Eisenstein series for SL 2 - Department of Mathematicsthyang/EisensteinFinal2010.pdf · Eisenstein series may be attached to many diﬀerent quadratic spaces—the phenomena of ‘matching’,

SCIENCE CHINAMathematics

. ARTICLES . September 2010 Vol. 53 No. 9: 2275–2316

doi: 10.1007/s11425-010-4097-1

c⃝ Science China Press and Springer-Verlag Berlin Heidelberg 2010 math.scichina.com www.springerlink.com

Eisenstein series for SL(2)Dedicated to Professor Wang Yuan on the Occasion of his 80th Birthday

KUDLA Stephen S.1 & YANG TongHai2,∗

1Department of Mathematics, University of Toronto, Toronto, Ontario, M5S2E4, Canada;2Department of Mathematics, University of Wisconsin Madison, Van Vleck Hall, Madison, WI 53706, USA

Email: [email protected], [email protected]

Received June 21, 2010; accepted July 26, 2010

Abstract This paper gives explicit formulas for the Fourier expansion of general Eisenstein series and local

Whittaker functions over SL2. They are used to compute both the value and derivatives of these functions at

critical points.

Keywords Eisenstein series, local Whittaker functions, Weil representation, derivative

MSC(2000): 11G15, 11F41, 14K22

Citation: Kudla S S, Yang T H. Eisenstein series for SL(2). Sci China Math, 2010, 53(9): 2275–2316, doi:

10.1007/s11425-010-4097-1

1 Introduction

The purpose of this paper is to derive explicit formulas for the Fourier coefficients of Eisenstein series

and their derivatives for the group SL(2) and its metaplectic cover. Eisenstein series play an important

role in number theory. They provide the first examples of modular forms and are one of the most im-

portant ingredients in the study automorphic L-functions, e.g., the Rankin-Selberg L-functions, doubling

L-functions, and the Langlands-Shahidi method. In particular, they are useful in proving analytic contin-

uation and functional equations of the L-functions, since the corresponding properties of Eisenstein series

are known in full generality due to the work of Langlands [26]. On the arithmetic side, the well-known

Siegel-Weil formula (e.g., [20,21,31,32]) asserts that the Fourier coefficients of a special value of an Eisen-

stein series can be used to count the number of representations of a number by a quadratic form, at least

on average. The deep relation between the Fourier coefficients of the derivative of a certain Eisenstein

series and the height pairing on a Shimura variety was implicit in the work of Gross and Zagier in their

beautiful Gross-Zagier formula [14], and was made explicit and generalized in the arithmetic Siegel-Weil

formula (see [17,22–24]). At the moment, the proof of these deep relations is a little ad hoc, as it amounts

to computing both quantities explicitly and comparing them. In any case, precise information about the

Fourier coefficients of Eisenstein series and their derivatives is important for such applications.

In this paper, we consider the simple case of the Eisenstein series for the group G = SL(2) over Qdefined for a global induced representation I(s, χ), where χ is a quadratic character of Q×\A×. We

include the case of the metaplectic cover. When the Eisenstein series has weight l ∈ 12Z, we make a

thorough study of the Fourier expansions of the values and derivatives at certain critical values s = s0.

∗Corresponding author

2276 KUDLA Stephen S. et al. Sci China Math September 2010 Vol. 53 No. 9

For example, when the Eisenstein series arises from a quadratic space V over Q of dimension m and

signature (p, q), the weight is l = 12 (p − q), and s0 = 1

2m − 1 is the point at which the Siegel-Weil

formula identifies the value of the Eisenstein series with the integral of a theta function. Of course, not

all Eisenstein series arise in this way—for example there are incoherent Eisenstein series—and a given

Eisenstein series may be attached to many different quadratic spaces—the phenomena of ‘matching’,

see [18].

On the basis of many examples and conjectures, we expect that the Fourier coefficients of the second

term in the Laurent expansion of an Eisenstein series at a critical point s0 should also have important

arithmetic meaning. This should be true even when the series is convergent at s0 and the value at s0 is a

familiar classical Eisenstein series like E2k(τ)! Of course, in this case, the classical holomorphic Eisenstein

series is usually defined without introducing s at all; nonetheless, the Fourier coefficients of the next term

in the Laurent expansion turn out to carry arithmetic information. Thus, the many explicit formulas

for the Fourier series of both values and derivatives provided by this paper — a kind of zoo of examples

— will be the basis for further investigations of their arithmetic significance. Some cases have already

played a role, e.g., in [7, 8, 12,18,30], and [9].

For m ∈ Q×, the m-th Fourier coefficient of an Eisenstein series constructed from a factorizable section

Φ(s) = ⊗Φp(s) ∈ I(s, χ) is a product of local Whittaker functions. For good finite primes p and for p = ∞,

these functions are well known, and so the essential point is to determine the values and derivatives of

the remaining local Whittaker functions at critical points s = s0. The basic technique is to express

such local Whittaker functions in terms of the Weil representation associated to local quadratic spaces.

This allows us to obtain a very general formula whose specializations give a wide range of examples. As

noted, in a number of cases, an interpretation of these formulas can be given in terms of the geometry

and arithmetic geometry of Shimura varieties attached to rational quadratic spaces of signature (n, 2),

including modular curves, Shimura curves, Hilbert modular surfaces, Siegel 3-folds, etc. We expect that

such interpretations can be given in general, and this is the motivation for this exploration.

In the simplest case, the classical Eisenstein series of weight l ∈ 2Z is given by

E(τ, s; l) =∑

γ=(a bc d

)∈Γ∞\SL2(Z)

1

(cτ + d)lIm(γτ)

12 (1+s−l)

for Re(s) > 1. If l > 2, then s0 = l − 1 > 1 and

E(τ, l − 1; l) = 1− 2l

Bl

∞∑m=1

σl−1(m)qm,

where, as usual, Bl is the l-th Bernoulli number and σl−1(m) is the divisor function. On the other hand,

writingd

dsE(τ, s; l)|s=s0 =

∑m∈Z

Al,m(v)qm,

we obtain the following formulas:

1. For m < 0,

Al,m(v) = − 2l

Blσl−1(|m|)(4πmv)1−l

∫ ∞

1

e−4π|m|vrr−ldr.

2. For m > 0,

Al,m(v) = − 2l

Blσl−1(m)

[1

2J(l − 1, 4πmv) +

1

2log(mπ)− 1

2

Γ′(l)

Γ(l)− ζ ′(l)

ζ(l)

+∑p|m

(−kp(m) +

kp(m) + 1

1− p(1−l)(kp(m)+1)− 1

1− p1−l

)log p

].

Here J(n, z) is given in Lemma 2.2, and kp(m) is given in (2.15) and (2.16).

KUDLA Stephen S. et al. Sci China Math September 2010 Vol. 53 No. 9 2277

3. The constant term is given by

Al,0(v) =1

2log v +

π(−i)l(2v)1−l

l − 1

ζ(l − 1)

ζ(l).

The quantities appearing here should be related to the arithmetic volumes of a certain divisors on an

integral model of a Shimura variety attached to a rational quadratic space of signature (2l− 2, 2). Some

information in the case of signature (2, 2) is given in [3], while, in the case of signature (1, 2) where l = 32 ,

a precise version of the analogous statement is proved in [23].

A rough draft of this paper was written in 2001 in preparation for [23], [24], and [18]. Since then, some

of the formulas we obtain have been used in various places, as noted above.

Contents

1 Introduction 2275

1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2277

2 Eisenstein series 2278

3 Classical Eisenstein series 2284

4 Local Whittaker functions 2286

4.1 Generalized local densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2286

4.2 The case p = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2287

4.3 The case p = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2289

4.4 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2292

5 Local examples 2293

5.1 The unary case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2293

5.2 The case of a quadratic extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2293

5.3 Quaternion algebras—the odd case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2296

5.4 Quaternion algebras—the even case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2298

6 Cohen and Zagier Eisenstein series 2299

7 An incoherent Eisenstein series of weight 1 2303

8 Quaternion algebras—ternary forms 2305

9 Quaternion algebras—quaternary forms 2308

10 Integrals of Borcherds forms 2312

1.1 Notation

We write Q, Qp, R and C for the fields of rational, p-adic, real and complex numbers. Let Z =∏p Zp,

where Zp ⊂ Qp is the ring of p-adic integers, write Q = Z ⊗Z Q and A = R × Q. We fix the standard

additive character ψ : A/Q → C× that is trivial on Z and has restriction ψ∞(x) = e(x) = e2πix to R. Let(·, ·)A be the global Hilbert symbol on the idele group A×, and, for a prime p 6 ∞, let (·, ·)p be the local

Hilbert symbol on Q×p , where Qp = R if p = ∞.

A quadratic lattice L, (·, ·) over a domain R is a finite rank free R-module L with a symmetric R-biinear

form (·, ·) : L × L → R. Its associated Gram matrix S = S(L) is defined as S = ((ei, ej)) where ei is a

R-basis of L, and its determinant detL is defined to be detS ∈ R/R×,2.


If R = Qp and L = V is a vector space of dimension n, we let χV be the quadratic character of Q×p

defined by

χV (x) = (x, (−1)n(n−1)

2 det(V ))p (1.1)

and let ϵ(V ) = ±1 be the Hasse invariant of V .

