The aim of this paper is to present a function field analogue of the classi- cal Kronecker limit formula

FU-TSUN WEI

Abstract. The aim of this paper is to present a function field analogue of the classi- cal Kronecker limit formula. We first introduce a “non-holomorphic” Eisenstein series on the Drinfeld half plane, and connect its “second term” with Gekeler’s discriminant func- tion. One application is to express the Taguchi height of rank 2 Drinfeld modules with complex multiplication in terms of the logarithmic derivative of the corresponding zeta functions. Moreover, from the integral form of the Rankin-typeL-function associated to two “Drinfeld-type” newforms, we then derive a formula for a non-central special deriv- ative of the L-function in question. Adapting the classical approach, we also obtain a Kronecker-type solution for Pell’s equation over function fields.

1. Introduction

Forz∈Cwith the imaginary part Im(z)>0, recall the non-holomorphic Eisenstein series:

E(z, s) :=

0

X

c,d∈Z

Im(z)^s

|cz+d|^2s, Re(s)>1.

Here⁰ means that cand dare not both zero. ExtendingE(z, s) to a meromorphic function on the complexs-plane, the classical Kronecker limit formula is stated as follows:

(1.1) E(z, s) = π

s−1 +π 2γ−2 ln 2−ln(Im(z)|∆(z)|¹²²)

+O(s−1).

Here γ is the Euler constant and ∆(z) is the modular discriminant function (a weight 12 modular cusp form for SL2(Z)). From the functional equation ofE(z, s):

π^−sΓ(s)E(z, s) =:E(z, s) =e E(z,e 1−s), the equation (1.1) can be reformulated to:

(1.2) E(z,0) =−1 and ∂

∂sE(z, s)

_s=0=−ln(Im(z)|∆(z)|¹²²).

This formula is applied (also by Kronecker) to a “modular” solution of Pell’s equations. To- gether with the Chowla-Selberg formula, the equation (1.2) connects the “Faltings height”

of elliptic curves having complex multiplication with special Γ-values (cf. Colmez [2], Sec- tion 0.6). Moreover, by Rankin-Selberg method, E(z, s) shows up in the integral form of the Rankin-typeL-functionL(f1×f2, s) associated to two normalized Hecke eigenformsf1

andf₂ with the same weight for SL₂(Z). Via the equation (1.2), the non-central derivative

∂

∂sL(f1×f2, s)

_s=0 can be expressed as the Petersson inner producthf1·ln(Im(·)|∆|¹²²), f2i.

In this paper, we shall present a precise analogue of the Kronecker limit formula in the func- tion field context, and explore its various arithmetic applications.

Letkbe an arbitrary global function field with finite constant fieldFq. Fix a place∞of k, regarded as the place at infinity, and the corresponding absolute value is denoted by| · |_∞.

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

Key words and phrases. Function field, Kronecker limit formula, Eisenstein series, Drinfeld modules.

This work was supported by the grants from Ministry of Science and Technology, Taiwan.

1

Letk_∞ be the completion of k at ∞ and C∞ be the completion of an algebraic closure of k_∞. We setH:=C∞−k_∞, called the Drinfeld half plane. For eachz∈H, the “imaginary part” ofzis the distance betweenzandk_∞:

|z|i:= inf

α∈k∞

|z−α|∞.

LetA be the subring ofkconsisting of the functions regular away from ∞. For each (frac- tional) ideala ofA, we introduce the following “non-holomorphic” Eisenstein series

E^a(z, s) :=

0

X

c∈A,d∈a

|z|^s_i

|cz+d|^2s_∞, Re(s)>1.

Our main theorem is stated in the following (cf. Theorem 3.1):

Theorem 1.1. (1)E^a(z, s)has meromorphic continuation to the wholes-plane.

(2) Let eE^a(z, s) be the modified Eisenstein series defined in Theorem 3.1 (2). We have the functional equation

Ee^a(z, s) =Ee^a

−1(z⁻¹,1−s).

(3)(Kronecker limit formula) E^a(z,0) =−1 and ∂

∂sE^a(z, s)

_s=0 =−ln |z|i· |∆^a(z)|

2 q2 deg∞ −1

∞

.

Here∆^a(z)is Gekeler’s discriminant function (cf. Section 2.4), which is a Drinfeld modular form of weightq^{2 deg∞}−1forGL(Λ^a)whereΛ^a :=A⊕a⊂k_∞² .

We remark that Theorem 1.1 (3) provides a geometric interpretation of the special deriva- tive of the non-holomorphic Eisenstein series in question. Indeed, according to the theory of Drinfeld, the lattice Λ^a(z) :=Az+a⊂C∞forz∈Hcorresponds to a unique rank 2 Drinfeld A-moduleφ^a,z overC∞ (see Section 2.3). Givenz0 ∈ H, suppose φ^a,z0 is isomorphic (over C∞) to a DrinfeldA-moduleφe^a,z0 defined over the algebraic closurekofkin C∞. Then the discriminant ∆^a(z0) of φ^a,z0 is in fact an algebraic multiple of a “period” of φe^a,z0 raised to the (q^{2 deg∞}−1)-th power (cf. Chang [3]), which is transcendental overk by Yu [23].

Our formula is closely parallel to the classical one, however, the approach is quite differ- ent. Note that E^a(z, s) is a C-valued function but ∆^a(z) lies in the positive characteristic world. Via the building map from the Drinfeld half plane H to the Bruhat-Tits tree as- sociated to GL₂(k_∞), we first connect E^a(z, s) with “adelic” Eisenstein series on GL₂ (cf.

Proposition 3.4). Since the analytic behavior of adelic Eisenstein series is well-understood, the meromorphic continuation and the functional equation ofE^a(z, s) are then easily verified.

