second) limit formula of Kronecker expresses the “second term” of the non-holomorphic Eisenstein series by the “order” of the modular discriminants (resp

ON KRONECKER TERMS OVER GLOBAL FUNCTION FIELDS

FU-TSUN WEI

Abstract. We establish a general Kronecker limit formula of arbitrary rank over global function fields with Drinfeld period domains playing the role of upper-half plane. The Drinfeld-Siegel units come up as equal characteristic modular forms replacing the classical

∆. This leads to analytic means of deriving a Colmez-type formula for “stable Taguchi height” of CM Drinfeld modules having arbitrary rank. A Lerch-Type formula for “to- tally real” function fields is also obtained, with the Heegner cycle on the Bruhat-Tits buildings intervene. Also our limit formula is naturally applied to the special values of both the Rankin-SelbergL-functions and the Godement-JacquetL-functions associated to automorphic cuspidal representations over global function fields.

1. Introduction

The celebrated first (resp. second) limit formula of Kronecker expresses the “second term”

of the non-holomorphic Eisenstein series by the “order” of the modular discriminants (resp.

modular units) at the archimedean place. This formula reveals very interesting stories in arithmetic geometry concealed inside the Euler-Kronecker constants of quadratic number fields (cf. Colmez [5], Hecke [19]) and “non-central” special L-values coming from classical modular forms (cf. Beilinson [2]). The aim of the present paper is to take up the study of this phenomenon in the function field setting. We first establish an “adelic” Kronecker limit formula of arbitrary rank in the mixed characteristic context, and then explore the arithmetic of various “Kronecker terms” over global function fields.

1.1. Kronecker limit formula for arbitrary rank. Letkbe a global function field with a finite constant fieldFq. Fix a place∞ofk, referred to the infinite place ofk. Letk_∞ be the completion ofkat∞andC∞the completion of a chosen algebraic closure ofk∞. LetH^rbe the Drinfeld period domain of rankr, which admits a “Möbius” left action ofGLr(k_∞). Let A(resp.A^∞) be the (resp. finite) adele ring of k. PutH^r_A:=H^r×GLr(A^∞). ThenGLr(k) acts onH^r

A diagonally from the left, and GL_r(A^∞)acts on the second component ofH^r

A by right multiplication. Let S((A^∞)^r)be the space of Schwartz functions (i.e. locally constant and compactly supported) on(A^∞)^r. For eachϕ^∞∈S((A^∞)^r), we introduce the following

“non-holomorphic” Eisenstein series onH^r

A: forz_A= (z₁:· · ·:z_r−1: 1), g^∞

∈H^r

Awe set E(z_A, s;ϕ^∞) := X

06=x=(x1,...,xr)∈k^r

ϕ^∞(xg^∞)· Im(z_A)^s

|x₁z₁+· · ·+x_r−1z_r−1+x_r|^rs_∞.

HereIm(z_A)is the “total imaginary part” of z_A(cf. the equation (2.5) and (3.3)), and| · |_∞ is the normalized absolute value onC∞ (cf. Section 2.1). One can (formally) check that

E(γ·z_A, s;ϕ^∞) =E(z_A, s;ϕ^∞), ∀γ∈GLr(k).

2010Mathematics Subject Classification. 11M36, 11G09, 11R58.

Key words and phrases. Function field, Drinfeld period domain, Bruhat-Tits building, Kronecker limit formula, Drinfeld-Siegel unit, Mirabolic Eisenstein series, CM Drinfeld module, Taguchi height, Colmez-type fomula, SpecialL-value.

1

For instance, letAbe the ring of integers ofk(with respect to∞). Taking a rankrprojective A-moduleY ⊂k^r, let1

Yb be the characteristic function ofYb, the closure ofY in(A^∞)^r. Then we may write

E (z,1), s;1

Yb

= X

06=(c1,...,c_r−1,d)∈Y

Im(z)^s

|c1z1+· · ·+c_r−1z_r−1+d|^rs_∞ =:E^Y(z, s), ∀z∈H^r. The main theorem of this paper is presented in the following:

Theorem 1.1.

(1) The Eisenstein series E(z_A, s;ϕ^∞) is an “extension” of the “mirabolic” Eisenstein series onGL_r(A) associated toϕ^∞ toH^r

A through the building map.

(2) Let S((A^∞)^r)_Z be the subspace of Z-valued Schwartz functions in S((A^∞)^r). For z_A∈H^r_A andϕ^∞∈S((A^∞)^r)_Z, we have E(z_A,0;ϕ^∞) =−ϕ^∞(0)and

∂

∂sE(z_A, s;ϕ^∞) _s=0

=−ϕ^∞(0)·ln Im(z_A)− r

q^rdeg∞−1·ln|u(z_A;ϕ^∞)|_∞,

whereu(·;ϕ^∞)is the Drinfeld-Siegel units onH^r_Aassociated toϕ^∞(which is an “equal characteristic” C∞-valued modular form onH^r

A, cf.Definition 4.2).

In order to state Theorem 1.1 (1) more concretely, we first recall basic properties of mirabolic Eisenstein series on GL_r(A). Given a unitary Hecke character χ on k^×\A^× and ϕ∈S(A^r), the mirabolic Eisenstein series onGLr(A)associated toχandϕcan be expressed as follows (cf.Remark3.3):

E(g, s;χ, ϕ) =|detg|^s_A Z

k^×\A^×



 X

06=x∈k^r

ϕ(a⁻¹xg)



χ(a)|a|^−rs

A d^×a, ∀g∈GL_r(A), Re(s)>1.