Let G = SL(2) over Q, and let P = NM be its standard Borel subgroup, where

N =

{n(b) =

(1 b

0 1

): b ∈ Q

}, M = {m(a) = diag(a, a−1) : a ∈ Q×}. (1.2)

There is an Iwasawa decomposition G(A) = P (A)K, where K = K∞Kf , K∞ = SO(2), Kf = SL2(Z) =∏p SL2(Zp). For τ = u+ iv ∈ H, the upper half-plane, and for θ ∈ R, we write

gτ = n(u)m(v12 ) ∈ P (R), kθ =

(cos θ sin θ

− sin θ cos θ

). (1.3)

Let

1 −→ C1 −→ G′A −→ G(A) −→ 1 (1.4)

be the metaplectic extension of G(A) by C1. As in the appendix to [24, Chapter VIII], we fix a section

of (1.4) and identify G′A = G(A) × C1 via [24, (8.5.32)] with multiplication given by [g1, z1][g2, z2] =

[g1g2, z1z2 c(g1, g2)], where the global cocycle c(g1, g2) is given in [24, (8.5.31)]. The cocycle is a product

of local cocycles that define the extensions G′p obtained by restricting G′

A to the image of G(Qp) in

G(A). We identify G(Q) = SL2(Q) with a subgroup G′Q of G′

A via the canonical splitting homomorphism

G(Q) → G′A. For the compact open subgroup

K0(4) = K0(4)2 ×∏p=2

Kp (1.5)

of G(Q) there is a splitting homomorphism K0(4) → G′A. There is also a splitting homomorphism

P (A) → G′A so that, if P ′

A denotes the inverse image of P (A) in G′A, we have

P (A)× C1 ≃ P ′A, (p, z) 7→ [p, z]. (1.6)

We will frequently abuse notation and write n(b), m(a), and w for the elements [n(b), 1], [m(a), 1], and

[w, 1]. We will write K ′ for the inverse image of K in G′A.

Similarly, let K ′∞ be the preimage of K∞ = SO(2) in G′

∞. K ′∞ ≃ SO(2) · C1, where SO(2) is the

double cover of SO(2). We identify the character group of SO(2) with 12Z and note that the value of the

character νl on −1 ∈ C1 ∩ SO(2) is (−1)2l. For l ∈ Z, the character νl factors through SO(2) and we

write νl(kθ) = eilθ.

By strong approximation, G(A) = G(Q)G(R)K0 for any compact open subgroup K0 ⊂ G(Q) and so

G′A = G′

QG′∞K0, where we suppose that K0 ⊂ K0(4) and identify it with its image under the splitting

homomorphism.

For a (finite) set of finite primes S, we write ζS(s) =∏p/∈S(1−p−s)−1, and similarly for other functions

defined by products over primes. Similarly for an integer D, ζD(s) =∏p-D(1− p−s)−1.

For m ∈ Q×, let χm be the quadratic Dirichlet character associated with Q(√m). For m ∈ Q×

p , let

χm be the quadratic character of Q×p associated with Qp(

√m)/Qp. On the other hand, for a Dirichlet

character χ, let χp be the associated quadratic character of Q×p . We trust that the reader will easily

distinguish the two different meanings of χp in context.

2 Eisenstein series

An idele class character χ of Q×\A× defines two principal series representations of G′A, one genuine (odd)

and the other factoring through G(A) (even), both of which are denoted by I(s, χ) in this paper. A


section Φ(s) ∈ I(s, χ) is a smooth function on G′A such that

Φ(p′g′, s) = χ(a) |a|s+1Φ(g′, s)

{1, in the even case,

z, in the odd case,(2.1)

where p′ = [n(b)m(a), z] ∈ P ′A. Notice that we use the normalization in [24, p. 287]. Different authors have

different normalizations, we refer to [24, Section 8.4.2] for discussion on this. The formula [24, (8.5.10)]

implies that [p, z] = [p, z]L for any p ∈ PA, where [ , ]L is the Leray cocycle defined in [24].

A section Φ(s) ∈ I(s, χ) is called standard if its restriction to the maximal compact subgroup K ′ ⊂ G′A

is independent of s. It is called factorizable if Φ(s) = ⊗pΦp(s) for the decomposition of the induced

representation I(s, χ) = ⊗′pIp(s, χ). Here Ip(s, χ) is the corresponding induced representation of G′

p. The

Eisenstein series associated to a standard section Φ(s) ∈ I(s, χ) is given by

E(g′, s,Φ) =∑

γ∈P (Q)\G(Q)

Φ(γg′, s). (2.2)

This series is absolutely convergent for Re (s) > 1 and has a meromorphic continuation to the whole

complex s-plane. Furthermore, it has a functional equation

E(g′,−s,M(s)Φ) = E(g′, s,Φ), (2.3)

where M(s) : I(s, χ) → I(−s, χ−1) is the intertwining operator, defined in the half-plane of absolute

convergence by

M(s)Φ(g′, s) =

∫AΦ(wn(b)g′, s) db,

for w =(0 −11 0

)and for db the self-dual measure with respect to ψ. The Eisenstein series has a Fourier

expansion

E(g′, s,Φ) =∑m∈Q

Em(g′, s,Φ), (2.4)

where

Em(g′, s,Φ) =

∫Q\A

E(n(b)g′, s,Φ)ψ(−mb) db, (2.5)

for vol(Q\A, db) = 1. When m = 0 and Φ(s) = ⊗pΦp(s) is factorizable, the m-th Fourier coefficient has

a product expansion

Em(g′, s,Φ) =∏p6∞

Wm,p(g′p, s,Φp), (2.6)

where

Wm,p(g′p, s,Φp) =

∫Qp

Φp(wn(b)g′p, s)ψp(−mb) db (2.7)

is the local Whittaker function. Here db is the self-dual measure on Qp for ψp. On the other hand, the

constant term is

E0(g′, s,Φ) = Φ(g′, s) +

∏p6∞

W0,p(g′, s,Φ) = Φ(g′, s) +M(s)Φ(g′, s). (2.8)

Recall that the poles of the Eisenstein series are precisely those of its constant term.

From now on, we suppose that χ is a quadratic character and write

χ(x) =

{(x, κ)A, in the even case,

(x, 2κ)A, in the odd case(2.9)

for a square free integer κ.


The archimedean component Φ∞(s) will be taken as follows. For l ∈ 12Z satisfying(−1)l = sign(κ), in the even case,

l ≡ 1

2sign(κ) (mod 2), in the odd case,

(2.10)

let Φl∞(s) be the normalized eigenfunction of weight l in I∞(s, χ), i.e.,

Φl∞(g′k′, s) = νl(k′)Φl∞(g′, s), Φl∞(1, s) = 1 (2.11)

for k′ ∈ SO(2), where νl is the character of K′∞ of weight l as defined in Section 1.1. The functions Φl∞(s)

for l satisfying (2.10) span the K ′∞-finite functions in I∞(s, χ).

Now take Φ(s) = Φl∞(s) ⊗ Φf (s) and note that the section Φf (s) = ⊗p<∞Φp(s) is invariant under

some open subgroup K0 of K0(4). By strong approximation, E(g′, s,Φ) is determined by the Eisenstein

series

E(τ, s,Φl∞ ⊗ Φf ) := v−l2E(g′τ , s,Φ

l∞ ⊗ Φf ), (2.12)

which is a non-holomorphic modular form of weight l. Thus, it is enough to determine the Fourier

expansion of this series,

Em(τ, s,Φl∞ ⊗ Φf ) =Wm,∞(τ, s,Φl∞)∏p<∞

Wm,p(s,Φp), m = 0, (2.13)

where, for p <∞,

Wm,p(s,Φp) =Wm,p(1, s,Φp). (2.14)

We write

4κm = dc2 (2.15)

so that d is the fundamental discriminant of Q(√κm). For a square free positive number D, we define

bp(κm, s;D) as follows.

Let k = kp(c) = ordp(c) and X = p−s. Let vp = 1, 0, or − 1, depending on whether Q(√κm)/Q is

split, ramified, or inert at p.

1. If p - D and ordpc > 0, then

bp(κm, s;D) =1− vpX + pkvpX

1+2k − pk+1X2k+2

1− pX2. (2.16)

2. If p|D and ordpc > 0, then

bp(κm, s;D)=(1− vpX)(1− p2X2)−vppk+1X2k+1 + pk+2X2k+2+vpp

k+1X2k+3−p2k+2X2k+4

1− pX2. (2.17)

3. If ordpc < 0, then bp(κm, s;D) = 1.

Clearly, bp(κm, s;D) depends only on p-adic properties of κ, m, and D. According to [23, (8.10)], it

satisfies the following function equation

|cD|−sp bp(κm, s;D) = |cD|s−1p bp(κm, 1− s;D). (2.18)

We will simply write bp(κm, s) = bp(κm, s;D) for p - D.

We also need the divisor function

σs(m,χ) =∑d|m

χ(d)ds =∏p|m

σs,p(m,χ), with σs,p(m,χ) =

ordpm∑r=0

(χp(p)ps)r. (2.19)


Proposition 2.1. Assume that χp is unramified and that p = 2 in the odd case. Let Φ0p(s) be the

spherical, i.e., Kp-invariant, section in Ip(s, χ) with Φ0p(e, s) = 1. Then

Wm,p(s,Φ0p) =

σ−s,p(m,χ)

Lp(s+ 1, χ), in the even case,

Lp(s+12 , χκm)

ζp(2s+ 1)bp

(κm, s+

1

2

), in the odd case.

Here bp(κm, s) is given by (2.16). In particular,

W0,p(s,Φ0p) =

Lp(s, χ)

Lp(s+ 1, χ), in the even case,

ζp(2s)

ζp(2s+ 1), in the odd case.

Next, we consider the archimedean local Whittaker function.

Wm,∞(τ, s,Φl∞) = v−12 l

∫RΦl∞(wn(b)g′τ , s)ψ∞(−mb) db.

Let

Ψ(a, b; z) =1

Γ(a)

∫ ∞

0

e−zr(r + 1)b−a−1ra−1dr

be the standard confluent hypergeometric function of the second kind [27], where a > 0, z > 0 and b is

any real number. It satisfies the functional equation [27, p. 265]

Ψ(a, b; z) = z1−bΨ(1 + a− b, 2− b; z). (2.20)

For convenience, we also define

Ψ(0, b; z) = lima→0+

Ψ(a, b; z) = 1.