To prove Theorem 1.1 (3), we introduce the following Eisenstein series of “Jacobi-type”:

E^a(z, w, s) := X

λ∈Λ^a(z)

|z|^s_i

|λ−w|^2s_∞, ∀z∈Handw∈C∞−Λ^a(z), which satisfies

E^a

az+b cz+d, w

cz+d, s

=E^a(z, w, s), ∀ a b

c d

∈GL(Λ^a).

The non-archimedean property of|·|∞implies thatE^a(z, s)−E^a(z, w, s) is entire with respect to the variables, which gives the meromorphic continuation of E^a(z, w, s). Our Kronecker limit formula then follows from a product expansion of Gekeler’s discriminant function in Lemma 2.3 and the equality below (cf. Section 3.2)

(|a|^2s_∞−1)E^a(z, s) = X

06=w∈¹_aΛ^a(z)/Λ^a(z)

E^a(z, w, s), ∀a∈A−Fq.

There have been other analogues of the Kronecker limit formula for different types of Eisenstein series over function fields. Gekeler [7] first observed that, when the base fieldk is rational, thevan der Put logarithmic derivatives r(∆^a) of ∆^a is related to a conditionally convergent complex-valued Eisenstein series on the Bruhat-Tits tree. In [15], Kondo studies the Jacobi-type Eisenstein series of “arbitrary rank,” and connects the special derivative of this series with the discriminant function multiplied with the Drinfeld exponential function of the givenA-lattice. His formula is a general version of (3.1) forE^a(z, w, s) in Remark 3.5 (2) for arbitrary rank case. Kondo’s argument in [15] is to compare the Fourier coefficients of both sides directly. Thus our limit formula gives an alternative approach to Kondo’s result for the case of rank 2. In [16, Section 4], P´al introduces another type of Eisenstein series, and connects the values of these series ats= 0 with the van der Put logarithmic derivatives of a special family of invertible holomorphic functions onH(cf. [16, Kronecker limit formula 4.8]). One application of P´al’s formula in [16] is to determine the Fourier coefficients ofr(∆^a) (cf. [16, Proposition 5.8]). From our limit formula in Theorem 1.1 (3), we may expressr(∆^a) explicitly in terms of the derivative of an adelic Eisenstein series ats = 0 (cf. Remark 3.5 (4)). This generalizes Gekeler’s formula in [7, 2.8 COROLLARY] to arbitrary global function fields. Consequently, the standard results on the Whittaker functions of adelic Eisenstein series then provides an alternative way to calculate the Fourier coefficients ofr(∆^a).

One application of Theorem 1.1 is to derive a Colmez-type formula for the “Taguchi height”

of rank 2 DrinfeldA-modules with “complex multiplication.” This height (introduced in [21]

for the case when A is a polynomial ring) is viewed as a natural analogue of the Faltings height of abelian varieties over number fields. LetK/k be a quadratic field extension which is “imaginary” (i.e.K/k is separable and ∞ does not split in K) and O_K be the integral closure ofAinK. For each integral idealcofA, letO_c:=A+cO_K be the quadraticA-order of conductorc. Applying our Kronecker limit formula, we obtain that (cf. Corollary 4.5):

Corollary 1.2. For every rank2 DrinfeldA-moduleφoverkwith complex multiplication by Oc, the “Taguchi height” ofφis equal to

ehTag(φ) =−1

4lnkd(Oc/A)k − 1 2· ζ_O⁰

c(0) ζO_c(0). Herekak:= #(A/a)for every integral ideal a of A,

d(Oc/A) :=c²· Y

primepCA pis ramified inK

p, and ζO_c(s) := X

invertible ideal A⊂Oc

#(Oc/A)^−s.

Note thatd(Oc/A) is the discriminant ideal forOc/Awhenqis odd. In particular, suppose kis a rational function fieldFq(t) withqodd andA=Fq[t]. LetD∈A−Fq be a square-free polynomial such thatK=k(√

D) is imaginary. We have ζ_O⁰

K(s)

ζO_K(s) =ζ_A⁰ (s)

ζA(s)+L⁰_A(s, ^D_· ) LA(s, ^D_·

) where ^D_·

is the Legendre quadratic symbol and LA(s,

D

·

) := X

monicm∈A

D m

q^−degms.

Then for every rank 2 DrinfeldA-moduleφover kwith complex multiplication by OK, the Taguchi heightehTag(φ) can be written as follows (cf.Remark 4.6 (2)):

ehTag(φ) = 1

4ln|D|_∞− qlnq

2(q−1) + 1 2# Pic(O_K)

X

monicm∈A degm<degD

D m

·ln

m D ∞.

By Ihara’s estimation of the Euler-Kronecker constant of K in [13, upper bound (0.6) and lower bound(0.12)], we thereby obtain the following asymptotic formula:

eh_Tag(φ) = 1

4ln|D|_∞+O(ln ln|D|_∞) for|D|_∞0.

Another application of Theorem 1.1 is on special values of Rankin-typeL-functions asso- ciated to “Drinfeld-type” automorphic forms. These forms, viewed as analogue of classical weight 2 modular forms, are very useful tools in function field arithmetic (cf. [4], [8], and [22]).

We refer the readers to Section 5.1 for the precise definitions and the analytic properties to be used. Let f₁ andf₂ be two normalized Drinfeld-type newforms of square-free levels N₁ andN2, respectively. The Rankin-Selberg method enables us to express the Rankin-typeL- functionL(f1×f2, s) as a Petersson inner producthf1EN₁,N₂, f2i, where the functionEN₁,N₂

comes from our non-holomorphic Eisenstein series (see Section 5.2). Therefore, the Kronecker limit formula in Theorem 1.1 (3) leads us to (cf. Theorem 5.5):

Corollary 1.3. Given two normalized Drinfeld-type newforms f1 and f2 with square-free levelsN₁ andN₂, respectively, letΛ(f₁×f₂, s)be the Rankin-typeL-function modified as in Theorem 5.5. Then

Λ(f₁×f₂,0) =−hf₁, f₂i and ∂

∂sΛ(f₁×f₂, s)

_s=0=hf₁·η_N1,N2, f₂i.