Here the Haar measured^×ais normalized so that the maximal compact subgroupO^×

A ofA^× has volume one. Forϕ^∞∈S((A^∞)^r), define the “finite” mirabolic Eisenstein series associated toϕ^∞by:

E^∞(g, s;ϕ^∞) := q−1

# Pic(A)· X

χ∈bI^∞

E(g, s;χ, ϕ^∞⊗1O^r_∞), ∀g∈GLr(A),

whereIb^∞ is the Pontryagin dual group of the finite idele class group I^∞:=k^×\A^∞,×, and 1O_∞^r is the characteristic function ofO^r_∞⊂k_∞^r whereO_∞ is the maximal compact subring of k_∞. Note that the normalized factor(q−1)·# Pic(A)⁻¹ comes from the volume ofI^∞with respect to the chosen Haar measure (cf. the equation (3.1)). On the other hand, letB^r(R)be the realization of the Bruhat-Tits buildingB^r ofPGL_r(k_∞), and denote byλ:H^r→ B^r(R) the building map (cf. Definition 2.6). PutB_A^r(R) :=B^r(R)×GLr(A^∞), and extendλnaturally to aGLr(k)×GLr(A^∞)-biequivariant mapλ_AfromH^r_AtoB_A^r(R). We observe thatE(·, s;ϕ^∞) actually factors throughλ_A (cf. Proposition 3.7), and satisfies (cf. the equation (3.4)):

E(z_A, s;ϕ^∞) = (1−q^{−(rdeg∞)s})· E^∞(gz_A, s;ϕ^∞) (1.1)

for everyz_A∈H^r_Awith λ(z_A) = [gz_A]∈ B_A^r(Z)∼= GLr(A)/k^×_∞GLr(O_∞). In other words, the equality (1.1) links our non-holomorphic Eisenstein series with automorphic Eisenstein series onGLr(A)in a conceptual way.

Remark 1.2.

(1) From the analytic behavior of mirabolic Eisenstein series (recalled in Theorem 3.2), the equality (1.1) says that E(z_A, s;ϕ^∞) converges absolutely for Re(s) > 1, and has a meromorphic continuation to the whole complex s-plane satisfying a “weak”

functional equation with the symmetry between values atsand1−s(cf. Proposition 3.7 andRemark3.8).

(2) Let∆^Y(z)be the Drinfeld-Gekeler discriminant function associated toY onH^r, which is a Drinfeld modular form of weight q^rdeg∞−1 on H^r (cf. Section 2.4). Then for every z ∈H^r we haveu((z,1),1

Yb) = ∆^Y(z). Accordingly, the above theorem leads to a precise function field analogue of the Kronecker (first) limit formula:

E^Y(z,0) =−1 and ∂

∂sE^Y(z, s) s=0

=−ln

Im(z)· |∆^Y(z)|

r qrdeg∞ −1

∞

, ∀z∈H^r. (3) In the number field case, there seems no ideal candidates of the symmetric space

forGLr when r >2. Contrarily, the period domain of Drinfeld, together with the building map to the Bruhat-Tits building, provide a perfect choice of the symmetric space forGLr over global function fields. Our treatment of higher rank Eisenstein series is thereby well-developed. This framework over global function fields is so natural that we can get hold of the Kronecker terms, just as the classicalGL₂ case.

In other words, Kronecker terms can be understood completely even forr >2in the function field setting.

1.1.1. Outline the proof of Theorem 1.1 (2). Our approach is completely different from the classical one. Note thatE(z_A, s;ϕ^∞)isC-valued, andu(z_A;ϕ^∞)lies in the positive character- istic world. The building mapλ_A:H^r_A→ B^r_A(R)is the main bridge in the mix characteristic scene. In particular, we pin point that the building map strips out all the transcendentals in the Drinfeld period domain leaving only an elegant discrete structure.

Another key ingredient of our proof is an explicit description of the meromorphic contin- uation ofE^Y(z, s)(cf. Lemma 3.10). This enables us to derive a Stieltjes-type formula of all the Taylor coefficients ofE^Y(z, s)ats= 0 (cf. Corollary 3.11).

Remark 1.3. In [9], Gekeler first connects an “improper” Eisenstein series on GL₂(k_∞)over rational function fields with Drinfeld discriminant functions (cf. [9]). His result is then gen- eralized by Pál [27, Section 4] to a special family of modular units (in the rank2 case), and also by Kondo [23], Kondo-Yasuda [24, Section 3.5] who target at Jacobi-type Eisenstein series with arbitrary rank. One purpose of this paper is to give a complete account of this phenomenon in adelic settings from the point of view of automorphic representation theory.

Comparing with [32] on the rank2case, there are many new terminologies and approaches when dealing with higher ranks. For instance:

• We introduce a concept of the “total imaginary part” ofz∈H^rfor arbitrary rank (cf.

Section 2.5.1). This new notion is in fact very essential in the whole paper.

• The connection of our non-holomorphic Eisenstein series and automorphic (mirabolic) Eisenstein series (in Theorem 1.1 (1)) is much more conceptual.

• The adelic formulation in Theorem 1.1, together with the new input of Schwartz functionsϕ^∞, enables us to utilize the well-developed tools of the automorphic rep- resentation theory in our study on specialL-values.

Different from Kondo-Yasuda [24], our Eisenstein series totally reflect the whole combinatorial structures of the Bruhat-Tits building, and provide information not just for vertices.

1.1.2. Lerch-type formula. Applying Theorem 1.1, we obtain a Lerch-type formula of mirabolic Eisenstein series onGL_r(A)over function fields. Indeed, let χbe a unitary Hecke character onk^×\A^× with χ(k_∞^×) = 1. Givenϕ^∞∈S((A^∞)^r)_Z, suppose (for simplicity) that either χ is non-trivial orϕ^∞ vanishes at0. Forg∈GLr(A), we set

ηχ(g;ϕ^∞) :=

Z

k^×\A^∞,×

χ(a^∞)·log_qdeg∞|u(z_Aa^∞;ϕ^∞)|_∞d^×a^∞,

wherez_A∈H^r

Ais any point satisfyingλ_A(z_A) = [g]∈ B_A^r(Z), and the Haar measured^×a^∞is normalized so that the maximal compact subgroupO^×

A^∞ ofA^∞,× has volume one. Then (cf.