So Ψ(a, b; z) is well defined for z > 0, a > min{0, b− 1}. Finally, for any number n, we define Ψn(s, z) =

Ψ( 12 (1 + n+ s), s+ 1; z). Then (2.20) implies

Ψn(s, z) = z−sΨn(−s, z). (2.21)

The formulae in the following lemma will be used in this paper.

Lemma 2.2. Let the notation be as above.

(i) For t > 0,

Ψ−l(l − 1, t) = 1, and Ψ′−l(l − 1, t) = −1

2log t+

1

2J(l − 1, t),

where

J(n, t) =

∫ ∞

0

e−zr(1 + r)n − 1

rdr.

(ii)

Ψl(l − 1, t) = t1−let∫ ∞

1

e−rtr−ldr.

Proof. The first part of (i) is clear. The second part of (i) is [23, (15.9)]. Part (ii) follows from the

calculation just before [23, (15.11)]. 2

Here is the relation between Ψ and the usual W -Whittaker function. Let Wν,µ(z) be the W -Whittaker

function defined in [1, (13.2.5)] for Re (12 + µ− ν) > 0 and Re z > 0:

Wν,µ(z) =e−

z2 zµ+

12

Γ(12 + µ− ν)

∫ ∞

0

e−tzt−12+µ−ν(1 + t)−

12+µ+νdt. (2.22)


To lighten the notation, for s ∈ C and v ∈ R \ {0}, we let

Wks (v) = |v|−k/2Wsgn(v)k/2,(1−k)/2−s(|v|). (2.23)

Notice that

Wk0 (v) =

{e−v/2, if v > 0,

e−v/2Γ(1− k, |v|), if v < 0,(2.24)

where Γ(a, x) =∫∞xe−tta−1dt denotes the incomplete Gamma function as in [1, p. 81]. It is easy to check

that

Ψ(a, b; v) = ev2 v−

b2W b−2a

2 , b−12(v). (2.25)

So, when l > 0, and v > 0, one has

Ψl(s, v) = W l1−l−s

2

(v) ev2 v−

1+s−l2 , (2.26)

Ψl(s, v) = W l1−l−s

2

(−v) e|v|2 |v|−

1+s−l2 . (2.27)

Proposition 2.3 [23, Proposition 15.1]. Let q = e(τ), (−i)l = e(−l/4), and α = s+1+l2 , β = s+1−l

2 .

(i) For m > 0,

Wm,∞(τ, s,Φl∞) = 2π (−i)l vβ (2πm)sΨ−l(s, 4πmv)

Γ(α)· qm.

(ii) For m < 0,

Wm,∞(τ, s,Φl∞) = 2π (−i)lvβ (2π|m|)s Ψl(s, 4π|m|v)Γ(β)

e−4π|m|v · qm.

(iii) For m = 0,

W0,∞(τ, s,Φl∞) = 2π (−i)ℓ v 12 (1−l−s)

2−sΓ(s)

Γ(α)Γ(β).

(iv) The special value at s0 = l − 1 is

Wm,∞(τ, l − 1,Φl∞) =

0, if m 6 0,

(−2πi)l

Γ(l)ml−1 qm, if m > 0.

Combining these facts, we obtain the following preliminary result.

Theorem 2.4. Let Φf (s) = ⊗Φp(s) ∈ I(s, χf ) be a factorizable standard section and let l ∈ 12Z satisfy

(2.10). Let S = S(Φf ) be a finite set of finite primes such that Φp(s) = Φ0p(s) for p /∈ S, and let

Wm,S(s,ΦS) =∏p∈S

Wm,p(s,Φp).

Then the Eisenstein series (2.12) has the following Fourier expansion.

(i) In the even case, where l is an integer,

E(τ, s,Φl∞ ⊗ Φf ) = vβΦf (1) + 2π(−i)lv(−l−β) 2−sΓ(s)

Γ(α)Γ(β)

LS(s, χ)

LS(s+ 1, χ)W0,S(s,ΦS)

+∑m=0

Wm,S(s,ΦS)Wm,∞(τ, s,Φl∞)σS−s(m,χ)

LS(s+ 1, χ).

(ii) In the odd case, where l ∈ 12Z− Z,

E(τ, s,Φl∞ ⊗ Φf ) = vβΦf (1) + 2π(−i)lv−l−β) 2−sΓ(s)

Γ(α)Γ(β)

ζS(2s)

ζS(2s+ 1)W0,S(s,ΦS)


+∑m=0

Wm,S(s,ΦS)Wm,∞(τ, s,Φl∞)LS(s+ 1

2 , χκm)

ζS(2s+ 1)bS(κm, s+

1

2

).

The normalized Eisenstein series has better analytic properties.

Corollary 2.5. Let

E∗(τ, s,Φl∞ ⊗ Φf ) = E(τ, s,Φl∞ ⊗ Φf )

{LS(s+ 1, χ), in the even case,

ζS(2s+ 1), in the odd case.

Then E∗(τ, s,Φl∞ ⊗ Φf ) is a holomorphic function of s with one possible exception when l = 12 where a

pole could occur at s = 12 .

The purpose of this paper is to compute the Fourier coefficients of the first two terms of the Laurent

expansion of the Eisenstein series at the critical point s0 = l − 1. By the functional equation (2.3), we

may and will assume that l > 1 and hence s0 > 0. Our intermediate results can also be used to compute

the Laurent expansion of the Eisenstein series at other ‘critical’ points s0. The key is to compute the

local Whittaker functions at ‘ramified’ primes p ∈ S. We first state a few consequences of Theorem 2.4

for l > 2. The cases l = 1, 3/2, and 2 are very interesting and will be dealt with separately in later

sections, after the local Whittaker functions have been computed.

Proposition 2.6. When l > 2, the special value E(τ, l − 1,Φl∞ ⊗ Φf ) is a holomorphic modular form

of weight l.

(i) In the even case, it has Fourier expansion

E(τ, l − 1,Φl∞ ⊗ Φf ) = Φf (1) +(−2πi)l

Γ(l)

1

LS(l, χ)

∑m>0

Wm,S(l − 1,ΦS)ml−1 σS1−l(m,χ) q

m.

(ii) In the odd case, it has Fourier expansion

E(τ, l − 1,Φl∞ ⊗ Φf ) = Φf (1) +(−2πi)l

Γ(l)

∑m>0

ml−1LS(l − 1

2 , χκm)

ζS(2l − 1)bS(κm, l − 1

2

)Wm,S(l − 1,ΦS) q

m.

The derivative is more delicate. For the moment, we still assume l > 2. First consider the case m < 0.

In this case Wm,∞(τ, l − 1,Φl∞) = 0. Moreover Lemma 2.2 and Proposition 2.3 imply that

W ′m,∞(τ, l − 1,Φl∞) = π (−i)l (2π|m|)1−lΓ(l − 1, 4π|m|v) qm. (2.28)

Thus, from Theorem 2.4, we have

Proposition 2.7. Assume that l > 2 and m < 0.

(i) In the even case,

E′m(τ, l − 1,Φl∞ ⊗ Φf ) · q−m

= π(−i)l(2π|m|)l−1Γ(l − 1, 4π|m|v)Wm,S(l − 1,ΦS)LS(l − 1, χ)

LS(l, χ)σS1−l(m,χ).

(ii) In the odd case,

E′m(τ, l−1,Φl∞ ⊗ Φf ) · q−m

=π(−i)l(2π|m|)l−1Γ(l − 1, 4π|m|v)Wm,S(l − 1,ΦS)LS(l − 1

2 , χκm)

ζS(2l − 1)bS(κm, l− 1

2

).

When m > 0 and Wm,S(l − 1,ΦS) = 0, a similar formula holds. Finally, when m > 0 and Wm,S(l −1,ΦS) = 0, one has Em(τ, l−1,Φl∞⊗Φf ) = 0 by Proposition 2.6. A simple calculation using Theorem 2.4,

Proposition 2.3, and Lemma 2.2 gives the following result.

Proposition 2.8. Assume that l > 2, m > 0 and that Wm,S(l − 1,Φf ) = 0.



E′m(τ, l − 1,Φl∞ ⊗ Φf )

Em(τ, l − 1,Φl∞ ⊗ Φf )=W ′m,∞(τ, l − 1,Φl∞)

Wm,∞(τ, l − 1,Φl∞)+W ′m,S(l − 1,Φf )

Wm,S(l − 1,Φf )

− LS,′(l, χ)

LS(l, χ)+∑p|m

(kp(m) + 1

(χp(p)ps)kp(m)+1 − 1− 1

χp(p)ps− 1

)log p.


E′m(τ, l − 1,Φl∞ ⊗ Φf )

Em(τ, l − 1,Φl∞ ⊗ Φf )=W ′m,∞(τ, l − 1,Φl∞)

Wm,∞(τ, l − 1,Φl∞)+W ′m,S(l − 1,Φf )

Wm,S(l − 1,Φf )

+LS,′(l − 1

2 , χκm)

LS(l − 12 , χκm)

− 2ζS,′(2l − 1)

ζS(2l − 1)+∑p|m

b′p(κm, l − 12 )

bp(κm, l − 12 ).

Moreover,W ′m,∞(τ, l − 1,Φl∞)

Wm,∞(τ, l − 1,Φl∞)=

1

2

[log(πm)− Γ′(l)

Γ(l)+ J(l − 1, 4πmv)

],

where J(n, z) is defined in Lemma 2.2.

For the constant term, we have the following proposition by Theorem 2.4 and the fact that Γ(β)−1 = 0

and has derivative 12 at s = l − 1.

Proposition 2.9. The derivative of the constant term at s = l − 1 is the following.