HereηN₁,N₂ is an automorphic form on GL2 induced from Gekeler’s discriminant function.

Following the classical story, we are also able to derive a function field version of Kronecker’s solution of Pell’s equations. Let L/k be a quadratic field extension which is “real” (i.e.

∞ splits in L). It is always possible to take an imaginary quadratic field K/k such that LK is a subfield of the Hilbert class field HO_K of OK. We denote the integral closure of A in L by OL. Let ξL : Pic(OK) → {±1} be the quadratic character associated to LK (i.e. kerξ_L ∼= Gal(H_OK/LK) via the Artin map). By a Shimura-type “reciprocity law” for Gekeler’s discriminant function (cf. Proposition 6.3), we observe that

uL:= Y

[A]∈Pic(OK)





∆(A)∆(A⁻¹)

∆(O_K)²

^ξL(A)

· ∆(A)∆(A⁻¹)

∆(O_K)²

!^ξL(A)

∈O_L^×.

Here· is the non-trivial automorphism ofK/k, and ∆(A) := α^{1−q2 deg∞}∆^a(z) when writing Aas the formα(Az+a) for an idealaofA,z∈H, andα∈C^×∞. Moreover, Theorem 1.1 (3) tells us that:

Corollary 1.4. u_L is not a root of unity. Moreover,

# O^×_L

< u_L>

=# Pic(OL)·# Pic(OK⁰)

# Pic(A)² · 2·#(O^×_K)·#(F^×q)²·(q_∞² −1)

#(O^×_K0) . HereK⁰/k is the imaginary quadratic subfield of LK/k different from K.

We point out that ifaO_K is a principal ideal ofO_K for every idealaofA, then

˜

uL:= Y

[A]∈Pic(OK)

∆(A)∆(A⁻¹)

∆(O_K)²

^ξL(A)

∈O^×_L

and uL = ˜u²_L. In particular, suppose k = Fq(t) with q odd and A = Fq[t]. Take a monic square-free polynomial D ∈ A with even degree and let L := k(√

D). We can choose K =k(√

D) where ∈ F^×q is a non-square element. Writing ˜uL as ˜uL =a+b√

D where a, b∈A, we then obtain a non-trivial solution (a, b) for the Pell’s equationX²−DY²= 1.

This paper is organized as follows. In Section 2, we set up notations and recall the needed properties of rank 2 Drinfeld modules, including Gekeler’s discriminant function and the building map on the Drinfeld half plane. Our analogue of the Kronecker limit formula is derived in Section 3. We connectE^a(z, s) with adelic Eisensteins series on GL₂in Section 3.1 and prove Theorem 1.1 in Section 3.2. In Section 4, we first introduce the Taguchi height of DrinfeldA-modules for generalA, and then express the height of rank 2 Drinfeld A-modules having complex multiplication byO_cas the logarithmic derivative of the zeta functionζ_Oc(s).

In Section 5, we study the non-central special value of the Rankin-typeL-functions associated to Drinfeld type automorphic forms. Basic properties of Drinfeld type forms are recalled in Section 5.1, and Corollary 1.3 is shown in Section 5.2. In Section 6, we apply Theorem 1.1 to get a Kronecker-type solution of Pell’s equations over function fields. Verifying a Shimura-type reciprocity law for Gekeler’s discriminant function, Corollary 1.4 is carried out in Section 6.2.

Acknowledgements. The author is deeply grateful to Jing Yu for his steady interest and encouragements, and to Ming-Lun Hsieh for his useful comments. He wish to thank the referee for valuable suggestions, and to Chih-Yun Chuang for helpful discussions on analytic number theory. This work is started during the Japan-Taiwan Joint conference on Number Theory 2014 in Kesennuma Oshima island, sponsored by Research Institute for Mathematical Sciences. The author would also like to thank the organizers for the invitation.

2. Preliminaries

2.1. Basic settings. LetFq be 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 andF^q is its algebraic closure 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, the residue field atv, andqv to be the cardinality of 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 ∈kv. LetA:=Q0

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

vO_v, the maximal compact subring ofA. For each elementα= (αv)v in the idele group A^×, the norm|α|_A is defined to be

|α|_A:=Y

v

|αv|v.

We can embed k (resp. k^×) into A (resp. A^×) diagonally, and have the product formula:

|α|_A = 1 for all α ∈ k^×. Throughout this paper, we fix a non-trivial additive character ψ= ⊗_vψ_v :A →C^× which is trivial on k (here ψ_v(a_v) :=ψ(0, ...,0, a_v,0, ...), for all a_v in kv). For each place v of k, letδv be the “conductor of ψ at v,” i.e. the maximal integer r such thatπ^−r_v Ov is contained in the kernel ofψv. It is known thatP

vδvdegv = 2gk−2, wheregk is the genus ofk. We callδ= (π^δ_v^v)v∈A^× a differential idele ofkassociated to ψ.

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

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

v6=∞O_v. 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) ideal a ofA, we denote by aCA ifa is an integral ideal. In this paper, every ideal is assumed to be non-zero. For each idealIofA, writingI=a⁻¹bwherea,bCAwe set

kIk:= #(A/b)

#(A/a).

In particular,kIk=|α|∞ whenI=αAforα∈k^×. Given two idealsa₁,a₂ ofA, let [a1,a2] :=a1∩a2 and (a1,a2) :=a1+a2.

Finally, we putπ^a:= (πv^ordv(a))v6=∞∈A^∞,× for each ideal aofA.

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 homomorphismφ:A→F{τ} satisfying that

φa=ι(a) +

rdega

X

i=1

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

(2) Given two DrinfeldA-modulesφ₁ andφ₂ overF, a homomorphism f :φ→φ⁰ over F is an element in F{τ} satisfyingf ·φa =φ⁰_a·f for everya∈A. f is called an isogeny if f is not zero. We denote the set of homomorphisms fromφto φ⁰ (resp. endomorphism ring of φ) overF by Hom_A(φ/F, φ⁰/F) (resp. End_A(φ/F)).