Corollary 4.7):

Corollary 1.4. Suppose eitherχ is non-trivial orϕ^∞ vanishes at 0. For everyg∈GLr(A) we have

E(g,0;χ, ϕ^∞⊗1O∞) = 1

1−q^rdeg∞ ·ηχ(g;ϕ^∞).

Remark 1.5.

(1) In Corollary 4.7, we also include the case whenχis the trivial character andϕ^∞(0)is non-vanishing. Then the corresponding mirabolic Eisenstein series may have a simple pole ats= 0. Our formula in Corollary 4.7 actually describes the first two terms of its Laurent expansion arounds= 0.

(2) Theorem 1.1 enables us to express the Kronecker term of zeta functions over “to- tally real” function fields as integrations of an “eta function” along the corresponding

“Heegner cycles” in B_A^r(Z), and leads us to a Lerch-type formula for the Dirichlet L-functions associated to “ring class characters” over totally real function fields (cf.

Theorem 6.4)

1.2. Colmez-type formula. Colmez [5] proposes a conjectural formula expressing explicitly the stable Faltings height of CM abelian varieties over number fields in terms of a precise linear combination of logarithmic derivatives of ArtinL-functions. This formula provides a very interesting arithmetic interpretation of the geometric invariant in question, took a stand near the center of arithmetic geometry ever since its discovery (cf. [5], [26], [35], [36], [1], and [3]). Here we apply Theorem 1.1 to derive an analogue of the Colmez formula for the stable

“Taguchi height” of Drinfeld modules.

Let ρ be a Drinfeld A-module of rank r over a finite extension F of k in C∞. The endomorphism ringEndA(ρ/¯k)can be identified with an A-order O of an “imaginary” field K/kwith the degree [K:k]dividingr. We callρCMif[K:k] =r.

In [31], Taguchi introduced a “metrized line bundle” L_ρ associated to ρ, and define the heighth_Tag(ρ/F)ofρbe the “degree” ofLρ(cf. Section 5.1 for an alternative definition). The stable Taguchi height ofρis defined by:

h^st_Tag(ρ) := lim

F: [F:k]<∞

hTag(ρ/F),

where the limit always exists from the fact that every DrinfeldA-module has potential stable reduction (cf. [16, Proposition 7.2]).

Suppose now thatρis a CM DrinfeldA-module. LetΛ_ρ⊂C∞be theA-lattice associated toρ. ViewingΛ_ρ as an O-module, we take an idealI ofOso that Λ_ρ and Ihave the same genus. LetζI(s)be the zeta function associated toI:

ζ_I(s) := X

invertible fractional idealIofO I⊂I

1 N(I)^s,

whereN(I) := #(I/I). Note thatζ_I(s) only depends on the genus ofI (as anO-module).

Our Colmez-type formula is stated as follows:

Theorem 1.6. Leth^st_Tag(ρ)be the stable Taguchi height of the CM DrinfeldA-moduleρ. We have:

h^st_Tag(ρ) =−lnDA(O)−1 r ·ζ_I⁰(0)

ζI(0).

HereDA(O)is the “lattice discriminant” ofO (as anA-lattice inC∞, cf. Remark 2.10).

Remark 1.7.

(1) This formula also provides a geometric interpretation forζ_I⁰(0)/ζ_I(0).

(2) In the proof of Theorem 1.6, we need to extend Hayes’ CM theory of Drinfeld modules to the case when the corresponding lattice has arbitrary genus. The details are attached in Appendix A for the sake of completeness.

(3) when the “CM” function fieldK is separable over k and tamely ramified at∞, we get (cf.Remark5.4 (2))

DA(O) =kd(O/A)k^2r1,

whered=d(O/A)⊂A is the discriminant ideal ofOoverAand kdk:= #(A/d).

(4) Hartl-Singh [15] investigate the Colmez conjecture forA-motives, and prove a product formula for the Carlitz module. Our formula in Theorem 1.6 in the case of the Carlitz module coincides with their result (cf. [15, Example 1.5]), but the analytic approach is completely different from theirs.

(5) Let OK be the integral closure of A in K. From the Ihara estimate of the Euler- Kronecker constant of the zeta functionζO_K(s)in [20, (0.6) and (1.2)], an asymptotic formula of the Taguchi height of Drinfeld modules with CM byO_K is worked out in Section 5.2.1.

1.3. Special values of automorphic L-functions. In the theory of automorphic repre- sentation, mirabolic Eisenstein series naturally occur as the kernel functions in the integral representations of automorphicL-functions (cf. [11] and [30]). From Theorem 1.1 (1), we may naturally apply our Kronecker limit formula to special values of Rankin-SelbergL-functions and Godement-JacquetL-functions over global function fields.

1.3.1. Rankin-SelbergL-functions. LetΠandΠ⁰be two automorphic cuspidal representations ofGLr(A)with unitary central charactersωandω⁰, respectively. Letχ:= (ω·ω⁰)⁻¹. Suppose χ

_k×

∞ = 1. We introduce the follwing multi-linear functional P^RS: Π×Π⁰×S((A^∞)^r)_Z→C (whereS((A^∞)^r)_Z consists ofZ-valued Schwartz functions inS((A^∞)^r)):

P^RS(f, f⁰, ϕ^∞) := 1 1−q^rdeg∞·

Z

A^×GL_r(k)\GL_r(A)

f(g)f⁰(g)η_χ(g;ϕ^∞)dg.

Hereηχ(·;ϕ^∞)is defined in the above of Corollary 1.4, anddgis chosen to be the Tamagawa measure (i.e. vol(A^×GLr(k)\GLr(A), dg) = 2, cf. [34, Theorem 3.3.1]). On the other hand, we have another multi-linear functionalP^RSonΠ×Π⁰×S((A^∞)^r)_Zcoming from the product of “local integrals” (cf. the equality (7.2)). LetL(s,Π×Π⁰)be the Rankin-SelbergL-function associated toΠ andΠ⁰ (following the definition in [30, Lecture 4, p. 137]). The Lerch-type formula in Corollary 1.4 results in:

Theorem 1.8. Let Π and Π⁰ be two automorphic cuspidal representations of GLr(A) with unitary central characters ω and ω⁰, respectively. Suppose Π⁰ is not isomorphic to the con- tragredient representation ofΠ and(ω·ω⁰)

_k×

∞ = 1. Then P^RS=L(0,Π×Π⁰)·P^RS.