E′0(τ, l − 1,Φl∞ ⊗ Φf ) =

1

2Φf (1) log v +

π(−i)l

l − 1

LS(l − 1, χ)

LS(l, χ)W0,S(l − 1,Φf )(2v)

1−l.


E′0(τ, l − 1,Φl∞ ⊗ Φf ) =

1

2Φf (1) log v +

π(−i)l

l − 1

ζS(2l − 2)

ζS(2l − 1)W0,S(l − 1,Φf )(2v)

1−l.

3 Classical Eisenstein series

In this section, as a first example, we reconsider the classical Eisenstein series for SL2(Z). In this

case the exceptional set S is empty and no further computation of local Whittaker functions is needed.

Let l be an even integer and let χ be the trivial character, so that I(s, χ) is the basic principle series

for G(A) = SL2(A) (the even case). We take the unramified section Φp(s) = Φ0p(s) with value 1 on

Kp = SL2(Zp) for all finite primes p and set

E(τ, s; l) = E(τ, s,Φl∞ ⊗ (⊗pΦ0p)).

Then it is easy to see from the definition that

E(τ, s; l) =∑

γ=(a bc d

)∈Γ∞\SL2(Z)

1

(cτ + d)lIm(γτ)

12 (1+s−l).

The following Fourier expansion follows immediately from Theorem 2.4.

Proposition 3.1. Set α = 1+s+l2 and β = 1+s−l

2 . Then

E(τ, s; l) = vβ +2π(−i)l2−svβ−sζ(s)Γ(α)Γ(β)ζ(s+ 1)

+(−i)l(2π)s+1vβ

ζ(s+ 1)Γ(α)

∞∑m=1

σs(m)Ψ−l(s, 4πmv)qm


+(−i)l(2π)s+1vβ

ζ(s+ 1)Γ(β)

∞∑m=1

σs(m)Ψl(s, 4πmv)qm.

Here σs(m) =∑d|m d

s and Ψl is defined by (2.9).

The case of weight l = 0 gives the classical Kronecker limit formula, from which one can derive the

modularity of the famous ∆ function [31].

Corollary 3.2 (Kronecker limit formula). Let

E(τ, s) := E(τ, 2s− 1; 0) =∑

γ∈Γ∞\SL2(Z)

Im(γτ)s.

Then E(τ, s) = 1 + log(v|∆(τ)| 16 )s+O(s2). Here ∆(τ) is the modular delta function.

Proof. (sketch) By Proposition 3.1, we have

E(τ, s) = vs +2π21−2sΓ(2s− 1)ζ(2s− 1)

Γ(s)2ζ(2s)v1−s +

(2π)2svs

ζ(2s)Γ(s)

∞∑m=1

σ2s−1(m)Ψ0(2s− 1, 4πmv)(qm + qm).

Since Γ(s) has a simple pole at s = 0 with residue 1, and Ψ0(−1, 4πmv) = 1, we have E(τ, 0) = 1 and

E′(τ, 0) = log v − π

3v − 2

∞∑m=1

σ−1(m)(qm + qm) = log(v|∆(τ)16 |).

When l > 2, taking the value at s = l − 1 gives the following well-known result.

Corollary 3.3. (i) When l > 2 is even,

E(τ, l − 1; l) =∑

γ∈Γ∞\SL2(Z)

1

(cz + d)l= 1− 2l

Bl

∞∑m=1

σl−1(m)qm.

Here Bl are the Bernoulli numbers.

(ii) Let E2(τ) = 1−24∑∞m=1 σ1(m)qm. Then E(τ, 1; 2) = − 3

πv−1+E2(τ) is a non-holomorphic modular

form of weight 2.

We remark that Corollary 3.3 (ii) is equivalent to the usual transformation formula

τ−2E2

(− 1

τ

)= E2(τ) +

12

2πiτ.

Corollary 3.4. For l > 2, let

d

dsE(τ, s; l)|s=l−1 =

∑m∈Z

Al,m(v)qm.

(i) For m < 0,

Al,m(v) = − 2l

Blσl−1(|m|)Γ(l − 1, 4π|m|v).

(ii) For m > 0,

Al,m(v) = − 2l

Blσl−1(m)

[1

2J(l − 1, 4πmv) +

1

2log(mπ)− 1

2

Γ′(l)

Γ(l)− ζ ′(l)

ζ(l)

+∑p|m

(− kp(m) +

kp(m) + 1

1− p(1−l)(kp(m)+1)− 1

1− p1−l

)log p

].

Here J(n, z) is given in Lemma 2.2.

(iii) The constant term is given by

Al,0(v) =

1

2log v +

π(−i)l(2v)1−l

l − 1

ζ(l − 1)

ζ(l), if l > 2,

1

2log v +

3

πv

(1

2log v +

1

2− γ + log 2 +

ζ ′(2)

ζ(2)

), if l = 2.


4 Local Whittaker functions

In this section, we compute the local Whittaker functions at finite primes. We will do this by expressing

the local sections and Whittaker functions in terms of the Weil representation. We begin by reviewing

some basic machinery, see [28].

For any p 6 ∞, let (V,Q) be a quadratic space of dimension n over Qp with character χV defined by

(1.1). Associated to the reductive dual pair (O(V ), Gp), there is a Weil representation ω = ωV,ψp of G′p

on the space S(V ) of Schwartz functions, and a G′p-equivariant map

λV : S(V ) −→ Ip(s0, χV ), λV (ϕ)(g′) = ω(g′)ϕ(0), s0 =

n

2− 1. (4.1)

We refer to [24, Section 8.5] for the explicit formula for ω. Here the even (resp. odd) case occurs when

dimV is even (resp. odd). The map λV is surjective except for the following cases:

1. When dimV = 4, χV = 1, and V is anisotropic, so that (V,Q) can be identified as the reduced

norm on the division quaternion algebra over Qp. In this case, R(V ) is irreducible of codimension one in

Ip(1, χV ).

2. When dimV = 3, and V is the trace zero elements of a division quaternion algebra over Qp with

scalar multiple of the reduced norm as its quadratic form. In this case, R(V ) is irreducible of infinite

codimension in Ip(12 , χV ) with the quotient also irreducible.

3. When dimV = 2 is anisotropic. Then s0 = 0 and Ip(0, χp) = R(V +) ⊕ R(V −), where V ± are two

dimensional quadratic space over Qp of character χV = χ = 1 and Hasse invariant ±1.

In view of this, one sees that every standard section Φp(s) ∈ Ip(s, χp) is associated to some λV (ϕp) for

some ϕp ∈ S(V ) and some Qp-quadratic space. For example, one has the following well-known lemma.

Lemma 4.1. (i) Let V be a quadratic space over R of signature (p, q), and let V = V +⊕V −, x = x++x−

be a fixed orthogonal decomposition of V as a direct sum of a positive definite and a negative definite

subspace. Let

ϕ∞(x) = e−2πQ(x+)+2πQ(x−) ∈ S(V ).

Then the standard section associated with λV (ϕ∞) is Φp−q2∞ (s).

(ii) Assume that p is a finite prime, and L is a unimodular Zp-lattice in V , i.e., L∗ = L, where

L∗ = {x ∈ V : (x, y) ∈ Zp, ∀ y ∈ L}.

Then the standard section associated to λV (char(L)) is the spherical section Φ0p(s) ∈ Ip(s, χp), i.e.,

Φ0p(g

′k, s) = Φ0p(g

′, s), Φ0p(1, s) = 1,

for any k ∈ Kp = SL2(Zp).A little explanation is needed about the meaning of last formula in the lemma. When p is odd,

k 7→ [k, 1] is a group homomorphism from Kp to G′p by [24, (8.5.30)], and we identify Kp with its image

in G′p. When p = 2, a lattice L cannot be unimodular unless m = dimV is even, in which case we are

dealing with the even case of the induced representation Ip(s, χp).

For a finite prime p and an even integral lattice L in V , i.e., Q(x) = 12 (x, x) ∈ Zp for every x ∈ L, we

write ϕµ = char(µ+L) for the characteristic function of the coset µ+L. Then S(V ) is generated by the

functions ϕµ as the even integral lattices L and the coset representatives µ ∈ L∗/L vary.

4.1 Generalized local densities

In this subsection, we assume p <∞. Let (V,Q) be a non-degenerate quadratic space overQp of dimension

n and character χ = χV . For ϕ ∈ S(V ), let Φ(s) ∈ I(s, χ) be the standard section associated with ϕ,

i.e., the standard section such that Φ(s0) = λV (ϕ) with s0 = n2 − 1. We will simply write Wm,p(s, ϕ) for

Wm,p(s,Φ). Let H = Z2p be the standard hyperplane quadratic lattice: Q((x, y)) = xy. For an integer


r > 0, Let Vr = Hr ⊗ Qp and V (r) = V ⊕ Vr. For a lattice L of V , we denote L(r) = L⊕Hr. Then we

have the following interpolation trick which we learned from Rallis, (see [17, Lemma A.3]).

Lemma 4.2. For any ϕ ∈ S(V ), ϕ(r) = ϕ⊗ char(Hr) ∈ S(V (r)) and ϕ give the same standard section

Φ(s) ∈ I(s, χ). In particular, let Φ(s) be the standard section associated with ϕ, then

Φ(g′, s0 + r) = ωV (r)(g′)(ϕ(r))(0).

From this lemma, one obtains easily

Wm,p(s0 + r, ϕ) = γ(V )

∫Qp

∫V (r)

ψ(bQ(r)(x))ϕ(r)(x) dV x ψ(−bt) db (4.2)

for any even integer r > 0. Here dV x is the self-dual Haar measure on V with respect to ψ((x, y)), and

γ(V ) =

{ϵp(V )γ

(1

2ψp

)nγ

(detV,

1

2ψp

)}−1

(4.3)

is the local splitting index defined in [19, Theorem 3.1], where ϵp(V ) is the Hasse invariant of V , and γ

is Weil’s local index (see [29,32]).