2.3. Drinfeld half plane. LetC∞ be the completion of a chosen algebraic closure of k_∞, and considerC∞as anA-field via the natural embeddingA ,→C∞. Given a rankrDrinfeld A-moduleφoverC∞. There exists a uniqueFq-linear entire function exp_φ onC∞ satisfying

exp_φ(aw) =φa exp_φ(w)

, ∀a∈A andw∈C∞.

It is known that Λφ:={λ∈C∞: exp_φ(λ) = 0} is a discrete projectiveA-submodule of rank r in C∞ (i.e. a rank r A-lattice in C∞). We call Λ_φ theA-lattice associated toφ. On the other hand, given a rankr A-lattice Λ⊂C∞, set

exp_Λ(w) :=w Y

06=λ∈Λ

1−w λ

.

This uniquely determines a rankrDrinfeldA-moduleφ^Λ overC∞ satisfying that (2.1) exp_Λ(aw) =φ^Λ_a(exp_Λ(w)), ∀a∈Aandw∈C∞.

In other words, we have the following bijection (cf. [5, Proposition 3.1])

{rankrDrinfeldA-modules overC∞} ∼={rankr A-lattices inC∞}.

For our purpose, we now focus on the case whenr= 2 and recall the analytic description of the moduli space for rank 2 Drinfeld A-modules over C∞. Let H := C∞−k∞, called the Drinfeld half plane. We have an action of GL2(k_∞) on H defined by fractional linear transformations:

a b c d

·z:=az+b

cz+d, ∀z∈Hand a b

c d

∈GL2(k∞).

Take an idealaofAand set Λ^a:=A⊕a⊂k². Forz∈H, let Λ^a(z) :=Az+a⊂C∞and φ^a,z denotes the rank 2 DrinfeldA-module overC∞ corresponding to Λ^a(z).

Theorem 2.2. (cf. [8, Section 2.5])The map (a, z)7→φ^a,z induces the following bijection



 a

[a]∈Pic(A)

GL(Λ^a)\H



←→ {rank-2 DrinfeldA-modules overC∞}/∼=.

Here

GL(Λ^a) :=

a b c d

∈GL₂(k_∞)

a, d∈A, b∈a, c∈a⁻¹, ad−bc∈F^×q

. 2.4. Gekeler’s discriminant function. Given an idealaofAanda∈A, let ∆^a_a(z) be the analytic function onHsatisfying

φ^a,z_a (x) =ax+· · ·+ ∆^a_a(z)x^{q2 dega}, ∀x∈C∞.

In other words, ∆^a_a(z) =l2 dega(φ^a,z_a ). It is observed that (cf. [6, Chapter V, 3.4 Example])

∆^a_a(γz) = (cz+d)^{q2 dega−1}·∆^a_a(z), ∀γ= ∗ ∗

c d

∈GL(Λ^a).

Moreover, the equation (2.1) implies

φ^a,z_a (x) = ∆^a_a(z)·x· Y

06=w∈_a¹Λ^a(z)/Λ^a(z)

x−exp_Λa(z)(w) .

Therefore we have:

Lemma 2.3. For everya∈A,

∆^a_a(z) = (−1)^{q2 dega−1}·a· Y

06=w∈¹_aΛ^a(z)/Λ^a(z)

exp_Λa(z)(w)⁻¹. Sinceφ^a,z_a ·φ^a,z_b =φ^a,z_ab =φ^a,z_b ·φ^a,z_a fora, b∈A, one gets

(2.2) ∆^a_a(z) ∆^a_b(z)^{q2 dega}

= ∆^a_ab(z) = ∆^a_b(z) ∆^a_a(z)^{q2 degb} . Given α=P

i≥ord∞(α)uiπ_∞ⁱ ∈ k^×_∞ where ui ∈ F∞, we callα monic ifu_ord∞(α) = 1. Take two monic elements a, b ∈ A such that gcd(ord_∞(a),ord_∞(b)) = 1, and choose `1, `2 ∈ Z such that`1 q∞^{2 ord∞(a)}−1

+`2 q∞^{2 ord∞(b)}−1

=q²_∞−1. Gekeler [6] introduces the following discriminant function

∆^a(z) := ∆^a_a(z)^{`1</sup>∆<sup>a</sup><sub>b</sub>(z)<sup>`2}, which is a nowhere-zero analytic function onHsatisfying that

∆^a(γz) = (cz+d)^q∞2−1·∆^a(z), ∀γ= ∗ ∗

c d

∈GL(Λ^a).

Proposition 2.4. (cf. [6, Chapter IV, Proposition 5.15]) ∆^a is independent of the chosen monica, b∈Aand`1, `2∈Z. In particular,(∆^a)^{q2 dega−1}= (∆^a_a)^q∞2−1 for everya∈A− {0}.

2.5. Building map. LetT be the Bruhat-Tits tree associated to PGL2(k_∞). The setV(T) of vertices of T is the collection of homothety classes of rank-2O_∞-lattices in k_∞² , and the set E(T~ ) (resp.E(T)) of (non-)oriented edges of T consists of (non-)order pairs ([L],[L⁰]) where [L], [L⁰]∈V(T) with π_∞L⁰ (L(L⁰. It is known that the realization T(R) of T is identified with the equivalent classes of norms onk_∞² as follows: suppose P ∈ T(R) belongs to the edge ([L],[L⁰]) withπL⁰(L(L⁰, sayP = (1−)[L] +[L⁰], 0≤ <1, then the norm ν_P onk_∞² corresponding toP is

νP(x) := sup{νL(x), q_∞ νL⁰(x)} with νL(x) := inf{|a|_∞:a∈k_∞, x∈aL}.