1.3.2. Godement-Jacquet L-functions. Let Π be an automorphic cuspidal representation of GLr(A)with unitary central character denoted by ω. Forf1, f2 ∈Π and Φ∈ S(Matr(A)), theGodement-Jacquet L-function associated tof1, f2 andΦis defined by (cf. [11, p. 12]):

L^GJ(s;f1, f2,Φ) :=

Z

GLr(A)

Φ(g)· hΠ(g)f1, f2iPet· |detg|^s_Adg, Re(s)> r.

Hereh·,·iPetis the Petersson inner product onΠ. We choose the Haar measuredgonGL_r(A) to be induced from the Tamagawa measure onA^×\GLr(A)and the measured^×aonA^×with vol(O^×

A, d^×a) = 1. It is known that thisL-function has analytic continuation to the whole complexs-plane and a functional equation with the symmetry between values atsandr−s.

On the other hand, identifyingMat_r(k)withk^r2 suitably (as in the identity (7.4)), we have a group homomorphismι: GL²_r= GLr×GLr→GL_r2 via the left and right multiplications:

(g₁, g₂)·X :=^tg₂Xg₁, ∀g₁, g₂∈GL_r andX∈Mat_r. Supposeω

k∞^× = 1. Define the multi-linear functionalP^GJ : Π×Π×S(Mat_r(A^∞))_Z→C by:

P^GJ(f1, f2, Φ^∞) := 1 1−q^r2deg∞·

Z Z

(A^×GL_r(k)\GL_r(A))²

f1(g1)·ηω⁻¹ ι(g1, g2);Φ^∞

·f2(g2)dg1dg2. Here η_ω−1(·;Φ^∞) is the function onGL_r2(A) coming from Drinfeld-Siegel units on H^r_A² (cf.

Corollary 1.4). Thus for f₁, f₂∈Π and Φ^∞ ∈S(Mat_r(A^∞))_Z, from the “doubling method”

of Piatetski-Shapiro and Rallis (cf. Proposition 7.2) and Corollary 1.4, we immediately get L^GJ(0;f₁, f₂, Φ^∞⊗1_Matr(O∞)) = P^GJ(f₁, f₂, Φ^∞).

(1.2)

LetL(s,Π)be the automorphicL-function associated toΠ. It is a fact that (cf. [12, Theorem 3.3 (2)])

L^GJ(s;f1, f2, Φ)

L(s−(r−1)/2,Π) ∈C[q^s, q^−s], ∀f1, f2∈ΠandΦ∈S(Matr(A)).

Using “local zeta integrals” at each place ofk, we obtain anther multi-linear functionalP^GJ onΠ×Π×S(Matr(A^∞))_Z(cf. the equality (7.6)). Then we arrive at:

Theorem 1.9. Let Π be an automorphic cuspidal representation of GLr(A) with a unitary central characterω. Supposeω

_k×

∞ = 1. The following equality holds:

P^GJ=L(1−r

2 ,Π)·P^GJ.

To gain an in-depth understanding of the specialL-valuesL(0,Π×Π⁰)andL((1−r)/2,Π), Theorem 1.8 and 1.9 reduces the technicalities to local calculations. More precisely, taking suitable test functions at each place, it is possible to determine the corresponding values of P^RS andP^GJ in concrete terms, which gives rise to explicit formulas for the special values L(0,Π×Π⁰)andL((1−r)/2,Π). This is actually a key ingredient in the study of Beilinson’s regulators for Drinfeld modular varieties, which will be explored in a subsequent paper.

Remark 1.10. Kondo-Yasuda [24] consider “partialL-functionsL^I,J(s,Π)”, and connect their special “derivatives” with an “Euler system” coming from rankr Drinfeld-Siegel units. Con- trarily, Theorem 1.9 illustrates a complete different phenomenon. Our formula states for the completeL-functionL(s,Π), and expresses the special L-value in question by an “inner product” with rankr² Drinfeld-Siegel units. We may expect, after further study, there is a natural link between Drinfeld-Siegel units having rank r and r² hidden behind the special L-value in question.

1.4. The content of the paper. We fix basic notations used throughout this paper in Section 2.1. The analytic theory of Drinfeld modules and Drinfeld period domain are reviewed in Section 2.2 and 2.3, respectively. Drinfeld-Gekeler discriminant functions are introduced in Section 2.4. In Section 2.5, we discuss the needed properties of the building map from H^r to the Bruhat-Tits building associated toPGLr(k_∞), and introduce “imaginary parts” of z∈H^r.

Section 3 is to understand the analytic behavior of our non-holomorphic Eisenstein series.

We first recall the well-known analytic properties of mirabolic Eisenstein series onGLr(A)in Section 3.1, and establish a natural identity (via the building map) between these automor- phic Eisenstein series with our non-holomorphic Eisenstein series in Section 3.2 and 3.3. In

Section 3.4, we present a Stieltjes-type formula ofE^Y(z, s)from an explicit description of its meromorphic continuation.

In Section 4. We first introduce the Drinfeld-Siegel units onH^r_A in Section 4.1, and prove our Kronecker limit formula in Section 4.2. In Section 4.3, we derive a Lerch-type formula for our non-holomorphic Eisenstein series onH^r_Aand mirabolic Eisenstein series onGLr(A).

In Section 5, we apply the Kronecker limit formula to prove a Colmez-type formula for the Taguchi height of CM Drinfeld modules. The definition of the Taguchi height of a Drinfeld module is recalled in Section 5.1, and our Colmez-type formula is derived in Section 5.2, together with a short discussion on the asymptotic behavior of the CM Taguchi height.