Choose a Zp-basis {ej : j = 1, . . . , n} of L and identify

L = ⊕Zpej ∼= Znp ,∑

µjej 7→ (µ1, . . . , µn).

Let dx =∏dxj be the standard Haar measure on L = Znp . Then

dV x = [L∗ : L]−12 dx = |detS|

12p dx

where S = ((ei, ej)) is the Gram matrix of L. So we have

Wm,p(s0 + r, ϕµ)

γ(V )|detS|12p

=Wp(s0 + r,m, µ), (4.4)

where

Wp(s0 + r,m, µ) =

∫Qp

∫µ+L(r)

ψ(bQ(r)(x)) dxψ(−mb) db. (4.5)

In the rest of this section, we compute this integral by extending the method in [33], where we deal

with the special case µ = 0.

4.2 The case p = 2

As in [33], we deal with the cases p = 2 and p = 2 separately. In this subsection, we assume that p = 2.

Let L ∼= Znp be an integral lattice and let S be its Gram matrix. Then S is GL2(Zp)-equivalent to the

matrix of the form S = diag(2ϵ1pl1 , . . . , 2ϵnp

ln), with ϵi ∈ Z×p and li > 0 being integers (warning: the

matrix S is twice of that used in [33]). In this case, L∗ = ⊕p−liZp. For µ = (µ1, . . . , µn) ∈ V = Qnp , wedefine Hµ = {i : 1 6 i 6 n, µi ∈ Zp} and

K0 = K0(µ) =

{∞, if µ ∈ L,

min(li + ordpµi : i /∈ Hµ), if µ /∈ L.(4.6)

Following [33], we further define

Lµ(k) = {i ∈ Hµ : li − k < 0 is odd}, lµ(k) = #Lµ(k),

dµ(k) = k +1

2

∑i∈Hµ,

min(li − k, 0), ϵµ(k) =

(−1

p

)[lµ(k)

2 ] ∏i∈Lµ(k)

(ϵip

),


where (α

p

)=

{(α, p)p, if α ∈ Z×

p ,

0, otherwise.

Finally, we define

f1(αpa) =

−1

p, if lµ(a+ 1) is even,(

α

p

)1√p, if lµ(a+ 1) is odd.

Theorem 4.3. Let the notation be as above and fix µ ∈ L∗.

(i) If m /∈ Q(µ) + Zp, then Wp(s+ s0,m, µ) = 0.

(ii) If m ∈ Q(µ) + Zp, set

tµ = tµ(m) = m−∑i/∈Hµ

ϵipliµ2

i , a = aµ(m) = ordptµ,

and let X = p−s.

If 0 6 a < K0, then

Wp(s+ s0,m, µ) = 1 +

(1− 1

p

) ∑0<k6a

lµ(k) even

ϵµ(k)pdµ(k)Xk + ϵµ(a+ 1)f1(tµ)p

dµ(a+1)Xa+1.

If a > K0, then

Wp(s+ s0,m, µ) = 1 +

(1− 1

p

) ∑0<k6K0

lµ(k) even

ϵµ(k)pdµ(k)Xk.

We remark that the sum in the definition of tµ is just the quadratic form Q(µ) with the terms for

which µi ∈ Zp omitted. In particular, one has tµ ≡ m−Q(µ) ∈ Zp.Proof. The proof is similar to that of [33, Theorem 3.1] and we give a sketch of it for the convenience

of the reader. For µ ∈ Qp/Zp, define

Iµ(t) =

∫µ+Zp

ψ(tx2)dx.

Then I0(t) = I(t) is given by [33, Lemma 2.1]. For µ /∈ Zp, a similar calculation gives a simpler formula:

Iµ(t) = ψ(tµ2)char(µ−1Zp)(t). (4.7)

Next, for µ = (µ1, . . . , µn) ∈ L∗ − L, set Jµ(t) =∫µ+L

ψ(tQ(x))dx. Then

Jµ(t) =∏i

Iµi(ϵiplit) =

∏i/∈Hµ

char(µ−1i Zp)(plit)ψ(tϵipliµ2

i )∏i∈Hµ

I(ϵiplit)

= char(p−K0Zp)(t)ψ( ∑i/∈Hµ

ϵipliµ2

i t

) ∏i∈Hµ

I(ϵiplit).

Now applying [33, Lemma 2.2] to∏i∈Hµ

I(ϵiplit), one sees for k > 0

Jµ(βp−k) =char(p−K0Zp)(p−k)ψ

( ∑i/∈Hµ

ϵipliµ2

iβp−k)[(

−βp

)δp

]lµ(k) ∏i∈Lµ(k)

(ϵip

)p

12

∑li<k(li−k). (4.8)

Clearly,

Jµ(b) = ψ

( ∑i/∈Hµ

ϵipliµ2

i b

)


for b ∈ Zp. Finally, one has

Wp(s0,m, µ) :=

∫Qp

∫µ+L

ψ(bQ(x))ψ(−bm)dx db =

∫Qp

Jµ(b)ψ(−bm)db

=

∫Zp

Jµ(b)ψ(−bm)db+∑

0<k6K0

δlµ(k)p

∏i∈Lµ(k)

(ϵip

)pdµ(k)

∫Z×p

(−βp

)lµ(k)ψ(−tµp−kβ) dβ

= 1 +∑

0<k6K0

lµ(k) even

ϵµ(k)pdµ(k)

∫Z×p

ψ(−tµp−kβ) dβ

+∑

0<k6K0

lµ(k) odd

ϵµ(k)pdµ(k)δp

(−1

p

)∫Z×p

(β

p

)ψ(−tµp−kβ) dβ.

Here we have used the fact that δ2p = (−1p ). Notice that

∫Z×p

ψ(−tµp−kβ)dβ =

1− p−1, if a > k,

−p−1, if a = k − 1,

0, if a < k − 1,

and (see [33, Lemma 2.4])

∫Z×p

(β

p

)ψ(−tµp−kβ) dβ =

p− 1

2 δp

(tµp

−p+1

p

), if a = k − 1,

0, otherwise.

Plugging these formulas back into the formula for Wp(s0,m, µ), one obtains the theorem for r = 0.

Replacing L by L(r), one obtains the theorem by Lemma 4.2.

4.3 The case p = 2

In this subsection, we assume that p = 2. By [10, Lemma 8.4.1], the Gram matrix S of L is Z2 equivalent

to

diag(ϵ12l1 , . . . , ϵH2lH )⊕

(M⊕i=1

ϵ′i2mi

(0 1

1 0

))⊕

N⊕j=1

ϵ′′j 2nj

(2 1

1 2

) (4.9)

where ϵh, ϵ′i, ϵ

′′j ∈ Z×

2 , lh > 1, mi, and nj are all non-negative integers. The associated quadratic form is

Q(x) =∑h6H

ϵh2lh−1x2h +

∑i6M

2mixiyi +∑j6N

2nj (x2j + xjyj + y2j ).

Of course, n = H + 2M + 2N . In this case, we identify L with Zn2 and then

L∗ =H⊕h=1

(2−lhZ2)⊕ (⊕i2−miZ22)⊕ (⊕j2−njZ2

2).

We need some more notation. For µ ∈ V = Qn2 , we write

µ = (µ1, . . . , µH , µ′1, . . . , µ

′M , µ

′′1 , . . . , µ

′′N ), with µh ∈ Q2, µ

′i, µ

′′j ∈ Q2

2,

Mµ = {i : 1 6 i 6M, µ′i ∈ Z2

2}, and Nµ = {j : 1 6 j 6 N, µ′′i ∈ Z2

2}.

We also define

Hµ = {h : 1 6 h 6 H, µh ∈ Z2}


and

Lµ(k) = {i ∈ Hµ : li − k < 0 is odd}, lµ(k) = #Lµ(k),

as in th last subsection. We have to redefine

dµ(k) = k +1

2

∑h∈Hµ

min(lh − k, 0) +∑i∈Mµ

min(mi − k, 0) +∑j∈Nµ

min(nj − k, 0).

We also redefine K0 = K0(µ) to be ∞ when µ ∈ L and to be the minimum of the following numbers

otherwise

{lh + ord2µh : ord2µh < −1}, {lh : ord2µh = −1},{mi + ord2µ

′i : i /∈Mµ}, {nj + ord2µ

′′j : j /∈ Nµ}. (4.10)

Here ord2(t1, t2) = min(ord2t1, ord2t2).

Furthermore, we need

pµ(k) = (−1)∑

j∈Nµmin(nj−k,0), ϵµ(k) =

∏h∈Lµ(k)

ϵh, δµ(k) =

{0, if lh = k for some h ∈ Hµ,

1, otherwise.

The following result is analogous to [33, Theorem 4.1].

Theorem 4.4. Let the notation be as above and fix µ ∈ L∗.

(i) If m /∈ Q(µ) + Zp, then W2(s+ s0,m, µ) = 0.

(ii) If m ∈ Q(µ) + Zp, then setting X = 2−s,

W2(s+ s0,m, µ) = 1 +∑

16k6min(K0,a+3)lµ(k) odd

δµ(k)pµ(k)2dµ(k)− 3

2

(2

ϵµ(k)ν

)Xk

+∑

16k6min(K0,a+3)lµ(k) even

δµ(k)pµ(k)2dµ(k)−1

(2

ϵµ(k)

)ψ

(ν

8

)char(4Z2)(ν)X

k.