Definition 2.5. Thebuilding mapλ:H→ T(Q) is defined by z∈H7−→ν_z:= (c, d)∈k_∞² 7→ |cz+d|∞

.

The right action of GL₂(k_∞) on k_∞² yields a left action on the set of norms on k²_∞ and then onT(R).

Proposition 2.6. (cf. [8, Proposition 1.5.3])The building map λisGL₂(k_∞)-equivariant.

Givenz∈H, theimaginary part of zis defined by

|z|i:= sup{|z−u|_∞:u∈k_∞}.

It is observed that (cf. [8, (1.1.5) Lemma])

|γz|i= |detγ|_∞

|cz+d|²_∞ · |z|i, ∀γ= ∗ ∗

c d

∈GL2(k_∞).

Via the left action of GL2(k_∞) on T(R), V(T) is identified with GL2(k_∞)/k^×_∞GL2(O_∞).

More precisely, letL₀:=O_∞² ⊂k²_∞. Then

V(T) ={[L0g⁻¹] :g∈GL2(k_∞)/k^×_∞GL2(O_∞)}.

Corollary 2.7. (cf. [7, 1.8 Lemma]) Given z ∈ H with |z|i =|z−u|_∞ = q_∞^−r+ for some u∈k_∞,r∈Z, and0≤ <1, we haveλ(z) =ν_Pz where

Pz= (1−)[L0g⁻¹_z ] +[L0˜g_z⁻¹]∈ T(Q), with gz=

π_∞^r u

0 1

andg˜z=

π^r−1_∞ u

0 1

.

3. Kronecker limit formula

Leta be an ideal ofA. For z ∈ Hand s ∈ C with Re(s) 0, we are interested in the following “non-holomorphic” Eisenstein series

E^a(z, s) :=

0

X

c∈A,d∈a

|z|^s_i

|cz+d|^2s_∞.

Here ⁰ means that c and dare not both zero. It is clear that E^a(γz, s) =E^a(z, s) for each γ∈GL(Λ^a). The main theorem of this section is stated below.

Theorem 3.1. (1)E^a(z, s)converges absolutely forRe(s)>1 and has meromorphic contin- uation to the wholes-plane.

(2) Suppose |z|i = |z −u|_∞ = q^−r+_∞ with u ∈ k_∞, r ∈ Z, and 0 ≤ < 1. We let

˜

z:=π¹⁻²_∞ (z−u) +u∈C∞ (which says |˜z|i =|˜z−u|_∞=q_∞^−r+1−), and

Ee^a(z, s) :=kak^s·











q^{(2gk−2+deg∞)s} q_∞^s −q^−s∞

·E^a(z, s) if = 0,

q^{(2gk−2+deg∞)s} q_∞^s+q∞^(1−)s−q∞^−s−q^(−1)s∞

· E^a(z, s) +E^a(˜z, s)

if 0< <1.

We have the following functional equation Ee^a(z, s) =Ee^a

−1(z⁻¹,1−s).

(3)(Kronecker limit formula) For everyz∈H,E^a(z,0) =−1and

∂

∂sE^a(z, s)

_s=0=−ln |z|i· |∆^a(z)|

2 q2

∞ −1

∞

.

We first connectE^a(z, s) with “adelic” Eisenstein series in Section 3.1, and give the proof of Theorem 3.1 in Section 3.2.

3.1. Adelic Eisenstein series. Let χ be a character on Pic(A) ∼= k^×\A^×/k^×_∞O^×_A. The principal series I(s, χ) consists of smooth (i.e. locally constant) functions Φ : GL2(A)→ C satisfying that fora, b∈A^× andg∈GL₂(A),

Φ

a ∗ 0 b

g

=χ(b)|a|^s_A|b|^−s_A Φ(g).

Let Φ⁰_χ(·, s)∈I(s, χ) be the function defined by Φ⁰_χ

a ∗ 0 b

κ, s

:=χ(b)|a|^s_A|b|^−s_A , ∀a, b∈A^× andκ∈GL2(O_A).

Theadelic Eisenstein series associated to Φ⁰_χ is E(g, s, χ) := X

γ∈B(k)\GL₂(k)

Φ⁰_χ(γg, s), ∀g∈GL₂(A).

Let

LA(s, χ) :=X

aCA

χ(a)

kak^s, Re(s)>1.

It is known thatLA(s, χ) has meromorphic continuation to the whole s-plane and satisfies the following functional equation (cf. [17, 7-19 Theorem])

L(s, χ) =e χ(δ)eL(1−s, χ⁻¹),

whereδis a (any) differential idele ofk,L(s, χ) :=e q^(gk−1)s(1−q_∞^−s)⁻¹L_A(s, χ) andg_k is the genus ofk. In particular, LA(s, χ) is holomorphic ats = 0 and vanishes if and only ifχ is non-trivial. We recall the analytic behavior ofE(g, s, χ) as follows.

Theorem 3.2. (cf. [1, Theorem 3.7.1 and 3.7.2, Proposition 3.7.2 and 3.7.5])

(1) E(g, s, χ) converges absolutely for Re(s) > 1 and has meromorphic continuation to the wholes-plane with the following functional equation

E(g, s, χ) =e χ(detg)E(g,e 1−s, χ⁻¹), whereE(g, s, χ) :=e L(2s, χe ⁻¹)E(g, s, χ).

(2) For everyg∈GL₂(A),E(g, s, χ)is holomorphic ats= 0. In particular,E(g, s, χ) = 1 ifχ is trivial.

The expression ofE(g, s, χ) below will be used later.

Lemma 3.3. Given an ideal a of A, r∈ Z, and u∈k_∞, letg = (g^∞_a , g_∞) ∈GL₂(A^∞)× GL2(k∞) = GL2(A)where

g^∞_a =

1 0 0 π^a−1

and g_∞=

π^r_∞ u

0 1

.