In Section 6, we expresses the Euler-Kronecker constants of zeta functions over ”totally real” (with respect to∞) function fields as integrations of finite mirabolic Eisenstein series along the corresponding “Heegner cycles” inB^r_A(Z). Consequently, we obtain a Lerch-type formula of the DirichletL-functions associated to ring class characters.

In Section 7, we study applications of our Kronecker limit formula to special values of au- tomorphicL-functions. Theorem 1.8 and 1.9 are demonstrated in Section 7.1 and Section 7.2, respectively.

Finally, we extend Hayes’ CM theory of Drinfeld modules to the case of arbitrary genus in Appendix A.

Acknowledgements. The author is very grateful to Jing Yu and Chieh-Yu Chang for their steady interest, encouragements, and very useful suggestions. He would also like to thank Mihran Papikian for helpful discussions. The author is deeply appreciate the anonymous referee for very careful reading and many useful comments to improve the manuscript. This work is supported by the Ministry of Science and Technology (grant no. 105-2115-M-007-018- MY2 and 107-2628-M-007-004-MY4) and the National Center for Theoretical Sciences.

2. Preliminaries

2.1. Basic settings. LetFqbe the finite field withqelements. Letkbe a global function field with constant fieldFq, i.e.kis a finitely generated field extension overFq with transcendence degree one andFq is algebraically closed ink. For each placevofk, the completion ofkatv is denoted byk_v, andO_v denotes the valuation ring ink_v. Choosing a uniformizer π_v in O_v once and for all, we setFv :=Ov/πvOv andqv := #(Fv). Let degv := [Fv :Fq], called the degree ofv. The absolute value onkv is normalized to:

|αv|v :=q_v^−ordv(αv)=q^{−degvordv(αv)}, ∀αv ∈k_v. LetA:=Q0

vkv, the adele ring of k and O_A :=Q

vOv, the maximal compact subring ofA. We embedk(resp.k^×) intoA(resp.A^×) diagonally. For each elementα= (α_v)_vin the idele groupA^×, the norm|α|_Ais defined to be

|α|_A:=Y

v

|αv|v.

Throughout this paper, we fix a non-trivial additive characterψ=⊗vψ_v :A→C^× which is trivial onk(hereψv:=ψ|k_v). For each placevofk, letδv be the “conductor ofψatv,” i.e.

the maximal integerrso thatπ_v^−rOv is contained in the kernel ofψv (cf. [33, Def. 4 in Chap.

II §5]). It is known that (cf. [33, Cor. 1 of Theorem 2 in Chap. VI])P

vδ_vdegv = 2g_k−2, wheregk is the genus ofk. We callδ= (π^δ_v^v)v∈A^× adifferential idele of kassociated to ψ.

Fix a place∞ofk, regarded as the place at infinity. We setA^∞:=Q0

v6=∞kv, called the finite adele ring ofk, andO_A^∞ :=Q

v6=∞Ov. LetAbe the ring of functions inkregular away from∞. Then the finite places ofk(i.e. the place not equal to∞) are canonically identified with non-zero prime ideals ofA. For a fractional ideala ofA, we denote byaCAifais an

integral ideal. In this paper, every ideal is assumed to be non-zero. For each fractional ideal IofA, writingI=a⁻¹bwherea,bCAwe set

kIk:= #(A/b)

#(A/a).

Note thatkαAk=|α|∞forα∈k^×. Finally, we putdega:=−deg∞ord∞(a)fora∈A.

2.2. Drinfeld modules. Let(F, ι)be anA-field, i.e.F is a field together with a ring homo- morphismι :A →F. TheFq-linear endomorphism ring End_Fq(Ga/F) is isomorphic to the twisted polynomial ring F{τ}, where τ :Ga/F →Ga/F is the Frobenius map (x7→x^q)and τ a=a^qτ for every a∈F.

Definition 2.1. Suppose anA-field(F, ι)and a positive integerris given.

(1)ADrinfeld A-module overF of rank ris a ring homomorphism ρ:A→F{τ} satisfying that

ρa=ι(a) +

rdega

X

i=1

li(ρa)τⁱ∈F{τ}, withlrdega(ρa)6= 0 ∀a∈A.

(2)Given two DrinfeldA-modulesρandρ⁰ overF, ahomomorphismf :ρ→ρ⁰ overF is an element inF{τ} satisfyingf·ρ_a =ρ⁰_a·f for everya∈A. We callf anisogenyiff 6= 0. We denote the endomorphism ring ofρoverF byEndA(ρ/F).

2.3. Drinfeld period domain. LetC∞ be the completion of a chosen algebraic closure of k_∞. We may viewC∞ as anA-field via the natural embedding A ,→C∞. Given a Drinfeld A-moduleρof rankroverC∞. There exists a uniqueFq-linear entire function exp_ρ onC∞

satisfying

exp_ρ(w) =w+

∞

X

i=1

ciw^qi and exp_ρ(aw) =ρa exp_ρ(w)

, ∀a∈A.

It is known that (cf. [13, Theorem 4.6.9]) Λ_ρ := {λ ∈ C∞ : exp_ρ(λ) = 0} is a discrete projectiveA-submodule of rankr inC∞ (i.e. anA-lattice of rankr inC∞). We callΛρ the A-lattice associated toρ. On the other hand, given anA-latticeΛof rankr inC∞, set

exp_Λ(w) :=w Y

06=λ∈Λ

1−w

λ

.

This uniquely determines a rankrDrinfeldA-moduleρ^Λ overC∞satisfying that (2.1) exp_Λ(aw) =ρ^Λ_a(exp_Λ(w)), ∀a∈A.

In other words, the correspondenceρ↔Λ_ρ gives us a bijection (cf. [7, Proposition 3.1]) {Drinfeld A-modules of rankroverC∞} ∼={A-lattices of rankrinC∞}.