Here

tµ = m−Q′(µ), a = ord2tµ, ν = νµ(m, k) = tµ23−k −

∑h∈Hµ,lh<k

ϵh,

and

Q′(z) =∑h/∈Hµ

ϵh2lhx2h +

∑i/∈Mµ

ϵ′i2miyi1yi2 +

∑j /∈Nµ

ϵ′′j 2nj (z2j1 + zj1zj2 + z2j2)

is the quadratic form on V obtained from Q(µ) by omitting the terms where µh, µ′i or µ

′′j ∈ Z2. When

there is no h ∈ Hµ with lh < k, set∑h∈Hµ,lh<k

ϵh = 0.

Proof. It is not hard to verify that

Iµ(t) :=

∫µ+Z2

ψ(tx2)dx =

I(t), if µ ∈ Z2,

ψ(tµ2) char(2−1Z2)(t), if ord2µ = −1,

ψ(tµ2) char((2µ)−1Z2)(t), if ord2µ < −1,

(4.11)

where I(t) = I0(t) is computed in [33, Section 4], and is given by

I(α2a) =

1, if a > 0,

0, if a = −1,

2a+12 ψ

(α

8

)(2

α

)a+1

, if a < −1.


Similarly, for µ = (µ1, µ2) ∈ Q22, one has

I ′µ(t) :=

∫µ+Z2

2

ψ(ty1y2) dy1dy2 =

{min(1, |t|), if µ ∈ Z2

2,

ψ(tµ1µ2) char(Z22)(tµ1, tµ2), if µ /∈ Z2

2,(4.12)

and

I ′′µ(t) :=

∫µ+Z2

2

ψ(t(z21 + z1z2 + z22)) dz1 dz2

=

1, if µ ∈ Z2

2, t ∈ Z2,

(−1)ord2t|t|−1, if µ ∈ Z22, t /∈ Z2,

ψ(t(µ21 + µ1µ2 + µ2

2)) char(Z22)(tµ), if µ /∈ Z2

2.

(4.13)

We verify (4.13) and leave the others to the reader. The case µ ∈ Z22 is [33, Lemma 4.2]. The case

tµ ∈ Z22 is also clear after replacing z by µ+ z. Now assume tµ /∈ Z2

2. We prove I ′′µ(t) = 0. By symmetry,

we may assume ord2(tµ1) 6 min(ord2(tµ1),−1). Choose b ∈ Z2 such that tb, tµ1b ∈ Z2, and tµ2b /∈ Z2.

Replacing z1 by b+ z1, and using the conditions on b, we have I ′′µ(t) = ψ(btµ2)I′′µ(t). So I

′′µ(t) = 0.

Next, unfolding the Whittaker integral as in Theorem 4.3 gives the desired formula. Indeed,

Wp(s0,m, µ) =

∫Q2

∫µ+L

ψ(bQ(x))ψ(−mb) dx db =∫Q2

Jµ(b)J′µ(b)J

′′µ (b)ψ(−mb) db.

Here Jµ, J′µ, and J

′′µ are given as follows. First,

Jµ(b) =∏h

∫µh+Z2

ψ(ϵh2lh−1bx2) =

∏h∈Hµ

I(ϵh2lh−1b)ψ(bQH(µ))char(2K0(H)Z2)(b),

where

QH(µ) =∑h/∈Hµ

ϵh2lh−1µ2

h,

and K0(H) is the minimum of lh + ord2µh for ord2µh < −1 and lh for ord2µh = −1. Plugging in the

formula for I(t), one sees that

Jµ(β2−k) = δ(k)

(2

ϵµ(k)

)(2

β

)lµ(k) ∏µ∈Hµ,lh<k

2lh−k

2 ψ

(ϵhβ

8

)ψ(β2−kQH(µ))char(2k−K0(H)Z2)(1). (4.14)

Similarly, one has

J ′µ(β2

−k) =∏i6M

∫µi+Z2

2

ψ(β2mi−kxy) dx dy =∏

i∈Mµ,mi<k

2mi−kψ(β2−kQM (µ))char(2k−K0(M)Z2)(1)

with

QM (µ) =∑i/∈Mµ

2miµi,1µi,2, K0(M) = min{mi + ord2µi : ord2mi < 0}.

Finally,

J ′′µ (β2

−k) =∏

j∈Nµ,nj<k

(−1)nj−k2nj−kψ(β2−kQN (µ))char(2k−K0(N)Z2)(1)

with QN and K0(N) similarly defined. Putting things together, and using [33, Lemma 4.2], one proves

the theorem for r = 0. Replacing L by L(r), one proves the theorem.


4.4 Consequences

Theorems 4.3 and 4.4 have the following implications.

Corollary 4.5. Let L be a unimodular lattice over Zp and let ϕ0 ∈ S(V ) be the characteristic function

of L. Then the standard section in I(s, χ) associated to λ(ϕ0) is the spherical section Φ0p(s), and the

Whittaker function Wm,p(s, ϕ0) is given by Proposition 2.1.

Proof. We assume p = 2 and leave the case p = 2 to the reader. So L is Zp-equivalent to Znp with

Q(x) = x21 + · · · + x2n−1 + ϵx2n with ϵ ∈ Z×p . It is easy to check from the definition that γ(V ) = 1 and

|detS|p = 1 in this case. Notice that µ = 0. So

Wm,p(s, ϕ0) =Wp(s,m, 0),

where Wp(s,m, µ) is given by (4.5). In this case, Kp is viewed as subgroup of G′p via its splitting. Since

Kp is generated by m(a), n(b) and w with a ∈ Z×p and b ∈ Zp. One checks that

ω(m(a))ϕ0(x) = χp(x)|a|n2p ϕ0(xa) = ϕ0(x),

ω(n(b))ϕ0(x) = ψp(bQ(x))ϕ0(x) = ϕ0(x),

ω(w)ϕ0(x) = γ(V )

∫L

ψ(−(b, x))db = ϕ0(x).

So ϕ0 and thus λ(ϕ0) is Kp-invariant. So Φ0p(s) is the standard section associated with λ(ϕ0). Now

Theorem 4.3 easily gives the formula for Wm,p(s, ϕ0), which is the same as that in Proposition 2.1. We

check the odd case, and leave the even case to the reader. Assume that n = dimV is odd. Then l0(k) is

even if and only if k is even. Moreover, d0(k) = (1− n/2)k,

ϵ0(k) =

{1, if k > 1 is even,

((−1)[n2 ]ϵ, p)p, if k > 1 is odd.

Recall that 4κm = dc2. If p|d, then χκm is ramified at p and a = ordpm = 1 + 2ordpc = 1 + 2k is odd.

Theorem 4.3 gives

Wp

(s+

n

2− 1,m, 0

)= 1 + (1− p−1)

∑0<l6k

(p2−nX2)k − p−1(p2−nX2)a+1

= (1− p1−nX2)∑

06l6k(p2−nX2)l.

So

Wp(s,m, 0) = (1− p−1X2)1−X2k+2

1−X2=Lp(s+

12 , χκm)

ζp(2s+ 1)bp

(κm, s+

1

2

),

as claimed.

If p - d, then χκm is unramified at p and a = 2k. Theorem 4.3 gives

Wp

(s+

n

2− 1,m, 0

)= 1 + (1− p−1)

∑0<l6k

(p2−nX2)l + vpp− 1

2 (p2−nX2)2k+1

2 ,

with vp = ϵ0(a + 1)( d4κ , p)p. The condition χV = χ implies ((−1)[n2 ]2ϵ, p)p = (2κ, p)p. So vp = (d, p)p =

χκm(p). The rest is a simple calculation.

Corollary 4.6. Let L be an even integral lattice in a non-degenerate quadratic space V of dimension

n. Fix µ ∈ L∗ and let s0 = n2 − 1.

(i) Wp(s0 + s,m, µ) is a polynomial in X = p−s with coefficients in Z[ 1p ].(ii) When s0 = 0, i.e., when n = 2, Wp(s,m, µ) is a polynomial of X = p−s with coefficients in Z

unless L is unimodular. When L is unimodular, L(s+1, χ)Wp(s,m, µ) is a polynomial in X = p−s with

coefficients in Z.


Proof. (sketch) The first claim follows from Theorems 4.3 and 4.4 and the fact that dµ(k)− 12 ∈ Z when

lµ(k) is odd and that dµ(k) ∈ Z when lµ(k) is even. Now assume that n = 2. The unimodular case

follows from Proposition 2.1. When L is not unimodular, and p = 2, one has S = diag(ϵ1pl1 , ϵ2p

l2), and

at least one li is positive. One sees that if lµ(k) is even and k > 0, one has dµ(k) = k or 12 l1 + l2. In

either case, dµ(k) > 0. Similarly f1(tµ)pdµ(a+1) ∈ Z. So Wp(s,m, µ) ∈ Z[X]. We leave the case p = 2 to

the reader.

5 Local examples

In this section, we will compute several local Whittaker functions using the formulas in section 4. The

results will be used later in this paper. Let (L,Q) be an even integral quadratic lattice over Qp of

dimension n, and let V = L⊗Zp Qp.

5.1 The unary case

We first consider the case L = Zp with quadratic form Q(x) = κx2, and κ ∈ Z×p . The associated quadratic

character is χV (x) = (2κ, x)p.

Proposition 5.1. Let L be as above. Then for any µ ∈ L∗/L,

Wp

(s− 1

2,m, µ

)=Lp(s, χκm)

ζp(2s)bp(κm, s)char(Q(µ) + Zp)(m).

Here bp(κm, s) = bp(κm, s; 1) is given by (2.16).

Proof. When p = 2, L is unimodular, and the claim is covered by Corollary 4.5. Now assume p = 2 so

that L∗/L = 12Z2/Z2, and there are two choices µ = 0 or 1

2 .

When µ = 12 , one has, in the notation of Theorem 4.4, that K0(µ) = 1, dµ(k) = k, and tµ = m − κ

4 .