We have

E(g, s, χ) = kak^s q−1

X

[n]∈Pic(A)

χ(n⁻¹)·





 1 knk^2s

0

X

c∈n−1,d∈n−1a ca+dA=n−1a

|π^r_∞|^s_∞

max(|cπ_∞^r |_∞,|cu+d|_∞)^2s





.

Proof. Without loss of generality, assume aCA. It is observed that we have the following two bijections (cf. [19, Proposition 10 in§2.3]):

Pic(A) ∼= B(k)\GL₂(k)/GL(Λ^a) [n] 7−→ γn:=

0 1 xn 1

wherexn∈k^× such thatxn·A= _n^m0 withm,n⁰CA, (m,n⁰) =A, and [n⁰,a] =n; and γ⁻¹_n B(k)γn∩GL(Λ^a)

GL(Λ^a) ∼= {(0,0)6= (c, d)∈n⁻¹×n⁻¹a:ca+dA=n⁻¹a}/F^×q

a b α β

∈GL(Λ^a) 7−→ (axn+α, bxn+β).

HereF^×q acts diagonally on n⁻¹×n⁻¹aby multiplication. We then obtain that E(g, s, χ) = kak^s· X

[n]∈Pic(A)

χ(n⁻¹) knk^2s ·





X

γ∈γn⁻¹B(k)γ_n∩GL(Λ^a)\GL(Λ^a)

Φ⁰_χ

(1, γnγ

π^r_∞ u

0 1

), s





= kak^s

q−1· X

[n]∈Pic(A)

χ(n⁻¹)·





 1 knk^2s

0

X

c∈n−1,d∈n−1a ca+dA=n−1a

|π^r_∞|^s_∞

max(|cπ_∞^r |_∞,|cu+d|_∞)^2s





.

LetPic(A) be the group of characters on Pic(A) and set\

E(g, s) := (q−1)

# Pic(A) X

χ∈Pic(A)\

LA(2s, χ⁻¹)E(g, s, χ).

The connection betweenE^a(z, s) and the adelic Eisenstein series E(g, s) is described in the following.

Proposition 3.4. For each ideal a of A and z ∈Hwith |z|i =|z−u|∞=q^−r+_∞ for some u∈k_∞,r∈Z, and0≤ <1, we have

kak^sE^a(z, s) = q^(−1)s_∞ −q_∞^(1−)s

q^−s∞ −q^s_∞ E(g_a,z, s) +q_∞^−s−q^s_∞

q^−s∞ −q_∞^s E(g_a,z⁰ , s), wherega,z andg_a,z⁰ are in GL2(A^∞)×GL2(k∞) = GL2(A)defined by

g_a,z =

1 0 0 π^a−1

,

π_∞^r u

0 1

, g_a,z⁰ =

1 0 0 π^a−1

,

π^r−1_∞ u

0 1

.

Proof. First, we writeE^a(z, s) as E^a(z, s) = X

06=cCA 0

X

c∈c,d∈ca ca+dA=ca

|z|^s_i

|cz+d|^2s_∞

= X

[n]∈Pic(A)

ζ[n⁻¹](2s)·





 1 knk^2s

0

X

c∈n−1,d∈n−1a ca+dA=n−1a

|z|^s_i

|cz+d|^2s_∞





,

= X

[n]∈Pic(A)

ζ[n⁻¹](2s)·





 1 knk^2s

0

X

c∈n−1,d∈n−1a ca+dA=n−1a

|z|^s_i

max{|cz|i,|cu+d|_∞}^2s





,

whereζ_[n](s) is the partial zeta functionP

06=cCA,c∈[n]kck^−s. Since ζ_[n⁻¹_](s) = 1

# Pic(A) X

χ∈Pic(A)\

χ(n⁻¹)LA(s, χ⁻¹),

By Lemma 3.3, the result is then straightforward.

3.2. Proof of Theorem 3.1. Given z ∈ H with |z|i = |z−u|_∞ =q^−r+ where u ∈ k_∞, r∈Z, and 0≤ <1, Proposition 3.4 shows that

eE^a(z, s) = (q−1)

# Pic(A)·









 X

χ∈Pic(A)\

E(ge a,z, s, χ), if= 0, X

χ∈Pic(A)\

E(ge a,z, s, χ) +Ee(g⁰_a,z, s, χ), if 0< <1.

On the other hand, ka⁻¹k^s·E^a

−1(z⁻¹, s) = ka⁻¹k^s· X

c∈A,d∈a⁻¹

|z|^s_i

|c+dz|^2s_∞

= X

[n]∈Pic(A)

ζ_[n−1a](2s)· kak^s

knk^2s · X

c∈n−1a,d∈n−1 cA+da=n−1a

|z|^s_i

|c+dz|^2s_∞.

= (q−1)

# Pic(A) X

χ∈Pic(A)\

χ(a⁻¹)LA(2s, χ)

· q^(−1)s∞ −q∞^(1−)s

q^−s∞ −q_∞^s E(g_a,z, s, χ⁻¹) +q^−s_∞ −q^s_∞

q∞^−s−q_∞^s E(g_a,z⁰ , s, χ⁻¹)

! ,

which implies that

Ee^a−1(z⁻¹, s) = (q−1)

# Pic(A)·









 X

χ∈Pic(A)\

χ(a⁻¹)·Ee(g_a,z, s, χ⁻¹), if= 0, X

χ∈Pic(A)\

χ(a⁻¹)·

Ee(ga,z, s, χ⁻¹) +E(ge _a,z⁰ , s, χ⁻¹)

, if 0< <1.

Thus the meromorphic continuation and functional equation of E^a(z, s) follow from Theo- rem 3.2 (1). Note that forχ ∈Pic(A),\ LA(0, χ) =−^{# Pic(A)}_q−1 ifχ is trivial and 0 otherwise.

Hence by Theorem 3.2 (2) and Proposition 3.4 we obtain thatE^a(z,0) =−1 for everyz∈H.