We now recall the analytic description of the moduli space for rankrDrinfeld A-modules overC∞. Given a= (a1:· · ·:ar)∈P^r−1(k_∞), letHea⊂C^r∞ (resp.Ha⊂P^r−1(C∞)) be the k_∞-rational hyperplane corresponding toa, i.e.

Hea:=n

(z1, ..., zr)∈C^r∞:

r

X

i=1

aizi= 0o

andHa:=n

(z1:· · ·:zr)∈P^r−1(C∞) :

r

X

i=1

aizi= 0o .

Let

He^r:=C^r∞− [

a∈P^r−1(k∞)

Hea and H^r:=P^r−1(C∞)− [

a∈P^r−1(k∞)

Ha (=He^r/C^×∞).

We callH^r theDrinfeld period domain of rank r. Note thatHe^r andH^r are equipped with a (compatible) left action ofGL_r(k_∞): givenz˜= (z₁, ..., z_r)∈He^r and the corresponding point

z= (z₁ :· · · :z_r)∈H^r, for g= (a_ij)_1≤i,j≤r ∈GL_r(k_∞) we putg·ez:= (z₁⁰, ..., z⁰_r)∈He^r and g·z:= (z⁰₁:· · ·:z⁰_r)∈H^r where





 z₁⁰

... z_r⁰





=







a₁₁ · · · a_1r ... ... a_r1 · · · a_rr











 z₁

... z_r





.

Note that every z ∈ H^r has a unique representative z = (z1 : · · · : z_r−1 : 1). For each g=

∗ ∗ c₁· · ·c_r−1 d

∈GL_r(k_∞), we set

j(g, z) :=c1z1+· · ·c_r−1z_r−1+d.

LetY ⊂k^rbe a projectiveA-module of rank r. Forz= (z1:· · ·:zr−1: 1)∈H^r, put Λ^Y_z :={a1z1+· · ·+ar−1zr−1+ar⊂C∞: (a1, ..., ar)∈Y}.

Observe that

Λ^{Y γ}_γz⁻¹=j(γ, z)⁻¹·Λ^Y_z, ∀γ∈GLr(k).

(2.2)

Letρ^Y,z denote the rank rDrinfeldA-module overC∞corresponding to theA-latticeΛ^Y_z. Theorem 2.2. (cf. [7])The map(Y, z)7→ρ^Y,z induces the following bijection

M^(r)_A :=



 a

[Y]∈P^rA

GL(Y)\H^r



←→ {rank-rDrinfeld A-modules overC∞}/∼=. HerePA^r is the set of isomorphism classes of projectiveA-modules of rank r.

2.4. Drinfeld-Gekeler discriminant function. Givenz= (z1:· · ·:z_r−1: 1)∈H^r and a projectiveA-moduleY of rankrin k^r, let

∆^Y_a(z) :=lrdega(ρ^Y,z_a ), ∀a∈A.

Then the equation (2.2) implies (cf. [8, Chapter V, 3.4 Example])

∆^Y_a(γz) =j(γ, z)^qrdega−1·∆^Y_a(z), ∀γ∈GL(Y).

Moreover, the functional equation (2.1) implies ρ^Y,z_a (x) = ∆^Y_a(z)·x· Y

06=w∈¹_aΛ^Y_z/Λ^Y_z

x−exp_ΛY z(w)

, ∀a∈A− {0}.

Therefore we have the following product formula of∆^Y_a(z):

Lemma 2.3. For everya∈A,

∆^Y_a(z) =a· Y

06=w∈_a¹Λ^Y_z/Λ^Y_z

exp_ΛY z(w)⁻¹. Since

ρ^Y,z_a ·ρ^Y,z_b =ρ^Y,z_ab =ρ^Y,z_b ·ρ^Y,z_a , ∀a, b∈A, one gets

(2.3) ∆^Y_a(z)·∆^Y_b(z)^qrdega= ∆^Y_ab(z) = ∆^Y_b (z)·∆^Y_a(z)^qrdegb.

Take two elementsa₁, a₂∈Asuch that gcd(ord_∞(a₁),ord_∞(a₂)) = 1, and choose`<sub>1</sub>, `₂∈Z such that`₁ q_∞^rord∞(a1)−1

+`₂ q_∞^rord∞(a2)−1

=q^r_∞−1. Set

∆^Y(z) := ∆^Y_a1(z)^{`1</sup>·∆<sup>Y</sup><sub>a2</sub>(z)<sup>`2}, which is a nowhere-zero analytic function onH^rsatisfying that

∆^Y(γz) =j(γ, z)^qr∞−1·∆^Y(z), ∀γ∈GL(Y).

Proposition 2.4. (cf. [8, Chapter IV, Proposition 5.15]) The function ∆^Y is, up to multi- plying with (q_∞^r −1)-th roots of unity, independent of the chosen a1, a2 ∈A and`1, `2 ∈Z. In particular, one has

(∆^Y)^qrdega−1= (∆^Y_a)^qr∞−1, ∀a∈A− {0}.

The following “norm compatibilities” are straightforward.

Lemma 2.5. Given two projective A-modules Y and Y⁰ of rank r in k^r with Y⁰ ⊂Y, for z∈H^r one has

exp_ΛY

z(w) = exp_ΛY0

z (w) Y

06=u∈Λ^Y_z/Λ^Y_z⁰

1−exp_ΛY0 z (w) exp_ΛY0

z (u)

! ,

and

∆^Y(z) = ∆^Y0(z)^{#(Y /Y0)} Y

06=u∈Λ^Y_z/Λ^Y_z⁰

exp_ΛY0

z (u)^qr∞−1.

2.5. Building map. LetB^r be the Bruhat-Tits building associated toPGL_r(k_∞). The set V(B^r) of vertices of B^r consists of all the homothety classes of O_∞-lattices in k^r_∞. Take L^o:=O_∞^r ⊂k_∞^r , the standardO_∞-lattice ink^r_∞. Via the left action of GLr(k_∞)onB^r, the setV(B^r)can be identified withGL_r(k_∞)/k_∞^× GL_r(O_∞):

V(B^r) ={[L^og⁻¹] :g∈GLr(k_∞)/k^×_∞GLr(O_∞)}.