Theorem 4.4 gives

W2

(s− 1

2,m, µ

)=

[1 + ψ

(4m− κ

8

)X

]char(Z2)(tµ).

Write 4κm = dc2 such that dZ2 is the discriminant of Q2(√κm)/Q2. Then tµ ∈ Z2 implies ord2c = 0,

ψ

(4m− κ

8

)= (2, d)2 = χκm(2),

and b2(κm, s) = 1. Therefore

W2

(s− 1

2,m, µ

)=L2(s, χκm)

ζ2(2s)b2(κm, s)char(Q(µ) + Z2)(m).

The case µ = 0 is handled in the same way. One can also use [23, Proposition 13.4]. Indeed, let

L′ = {X ∈M2(Z2) : trX = 0}

with quadratic form Q(X) = −κdet(X). Then L′ is equivalent to L(1) = L⊕H and thus (Lemma 4.2)

Wm,2(s, char(L)) =Wm,2(s, char(L′)),

see [23, Proposition 13.4], then gives the answer.

5.2 The case of a quadratic extension

In this subsection, let k = Qp(√κ) be a quadratic extension of Qp with ordpκ = 0, 1, and L = Ok with

quadratic form Q(x) = ϵxx where ϵ ∈ Z×p . The associated quadratic character is χV = χκ, the quadratic

character of Q×p associated with k.


When k is unramified over Qp, L is unimodular and this case is covered by Corollary 4.5. Now we

assume that k is ramified so that L = Zp + Zp√κ ∼= Z2

p with quadratic form Q(x) = ϵx21 − ϵκx22. Then

L∗ ≃ 12Zp ⊕

12κZp, and [L∗ : L] is equal to p, 8, or 4 according to whether p = 2, p = 2|κ, or p = 2 - κ

respectively.

Proposition 5.2. Let the notation and assumption be as above. Let µ = µ1 + µ2√κ ∈ L∗/L, and

m ∈ Q×p .

(i) If m /∈ Q(µ) + Zp, then Wp(s,m, µ) = 0.

(ii) If m ∈ Q(µ) + Zp, let a = ordpm > −f , where f = ordp(dk/Qp). Then, for X = p−s,

Wp(s,m, µ) =

{1, if a = −f,1 + χκ(mϵ)X

a+f , if a > −f.

Proof. We verify the complicated case p = 2 and leave the case p = 2 to the reader. This amounts

to a tedious calculation using Theorem 4.4. We first assume κ = 2κ0 ∈ 2Z×2 so f = 3. In this case,

Q(x1+x2√κ) = ϵx21−2ϵκ0x

22, so l1 = 1, l2 = 2 in the notation of Theorem 4.4. Write µ = µ1+µ2

√κ ∈ L∗.

Case 1. We first assume µ ∈ L. In this case, we have K0(µ) = ∞,

dµ(k) =

1, if k = 1,

3

2, if k > 2,

and lµ(k) = 1. Furthermore, for m = m02a with m0 ∈ Z×

2 one has

νµ(m, k) = m023−k+a − ϵ+ ϵκ0 ∈ Z×

2 ⇔ k = a+ 3.

When k = a + 3, νµ(m, k) = m0 − ϵ + ϵκ0. On the other hand, ϵµ(a + 3) = ϵ or −κ0ϵ depending on

whether a is odd or even. Set o(a) = 0 or 1 according to whether a is odd or even. Then one has by

Theorem 4.4

W2(s,m, µ) = 1 + (2, ϵm0 − 1 + κ0)2(2,−κ0)o(a)2 Xa+3.

Here (a, b)2 is the local Hilbert symbol. Now the desired formula follows from the identity

χκ(mϵ) = (2,−κ0)o(a)2 (2, ϵm0 − 1 + κ0)2. (5.1)

Indeed, V represents m if and only if χκ(mϵ) = 1 by [17, Proposition 1.3]. If χκ(mϵ) = −1, then V does

not represent m and thus W2(0,m, µ) = 0, which forces

(2, ϵm0 − 1 + κ0)2(2,−κ0)o(a)2 = −1 = χκ(mϵ).

If χκ(mϵ) = 1, then mϵ ∈ Nk/Qpk×. So there is an element x1 + x2

√κ ∈ O×

k such that

mϵ = N((√κ)a(x1 + x2

√κ)) = 2a(−κ0)a(x21 − 2κ0x

22),

and so

m0ϵ = (−κ0)a(x21 − 2κ0x22) ≡ (−κ0)1−o(a)(1− 2κ0x

22) (mod 8).

So

(m0ϵ+ κ0 − 1)(−κ0)o(a) ≡ −1 + 2x22 (mod 8) ≡ ±1 (mod 8).

This gives

(2,m0ϵ+ κ0 − 1)2(2,−κ0)o(a)2 = 1 = χκ(mϵ).

This proves the claim.

Case 2. Next we assume that µ1 ∈ 12 + Z2 and µ2 ∈ 1

2Z2. In this case, K0 = 1, δµ(1) = 1, a = −2,

and m = 14m0. Furthermore, lµ(1) = 0 is even

ν = νµ(m, 1) = 4(m−Q′(µ)) ≡ m0 − 4Q(µ) (mod 8),


and, by Theorem 4.4, one has

W2(s,m, µ) = 1 + ψ

(m0 − 4Q(µ)

8

)X.

A simple calculation shows

ψ

(m0 − 4Q(µ)

8

)= (2,m0ϵ+ κ0 − 1)2(2,−κ0)2 = χκ(mϵ)

by (5.1).

Case 3. Now we assume that µ2 ∈ 12 + Z2 and µ1 ∈ Z2. In this case, K0 = 2, a = −1 and m = 1

2m0.

Furthermore,

δµ(k) =

{0, if k = 1,

1, if k > 1.

One also checks lµ(2) = 1, dµ(2) =32 , ϵµ(2) = ϵ, and

νµ(m, 2) = 2(m+ 2ϵκ0µ22)− ϵ ≡ m0 + κ0ϵ− ϵ (mod 8)

So Theorem 4.4 gives

W2(s,m, µ) = 1 + (2, ϵm0 + κ0 − 1)2X2.

Now (5.1) gives the desired result in this subcase.

Case 4. Finally we assume µ2 ∈ 14Z2 − 1

2Z2. In this case, K0 = 0, and thus W2(s,m, µ) = 1 as

claimed. This proves the case 2|κ, i.e, f = 3.

Now we assume κ ≡ −1 (mod 4) (still the case p = 2) with f = 2. In this case, Q(x1 + x2√κ) =

ϵx21 − ϵκx22 with l1 = l2 = 1. Let µ = µ1 + µ2√κ ∈ L∗. There are also several cases.

Case 1. First assume µ ∈ L. We have Hµ = {1, 2}, dµ(k) = 0 when k = 1 and dµ(k) = 1 when k > 1.

One has also lµ(k) ≡ 0 (mod 2), and

ϵµ(k) =

{−κ, if k > 1 even,

1, if k > 1 odd.

Finally,

νµ(m, k) = m023−k+a − ϵ+ ϵκ ∈ 4Z2

if and only if k = a+ 2. So Theorem 4.4 gives

W2(s,m, µ) = 1 + (2, ϵµ(a+ 2))2ψ

(2m0 − ϵ+ ϵκ

8

)Xa+2.

So the proposition in this case follows from the identity

(2, ϵµ(a+ 2))2ψ

(2m0 − ϵ+ ϵκ

8

)= χκ(mϵ). (5.2)

Just as in the proof of (5.1), χκ(mϵ) = −1 implies the left-hand side of (5.2) is also −1. Now assume

that χκ(mϵ) = 1. When a is even,

(2, ϵµ(a+ 2))2 = (2,−κ)2 = ψ

(1 + κ

8

)= ψ

(ϵ+ ϵκ

8

),

and so

(2, ϵµ(a+ 2))2ψ

(2m0 − ϵ+ ϵκ

8

)= ψ

(2m0 + 2ϵκ

8

)= ψ

(ϵm0 + κ

4

)= 1,


since m0ϵ is a norm from k. When a is odd, 2m0ϵ is a norm from k. Since ϖ = 1+√κ is a uniformizer,

one has

2m0ϵ = N(ϖ)(x21 − κx22) = (1− κ)(x21 − κx22) ≡ 1− κ (mod 8)

for some x1 + x2√κ ∈ O×

k . So

(2, ϵµ(a+ 2))2ψ

(2m0 − ϵ+ ϵκ

8

)= ψ

(2m0 − ϵ+ ϵκ

8

)= 1

in this case. This proves the claim and thus the proposition in this special case.

Case 2. Next we assume that µ1 ∈ 12 + Z2, µ2 ∈ Z2 or µ2 ∈ 1

2 + Z2, µ1 ∈ Z2. In this case, K0 = 1,

δµ(1) = 0, and a = ord2m = −2. So Theorem 4.4 asserts W2(s,m, µ) = 1 as claimed.

Case 3. Finally we assume that µ1, µ2 ∈ 12 + Z2. In this case, K0 = 1, dµ(1) = 1, and m = 1

2m0,

a = −1. Furthermore, ϵµ(1) = 1, and

νµ(m, 1) = 4(m−Q(µ)) = 2m0 − ϵ(2µ1)2 + ϵκ(2µ2)

2 ≡ 2m0 − ϵ+ ϵκ (mod 8).

So Theorem 4.4 and (5.2) assert W2(s,m, µ) = 1 + ψ( 2m0−ϵ+ϵκ8 )X = 1 + χκ(mϵ)X as claimed.