Now, forw∈C∞−Λ^a(z) we consider the following Eisenstein series of “Jacobi-type”:

E^a(z, w, s) := X

λ∈Λ^a(z)

|z|^s_i

|λ−w|^2s_∞,

which satisfies E^a

az+b cz+d, w

cz+d, s

=E^a(z, w, s), ∀ a b

c d

∈GL(Λ^a).

Then

E^a(z, w, s)−E^a(z, s) = |z|^s_i

|w|^2s_∞ + X

06=λ∈Λa(z)

|λ|∞≤|w|∞

|z|^s_i

|λ−w|^2s_∞ − |z|^s_i

|λ|^2s_∞

is entire (with respect to the variable s), which gives the meromorphic continuation of E^a(z, w, s). Moreover, E^a(z, w,0) = 0 and

∂

∂sE^a(z, w, s)

_s=0 = ∂

∂sE^a(z, s)

_s=0+ ln|z|i−2





ln|w|_∞+ X

06=λ∈Λa(z)

|λ|∞≤|w|∞

ln|1−w/λ|_∞







= ∂

∂sE^a(z, s)

_s=0+ ln|z|i−2 ln|exp_Λa(z)(w)|∞. Takinga∈A−Fq, it is clear that

(1− |a|^−2s_∞ )E^a(z, s) =|a|^−2s_∞ · X

06=w∈¹_aΛ^a(z)/Λ^a(z)

E^a(z, w, s).

Therefore

2 ln|a|∞·E^a(z,0) = X

06=w∈¹_aΛ^a(z)/Λ^a(z)

∂

∂sE^a(z, w, s) _s=0

= (|a|²_∞−1) ∂

∂sE^a(z, s)

_s=0+ ln|z|i

−2 X

06=w∈¹_aΛ^a(z)/Λ^a(z)

ln|exp_Λa(z)(w)|_∞. By Lemma 2.3, we then obtain that

∂

∂sE^a(z, s)

_s=0 = −



ln|z|i+ (|a|²_∞−1)⁻¹



2 ln|a|_∞−2 X

06=w∈¹_aΛ^a(z)/Λ^a(z)

ln|exp_Λa(z)(w)|_∞









= −ln |z|i· |∆^a_a(z)|

2 q2 dega−1

∞

= −ln |z|i· |∆^a(z)|

2 q2

∞ −1

∞

.

Remark 3.5. (1) Different from the classical case, the residue of E^a(z, s) at s = 1 depends upon the imaginary part ofzby Proposition 3.4.

(2) For everyz∈Handw∈C∞−Λ^a(z), we have

∂

∂sE^a(z, w, s)

_s=0 = − 2

q_∞² −1ln|∆^a(z)|_∞−2 ln|exp_Λa(z)(w)|_∞. (3.1)

Whena=A, this coincides with Kondo’s formula in [15, Theorem 1] in the case of rank 2.

(3) Given an arbitrary rank 2A-lattice Λ ⊂C∞, take α∈C∞, z∈H, andaCA such that Λ =α(Az+a). Define

E(Λ, s) := X

06=λ∈Λ

|λ|^−2s_∞ =|z|^−s_i |α|^−2s_∞ E^a(z, s),

which also has meromorphic continuation andE(Λ,0) =−1. On the other hand, letφ^Λbe the rank 2 DrinfeldA-module overC∞associated to Λ. Writingφ^Λ_a(x) =ax+· · ·+ ∆_a(Λ)x^{q2 dega} for eacha∈A, it is observed that

∆a(Λ) =α^{−q2 dega+1}∆^a_a(z).

Let ∆(Λ) := ∆^a(z)/α^q∞2−1. Then Theorem 3.1 (3) implies:

(3.2) ∂

∂sE(Λ, s)

_s=0=− 2

q_∞² −1ln|∆(Λ)|_∞.

(4) Let O(H)^× be the group of invertible holomorphic functions on H and H(T,Z) be the group of Z-valued harmonic cochains on T. Recall the van der Put logarithmic derivative r :O(H)^× → H(T,Z) defined by (cf. [8, (1.7.5) ]): for f ∈ O(H)^× and e∈ E(T~ ) with its origino(e) and terminust(e),

r(f)(e) := log_q∞ sup

z∈λ⁻¹(t(e))

|f(z)|∞

!

−log_q∞ sup

z∈λ⁻¹(o(e))

|f(z)|∞

! .

Here λ : H → T(Q) is the building map introduced in Section 2.5. Note that for each v∈V(T),|∆^a(z)|∞ remains the same whenz varies inλ⁻¹(v). Therefore our limit formula says

r(∆^a)(e) = q²_∞−1 2 lnq_∞ · ∂

∂s

E^a(z1, s)−E^a(z2, s)

s=0

−q_∞² −1

2 ·sgn(e),

for everyz₁∈λ⁻¹(o(e)),z₂∈λ⁻¹(t(e)), where sgn(e) =

(1, ifepoints to the end∞ofT,

−1, otherwise.

For each v ∈ V(T), take gv ∈GL2(k_∞) so that v = [L0g_v⁻¹] where L0 = O²_∞ ⊂k_∞² is the standardO_∞-lattice (cf. Section 2.5). Let

g_a,v^∗ :=

π^a 0 0 1

, gv

∈GL2(A^∞)×GL2(k_∞) = GL2(A).

By Proposition 3.4 and Theorem 3.1 (2), we may write r(∆^a)(e) =q^{2gk−2+2 deg∞}·

E(g^∗_a,o(e), s)− E(g_a,t(e)^∗ , s)

s=1−q_∞² −1

2 ·sgn(e).

Here E(g, s) is the adelic Eisenstein series defined in the above of Proposition 3.4. This generalizes Gekeler’s formula in [7, 2.8 COROLLARY] to arbitrary global function fields.

Via the standard results on the Whittaker functions of adelic Eisenstein series (cf. [1, the proof of Theorem 3.7.1]), the Fourier expansion ofr(∆^a) can be understood accordingly.