For0≤i < r, a (resp. non-)orientedi-simplex is ani-tuples([L₀], ...,[L_i])(resp. up to cyclic permutations), whereL0, ..., Li areO∞-lattices satisfying

L₀)· · ·)L_i)π_∞L₀.

We letC~ⁱ(B^r)(resp.Cⁱ(B^r)) be the set consisting of all the (resp. non-)orientedi-simplices.

It is known that the realization B^r(R) of B^r is identified with the equivalence classes of norms onk_∞^r as follows: suppose P ∈ B^r(R)belongs to the realization of an i-simplex, say ([L0], ...,[Li]) with L0 ) · · · ) Li ) π_∞L0. Write P = Pi

j=0j[Lj] with 0 ≤j ≤ 1 and Pi

j=0_j = 1. Then

νP := sup{q^−ξ_∞^jνL_j : 0≤j≤i}

with

ξ_j :=

j−1

X

`=0

_` and ν_L(x) := inf{|a|_∞:a∈k_∞ withx∈aL}.

Definition 2.6. Thebuilding mapλ:H^r→ B^r(Q)is defined by

z= (z1:· · ·:zr−1: 1)∈H^r7−→νz:= (a1, ..., ar)∈k_∞^r 7→ |a1z1+· · ·+ar−1zr−1+ar|∞ . The right action ofGLr(k_∞) on k_∞^r yields a left action on the set of norms on k^r_∞ and then onBr(R).

Proposition 2.7. (cf. [10, Proposition 1.5.3] and [6, (4.2) Proposition] The building map λ isGL_r(k_∞)-equivariant.

2.5.1. Imaginary part. Given z = (z₁ : · · · : z_r−1 : 1)∈ H^r, for 1 ≤ i < r, define the “i-th imaginary part” ofz by

Im(z)i := inf





 zi+ (

r−1

X

j=i+1

ujzj) +ur

∞ : ui+1, ..., ur∈k_∞





 .

Take ω = (ω₁ : · · · : ω_r−1 : ω_r) ∈ H^r where ω_i := z_i + (Pr−1

j=i+1u_ijz_j) +u_ir ∈ C∞ with uij ∈k∞so thatIm(z)i=|ωi|∞for1≤i < randωr:= 1. Then

ν_ω(x) = sup{|xiω_i|∞: 1≤i≤r}, ∀x= (x₁, ..., x_r)∈k^r_∞.

Indeed, supposeνω(x) =|x1ω1+· · ·+xrωr|_∞<sup{|xiωi|_∞: 1≤i≤r}. Takei0 minimal so that|xi0ω_i0|∞ = sup{|xiω_i|∞ : 1≤ i ≤ r}. Then x_i0 6= 0 and |xiω_i|∞ < |xi0ω_i0|∞ for i < i0, which implies

|xi₀ωi₀+· · ·+xrωr|_∞<|xi₀ωi₀|_∞. Expressing

ωi₀+xi₀+1

xi₀

ωi₀+1+· · ·+ xr

xi₀

=zi₀+u⁰_i0+1zi₀+1+· · ·+u⁰_r for some u⁰_i0+1, ..., u⁰_r∈k_∞, we get|zi₀+u⁰_i

0+1zi₀+1+· · ·+u⁰_r|_∞<|ωi₀|_∞= Im(z)i₀, a contradiction.

Write|ωi|∞ = q<sup>−`</sup><sub>∞</sub><sup>i+1−ξi</sup> where `i ∈ Z and ξi ∈Q with 0 ≤ξi <1 for 1 ≤ i < r. Let ξ0:= 0andξr:= 1(so`r= 0). Take a permutationσof{1, ..., r−1}so that

ξσ(1)≤ · · · ≤ξ_σ(r−1). We may putσ(0) := 0andσ(r) :=r. Set

gω=gω,0:=





 π^`_∞¹

. .. π_∞^`r





 and gω,i:=







 π^`

(i)

∞1

. .. π^`∞^(i)r









for1≤i < r,

where

`⁽ⁱ⁾_σ(j):=

(`_σ(j)−1, ifj≤i,

`_σ(j), otherwise.

TakeLω,i:=L^og_ω,i⁻¹⊂k^r_∞for1≤i < r. Then

ν_ω= sup{q∞^−ξσ(i)ν_Lω,i : 0≤i < r}.

Indeed, forx= (x1, ..., xr)∈k_∞, we have

ν_ω(x) = sup{|x_jω_j|_∞: 1≤j≤r}= sup{q^−ξ_∞^j−`j+1|x_j|_∞: 1≤j≤r}.

On the other hand, for0≤i < rwe observe that

νL_ω,i(x) = inf{|a|∞:a∈k∞withx∈aL^og_ω,i⁻¹}

= infn

|a|∞:a∈k_∞with(π^`

(i)

∞1 x₁, ..., π_∞^`(i)r x_r)∈aL^oo

= supn q^−`

(i)

∞j |x_j|_∞: 1≤j ≤ro

= supn q^−`

(i)

∞σ(j)|x_σ(j)|_∞: 1≤j≤ro . Therefore

supn

q^−ξ∞^σ(i)ν_Lω,i(x) : 0≤i < ro

= supn

q^−ξ∞^σ(i)·q^−`

(i)

∞σ(j)|x_σ(j)|_∞: 0≤i < r, 1≤j≤ro

= sup

sup

0≤i<r

n

q^{−ξσ(i)−`}

(i)

∞ σ(j)

o· |x_σ(j)|_∞: 1≤j ≤r

= supn

q^−ξ_∞^{σ(j)−`σ(j)+1}|x_σ(j)|_∞: 1≤j≤ro

(forj=r, notice that0 =−ξ₀=−ξ_r+ 1)

= ν_ω(x).

