a ] ( z ) := ζ F ( a − , n ) n − X j =0 ( Φ j ( z,
1) + Ψ j ( z, , for z ∈ h nF . (5.7)(2) Let k/ Q be a number field. For a character χ ∈ Hom( Cl k , C × ), let L ( s, χ ) be theHecke L -function associated to the character χ . Then we set γ k ( χ ) := L (0) (1 , χ ) . (5.8)Moreover, let Γ k ( s ) be the gamma function over k (4.16). We define the digammafunction ψ k ( s ) over k as ψ k ( s ) := dds log Γ k ( s ) = Γ ′ k ( s )Γ k ( s ) = X σ ∈ S k n σ ψ (cid:16) n σ s (cid:17) . (5.9) Theorem 5.2.2 (Kronecker’s limit formula for E L, [ a ] ( z, s )) . We have E (0) L, [ a ] ( z,
1) = H L, [ a ] ( z ) + c n − h F X χ ∈ Hom( Cl F , C × ) χ ([ a a − ]) ζ F ( a − , n ) L ( n, χ ) γ F ( χ )+ c n − κ F h F ζ F ( a − , n ) ζ F ( n ) ψ F (1) − ψ F ( n ) − log( N a ) − n − X k =1 n − kn log( N F/ Q ( y k )) ! + c n − κ F h F ζ F ( a − , n ) ζ F ( n ) ζ (1) F ( a − , n ) ζ F ( a − , n ) − ζ (1) F ( n ) ζ F ( n ) ! . (5.10)27 roof. This follows from the Fourier expansion formula (5.1) (Theorem 4.2.4), (5.2), andTheorem 5.1.1.
Corollary 5.2.3 (Automorphy of the function H L, [ a ] ) . The function H ∗ L, [ a ] ( z ) := H L, [ a ] ( z ) − c n − κ F h F ζ F ( a − , n ) ζ F ( n ) n − X k =1 n − kn log( N F/ Q ( y k )) (5.11) is an automorphic function on h nF , that is, we have H ∗ L, [ a ] ( γz ) = H ∗ L, [ a ] ( z ) for all γ ∈ Γ L .Proof. Since E L, [ a ] ( z, s ) is an automorphic function (Proposition 3.2.3), E (0) L, [ a ] ( z,
1) isalso an automorphic function on h nF . Then, by Theorem 5.2.2, we obtain H ∗ L, [ a ] ( γz ) − H ∗ L, [ a ] ( z ) = E (0) L, [ a ] ( γz, − E (0) L, [ a ] ( z,
1) = 0 for all γ ∈ Γ L . Remark .
By the identity (5.4), we see that the function H L, [ a ] ( z ) gives a generalization ofthe function − π log | η ( z ) | . In fact, in the case where n = 2, the sum P A ∈ Cl F H L, A ( z )coincides with the function h F ( z, a , b ) considered by Yamamoto [18, Theorem 2.5.1] upto some constant factors, and Corollary 5.2.3 gives a generalization of [18, Corollary2.5.2]. ζ E/F, [ a ] ( A − , s ) Let the notations be as in Section 2.1 and Section 3.3. That is,
E/F is an extension ofnumber fields of degree n ≥
2, and A ⊂ E is a fractional O E -ideal. In this section, wetake a basis w := t ( w , . . . , w n ) of E over F so that the lattice L ⊂ F n corresponding to A via the isomorphism w : F n ∼ → E is of the form L = a ⊕ · · · ⊕ a n for some anti-integral O F -ideals a i (1 ≤ i ≤ n ). Note that this is always possible by Proposition 2.2.5. Then, ̟ : T E/F → h nF is the Heegner object associated to w . Let a be an anti-integral O F -ideal,and let E L, [ a ] ( z, s ) be the Eisenstein series associated to the parabolic data a ֒ → L .Combining Theorem 3.3.3, Theorem 4.2.4, Theorem 5.1.1, and Theorem 5.2.2, weobtain the following properties of ζ E/F, [ a ] ( A − , s ). Theorem 5.3.1. (1)
The relative partial zeta function ζ E/F, [ a ] ( A − , s ) can be continuedmeromorphically to whole s ∈ C , and has a simple pole at s = 1 . (2) (Residue formula) We have ζ ( − E/F, [ a ] ( A − ,
1) = 2 r ( E ) (2 π ) r ( E ) R E w E p | d E | ζ F ( a − , n ) ζ F ( n ) (5.12)= κ E h E ζ F ( a − , n ) ζ F ( n ) , (5.13) where κ E := ζ ( − E (1) is the residue of the Dedekind zeta function ζ E ( s ) of E . (Kronecker’s limit formula) We have ζ (0) E/F, [ a ] ( A − ,
1) = n c − n − h F h E κ E κ F R E/F Z T E/F /U E/F H ∗ L, [ a ] ( ̟ ( t )) d × t E/F + 1 h F X χ χ ([ a a − ]) ζ F ( a − , n ) L ( n, χ ) γ F ( χ ) ! + κ E h E ζ F ( a − , n ) ζ F ( n ) ( nr F − r E ) log 2 + log | d E | | d F | n + log N a n N a · · · N a n − n ζ (1) F ( a − , n ) ζ F ( a − , n ) + n ζ (1) F ( n ) ζ F ( n ) ! . (5.14) Proof. (1) follows from Theorem 3.3.3 and Theorem 4.2.4. To prove (2) and (3), we usethe following identity: | ∆ w | (= | N F/ Q (det W ) | ) = 2 nr ( F ) − r ( E ) | d E | N A | d F | n N a · · · N a n . (5.15)Indeed, by the definition of W and the normalization of the Haar measures dx E , dx F (Section 3.1), we see | d E | N A = vol ( E ∞ / A ) = 2 r ( E ) − nr ( F ) | ∆ w | vol ( F n ∞ / a ⊕ · · · ⊕ a n )= 2 r ( E ) − nr ( F ) | ∆ w | | d F | n N a · · · N a n (5.16)Then, (2) and (3) follows from Theorem 5.1.1 and Theorem 5.2.2 combined with Theorem3.3.3. Acknowledgements
I would like to express my deepest gratitude to Professor Takeshi Tsuji for the constantencouragement and valuable comments during the study. I would also like to thankKazuki Hiroe for pointing to me about the Hecke’s result in the case of general numberfields [10]. Part of this paper is written during my stay at the Max Planck Institute forMathematics in 2018. The author is supported by JSPS Overseas Challenge Programfor Young Researchers.
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Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro,Tokyo, 153-8914 Japan
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