4. Application I: Taguchi height of Drinfeld modules

LetF be a finite extension of k (viewing as anA-field viaA ,→ F) andφ be a Drinfeld A-module of rankroverF. Recall that for eacha∈A, we write

φ_a=

rdega

X

i=1

l_i(φ_a)τⁱ∈F{τ}.

Denote byOF the integral closure ofAin F. For each prime ideal P ofOF, put ord_P(φ) := min{ordP(φ_a) : 06=a∈A}

where ord_P(φ_a) := min{ord_P(l_i(φ_a))/(qⁱ−1) :i≥1}. Let L_φ be the fractional ideal ofO_F such that for every primeP ofO_F,

ord_P(L_φ) =bord_P(φ)c.

For each placewofF withw-∞, set the local height atwby hTag,w(φ/F) :=−[Fw:Fq]·ordP_w(Lφ),

wherePwCO_F is the prime ideal corresponding to w, andFw:=O_F/Pw.

To define the local height at places ofF lying above∞, we first introduce the “volume”

associated to a givenA-lattice. Let Λ⊂C∞be a rankr A-lattice. Choose a “good”k_∞-basis {λi}1≤i≤r ofk∞·Λ satisfying that

(i) λ_i∈Λ for 1≤i≤r;

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

To show the existence of a “good” basis, by Riemann-Roch theorem it suffices to find ak-basis {λ_i}_1≤i≤r of k·Λ satisfying (ii). Take a non-trivial k-linear functional f : k·Λ → k, and extend to ak∞-linear functional on k∞·Λ (still denoted byf). Then the map

k_∞·Λ− {0} −→ R≥0

z 7−→ |f(z)|∞

|z|_∞

yields a continuous map from the projective space P(k_∞·Λ) to R≥0. Since k·Λ is dense in k_∞·Λ, there exists λ₁ ∈ k·Λ such that the above map takes its maximum at λ₁. By

induction, we have a basis {λ2, ..., λ_r} of kerf satisfying (ii). Then fora₁, ..., a_r∈k_∞ with a16= 0,

|f(λ₁)|_∞

|λ1|∞

= |f(a₁λ₁)|_∞

|a1λ1|∞

≥ |f(a₁λ₁+· · ·a_rλ_r)|_∞

|a1λ1+· · ·arλr|∞

= |f(a₁λ₁)|_∞

|a1λ1+· · ·arλr|∞

.

Hence

|a1λ₁+· · ·a_rλ_r|∞=

(|a1λ1|_∞ if|a2λ2+· · ·arλr|_∞≤ |a1λ1|_∞ max{|aiλi|_∞; 2≤i≤r} otherwise.

Therefore thek-basis{λi}1≤i≤rofk·Λ satisfies (ii).

We define theA-volume D_A(Λ) associated to a givenA-lattice Λ by:

DA(Λ) :=q^1−gk·

Q

1≤i≤r|λ_i|_∞

# Λ∩(O∞λ1+· · ·+O∞λr)

!^1/r

=

Q

1≤i≤r|λ_i|_∞

# Λ/(Aλ1+· · ·+Aλr)

!^1/r . Here{λi}_1≤i≤ris a “good” basis. The second equality is from the Riemann-Roch Theorem.

It is clear thatDA(Λ) is independent of the chosen good basis{λi}_1≤i≤r. In particular, given twoA-lattices Λ and Λ⁰ of rankrwith Λ⁰⊂Λ, we haveDA(Λ⁰) =DA(Λ)·#(Λ/Λ⁰)^1/r.

For each place ˜∞ ofF with ˜∞ | ∞, we embed F into C∞ via ˜∞ and let Λφ,∞˜ ⊂C∞ be theA-lattice associated toφ. Then Λφ,∞˜ is of rankr, and we set

hTag,∞˜(φ/F) :=−[F∞˜ :k_∞]·log_qDA(Λφ,∞˜),

Definition 4.1. (cf. [21, Section 5]) TheTaguchi height ofφ/F is defined by hTag(φ/F) := 1

[F :k]·



 X

w-∞

hTag,w(φ/F) + X

∞|∞˜

hTag,∞˜(φ/F)



.

Remark 4.2. (1) LetF⁰ be a finite extensionF⁰ overF. Given places ˜∞ofF and ˜∞⁰ of F⁰ with ˜∞⁰|∞ | ∞, it is clear that Λ˜ φ,∞˜ = Λφ,∞˜⁰⊂C∞, and

hTag,∞˜⁰(φ/F⁰) = [F_∞⁰_˜0 :F∞˜]·hTag,∞˜(φ/F).

For each primeP ofOF, one has ord_P⁰(φ) =e_P⁰_/P·ord_P(φ) for every primeP⁰ ofOF⁰ lying above P, where e_P0/P is the ramification index ofP⁰/P. Thus for placesw of F andw⁰ of F⁰ with w⁰|w-∞, we get

hTag,w⁰(φ/F⁰)≤[F_w⁰0 :Fw]·hTag,w(φ/F).

In particular, ifφ has stable reduction atP, then ord_P(φ) is an integer, which implies that h_Tag,w⁰(φ/F⁰) = [F_w⁰0 : F_w]·h_Tag,w(φ/F). In conclusion, we haveh_Tag(φ/F⁰)≤h_Tag(φ/F), and the equality holds whenφhas stable reduction everywhere.

(2) Since every DrinfeldA-moduleφoverF has potentially stable reduction everywhere (cf.

[11, Proposition 7.2]),

eh_Tag(φ) := lnq· lim

F0: F0/Ffinite

h_Tag(φ/F⁰) is well-defined.

(3) Let φand φ⁰ be two Drinfeld A-modules over k where kis the algebraic closure of k in C∞. Then

eh_Tag(φ) =eh_Tag(φ⁰) ifφandφ⁰ are isomorphic overk.