The above description ofν_ω says λ(ω) = X

0≤i<r

i[Lω,i]∈ B^r(R) with i :=ξ_σ(i+1)−ξ_σ(i). Note thatω=u·z where

u=







1 uij

. .. 1





.

For0≤i < rwe take

gz,i:=u⁻¹gω,i and Lz,i:=L^og_z,i⁻¹. (2.4)

Then:

Lemma 2.8. Forz∈H^r with Im(z)i=q_∞<sup>−`i+1−ξi</sup> where`i∈Z and0≤ξi<1, we have

λ(z) =

r−1

X

i=0

_i[L_z,i],

wherei andLz,i are taken as above for0≤i < r.

Define the total imaginary part ofz∈H^r by Im(z) :=

r−1

Y

i=1

Im(z)_i, (2.5)

and put[Im(z)]i :=|det(gz,i)|∞ for0≤i < r. The above lemma implies:

Corollary 2.9. (1) Forz1, z2∈H^r withλ(z1) =λ(z2), we getIm(z1) = Im(z2).

(2)Given s∈C, the following equality holds:

Im(z)^s νz(x)^rs =

r−1

X

i=0

c_z,i(s)· [Im(z)]^s_i

νL_z,i(x)^rs, ∀x∈k^r_∞− {0}, where

cz,i(s) :=q

Pr

j=1(ξσ(i)−ξj)s

∞ · q_∞^ris−1

q_∞^is−q^(i−r)s_∞ , 0≤i < r.

Proof. Note that ν_z = sup{q∞^−ξσ(i)ν_Lz,i : 0 ≤ i < r}. Given x ∈ k^r_∞− {0}, we have that νz(x) =q^−ξ∞^{σ(i0 )}νL_z,i0(x)with minimali0∈ {0, ..., r−1} if and only if

νL_z,j(x) =

(q⁻¹_∞ν_Lz,i

0(x), ifj < i₀, ν_Lz,i

0(x), otherwise.

Then

Im(z)^s

ν_z(x)^rs =q^rξ∞^{σ(i0 )s}· q⁻

Pr

j=1(`_j−1+ξ_j)s

∞

ν_Lz,i

0(x)^rs .

On the other hand, one has

r−1

X

i=0

c_z,i(s) [Im(z)]^s_i ν_Lz,i(x)^rs

= q⁻

Pr

j=1(`_j+ξ_j)s

∞

νL_z,i0(x)^rs ·

"ⁱ0−1

X

i=0

q_∞^rξσ(i)s· q^r_∞^is−1 q^is_∞−q^(i−r)s_∞ · q_∞^is

q∞^−rs

+

^r−1 X

i=i₀

q^rξ_∞^σ(i)s· q_∞^ris−1 q_∞^is−q^(i−r)s_∞ ·q^is_∞

#

= q⁻

Pr

j=1(`_j−1+ξ_j)s

∞

ν_Lz,i

0(x)^rs·(1−q∞^−rs)·

"

ⁱ0−1

X

i=0

q∞^rξσ(i+1)s−q∞^rξσ(i)s

+q^−rs_∞

^r−1 X

i=i0

q∞^rξσ(i+1)s−q∞^rξσ(i)s

#

= q⁻

Pr

j=1(`j−1+ξj)s

∞

νL_z,i0(x)^rs ·q∞^{rξσ(i0 )s}.

Therefore the result holds.

Remark 2.10. For anA-latticeΛ of rankrin C∞, thelattice discriminantofΛ, denoted by DA(Λ), is the “covolume” ofΛ(cf. [32, Section 4]): choose an “orthogonal”k_∞-basis{λi}_1≤i≤r ofk_∞·Λ, i.e.λ₁, ..., λ_r satisfy that

(i) λi∈Λfor1≤i≤r;

(ii) |a1λ1+· · ·+arλr|_∞= max{|aiλi|_∞; 1≤i≤r} for alla1, ..., ar∈k_∞. (iii) k_∞·Λ = Λ + (O_∞λ₁+· · ·O_∞λ_r).

Set

DA(Λ) :=q^1−gk·

Q

1≤i≤r|λi|∞

# Λ∩(O_∞λ1+· · ·+O_∞λr)

!1/r

=

Q

1≤i≤r|λi|∞

# Λ/(Aλ1+· · ·+Aλr)

!1/r

. It is clear thatDA(c·Λ) =|c|∞·DA(Λ)for everyc ∈C^×∞. In particular, for z ∈H^r and a rankrprojective A-moduleY ⊂k^r, we have

DA(Λ^Y_z)^r = kYk ·Im(z).

(2.6)

HerekYk:= #(A^r/aY)· kak^−r for every idealaofA so thataY ⊂A^r. Lemma 2.11. Given z∈H^rand γ∈GLr(k_∞), we have

Im(γ·z) = |detγ|_∞

|j(γ, z)|^r_∞ ·Im(z) Proof. Take γ⁰∈GLr(k)closed enough toγso that

Im(γ⁰·z) = Im(γ·z), |detγ⁰|_∞=|detγ|_∞, and |j(γ⁰, z)|_∞=|j(γ, z)|_∞. The result then follows from the equalities (2.2) and (2.6).

3. “Non-holomorphic” Eisenstein series

We first recall the basic properties of mirabolic Eisenstein series onGL_r(A)to be used.

3.1. Mirabolic Eisenstein series. Let χ : k^×\A^× → C^× be a unitary Hecke character.

Given a Schwartz function ϕ ∈ S(A^r), i.e. the function ϕ on A^r is locally constant and compactly supported, put

Φ(g, s;χ, ϕ) :=|det(g)|^s_A· Z

A^×

ϕ (0, ...,0, a⁻¹)g

χ(a)|a|^−rs_A d^×a, ∀g∈GL_r(A) The Haar measured^×aonA^× is chosen so that vol(O^×

A, d^×a) = 1. It is known that (cf. [22, (4.1)] or [30, p. 119]) the functionΦ(g, s;χ, ϕ)converges absolutely for everyg∈GL_r(A)and