5.3 Quaternion algebras—the odd case

Let B be a quaternion algebra over Qp with maximal order OB . Let V be the quadratic space consisting

of trace zero elements in B with the quadratic form Q(x) = κdetx, where κ ∈ Z×p and detx is the

reduced norm of x. When B =M2(Qp), we take OB =M2(Zp), and for an integer e > 0

Le =

{A =

(b a

c −b

): a, b ∈ Zp, c ∈ peZp

}. (5.3)

We also write L = L0 when e = 0. When B is the division algebra, we take L = V ∩ OB , and write, as

in (2.15),

4κm = dc2, (5.4)

where dZp is the discriminant of Qp(√κm)/Qp, and we write kp = ordpc. Careful calculation using

Theorems 4.3 and 4.4 gives the following proposition. The special case µ = 0 is given in [23, Proposition

8.1].

Proposition 5.3. Let the notation be as above. Then, for µ ∈ L∗/L,

Wp

(s+

1

2,m, µ

)= char(Q(µ) + Zp)(m)Lp(s+ 1, χκm)

× bp(κm, s+ 1;B)

{ζ(2s+ 2)−1, if B is split ,

1, if B is ramified .

Here bp(κm, s;B) = bp(κm, s;D) is given as in (2.16) and (2.17) where D is square free with p|D if and

only if B is a division algebra.

Proof. (sketch) When B =M2(Qp) is split, L = L0 ⊕H, where L0 = Zp with quadratic form κx2, and

H = Z2p is the standard hyperplane. The result follows from Proposition 5.1 and Lemma 4.2. When B is

a division algebra, L ∼= Z3p with quadratic form

Q(x1, x2, x3) =

{κ(βx21 + pz22 − βpx22), if p = 2,

−κ(x21 + x22 + x23), if p = 2.

Here β ∈ Z×p with (p, β)p = −1. Now the claim follows from Theorems 4.3 and 4.4 and a careful

calculation.


Proposition 5.4. Let B =M2(Qp). Assume e > 0 and µ ∈ Le. Then

Wp

(s+

1

2,m, µ

)= char(Zp)(m)

×

1 + (1− p−1)

∑16l6kp

pc(2l)X2l − pc(2kp+2)−1X2kp+2 if p|d,

1 + (1− p−1)∑

16l6kppc(2l)X2l + χκm(p)pc(2kp+1)− 1

2X2kp+1 if p - d.

Here c(l) = min(l/2, e− l/2).

Now we deal with the case µ ∈ L∗e − Le. Write

µ =

(x1 x2

pex3 −x1

), x1 ∈ 1

2Zp, x2, x3 ∈ p−eZp.

Let tm = tµ(m) be defined as in Theorems 4.3 and 4.4. It is given as follows. Let Ke = min(e+ordpx2, e

+ ordpx3). When p = 2, µ /∈ L means Ke < e. In this case tµ(m) = m− κpex2x3. When p = 2, there are

three cases for µ /∈ L. Accordingly, we have

tµ(m) = m−

Q(µ), if x1 /∈ Z2,Ke < e,1

4κ, if x1 /∈ Z2,Ke > e,

κp−ex2x3, if x1 ∈ Z2,Ke < e.

We write

4κtµ(m) = dµc2µ, (5.5)

where dµZp is the discriminant of Qp(√κtµ(m))/Qp. Let χκm be the quadratic character of Q×

p associated

to Qp(√κtµ(m)). Let kp = ordpcµ.

Proposition 5.5. Let the notation be as above and assume that x1 ∈ Z2 if p = 2. Then, for X = p−s,

Wp(s+1

2,m, µ) = char(Zp)(tµ(m))

[1 + (1− p−1)

∑16l6min(kp,

Ke2 )

(pX2)l +Tail

]

with

Tail =

−pkpX2kp+2, if p|d, 2kp + 1 < Ke,

χκm(p)pkpX2kp+1, if p - d, 2kp < Ke,

0, otherwise.

Proof. (sketch) We check the case p = 2 and omit the case p = 2. It is easy to check, in the notation

of Theorem 4.3,

lµ(k) =

{1, if k 6 Ke odd,

0, if k 6 Ke even,dµ(k) =

1

2k for k 6 Ke, ϵµ(k) =

(κ

p

), if k 6 Ke odd,

1, if k 6 Ke even.

So

Wp(s+1

2,m, µ) = char(Q(µ) + Zp)(m)

×

1 + (1− p−1)∑

16l6Ke2

(pX2)l, if a > Ke,

1 + (1− p−1)∑

16l6 a2

(pX2)l − p12 (a−1)Xa+1, if a < Ke odd,

1 + (1− p−1)∑

16l6 a2

(pX2)l + χκm(p)p12aXa+1, if a < Ke even.


Here a = ordp(tµ(m)). It is easy to check that

a =

{2kp, if p - dµ,2kp + 1, if p|dµ.

Now the proposition is clear. 2

Proposition 5.6. Assume p = 2, and µ ∈ L∗e − Le with x1 ∈ 1

2Z2 − Z2. Then

W2(m, s+1

2, µ) = char(Z2)(tµ(m))

1, if Ke = 0,

1 + χκm(2)X, if Ke > e,

1 + ψ(1

2tµ(m))X, if 0 < Ke < e.

Proof. In this case, K0(µ) = min(1,Ke), and thus the case Ke = 0 is trivial by Theorem 4.4. When

Ke > 0, K0(µ) = 1, Theorem 4.4 gives

W2(m, s+1

2, µ) = char(Z2)(tµ)

(1 + ψ

(1

2tµ

)X

).

Here tµ = tµ(m). When Ke > e, we have tµ = m− κ4 ∈ Z2. So ψ(

12 tµ) = 1 if and only if

4m ≡ κ (mod 8), i.e. 4κm ≡ 1 (mod 8),

which is the same as χκm(2) = 1. 2

5.4 Quaternion algebras—the even case

Let B be a quaternion algebra over Qp, and let V = B with the quadratic form Q(x) = detx, the reduced

norm of x. The associated quadratic character χV is trivial.

We first consider the case where B is the division algebra over Qp. We take L = OB to be the

unique maximal order of B. When p = 2, choose β ∈ Z×p such that (β, p)p = −1, then L ∼= Z4

p with

quadratic form Q(x) = x21 − βx22 − px23 + pβx24. When p = 2, L ∼= Z42 with quadratic form Q(x) =

x21 + x1x2 + x22 + 2(x23 + x3x4 + x24).

Applying Theorems 4.3 and 4.4, we obtain

Proposition 5.7. Let B be a division algebra over Qp and let L be as above. For µ ∈ L∗/L and

m ∈ Q(µ) + Zp,

Wp(s,m, µ) =

1, if µ /∈ L,

1− pX + pXa+1 −Xa+2

1−X, if µ ∈ L.

Here a = ordpm. In particular,

Wp(s, 0, µ) =ζp(s)

ζp(s− 1)

{0, if µ /∈ L,

1, if µ ∈ L.

When B =M2(Qp), we will consider the lattices for an integer e > 0

Le =

{X =

(a b

pec d

): a, b, c, d ∈ Zp

}.

In this case, L∗e/Le

∼= (Z/peZ)2. Notice that L0 is unimodular, and the associated Whittaker function is

covered in Corollary 4.5.

Proposition 5.8. (i) When µ ∈ Le,

Wp(s+ 1,m, µ) = char(Zp)(m)

(1 +

(1− 1

p

) ∑16k6kp

pd(k)Xk − pd(kp+1)−1Xkp+1

)


where kp = ordpm, d(k) = min(e− k, 0).

(ii) When µ = (µ1 µ2

peµ3 µ4) /∈ Le, let Ke = min(e+ ordpµ2, e+ ordpµ3). Then

Wp(s+ 1,m, µ) = char(Zp)(tµ)(1 +

(1− 1

p

) ∑16k6min(kp,Ke)

Xk +Tail

),

where

tµ = tµ(m) = m+ peµ2µ3, kp = ordp(tµ),

and

Tail =

0 if kp > Ke,

−1

pXkp+1 if kp < Ke.

Proof. (sketch) We deal with the case p = 2 and omit the case p = 2.

When µ ∈ Le, Theorem 4.4 gives

W2(s+ 1,m, µ) =

(char(Z2)(m) +

∑16k

2d(k)−1ψ(m2−k)char(2k−1Z2)(m)Xk

)

= char(Z2)(m)

(1 +

(1− 1

2

) ∑16k6kp

2d(k)Xk − 2d(kp+1)−1Xkp+1

)

as claimed. When µ /∈ Le, Theorem 4.4 gives

W2(s+ 1,m, µ) = char(Z2)(tµ)

(1 +

1

2

∑16k6K0

ψ(tµ2−k)char(2k−1Z2)(tµ)X

k

)

= char(Z2)(tµ)

(1 +

(1− 1

2

) ∑16k6min(kp,Ke)

Xk +Tail

)

as claimed. 2

Corollary 5.9. Let the notation be as in Proposition 5.8 with e = 1. Then

Wp(s, 0, µ) =

1 + (p− 2)X

(1−X), if µ ∈ L1,

1, if µ /∈ L1, Q(µ) ∈ Zp,0, if Q(µ) /∈ Zp.

6 Cohen and Zagier Eisenstein series

Given l ∈ 12Z and a quadratic character χ satisfying (2.10), there is a systematic way to construct

Eisenstein series using global quadratic lattices as follows. Let L be a quadratic integral Z-lattice of

even or odd dimension depending on whether 2l is even or odd such that χL = χ. We do not require

dimL = 2l in general. For µ ∈ L∗/L, let ϕµ = char(µ+ L) =∏p<∞ ϕµ,p and let Φµ(s) be the associated

standard section in If (s, χ) =⊗′

p<∞ Ip(s, χ). This way we obtain an Eisenstein series

E(τ, s,Φl,µ) = E(τ, s,Φl∞ ⊗ Φµ). (6.1)

Using the results in previous sections, we can give an explicit formula for the Eisenstein series for any

given lattice. We will give some examples in the next few sections. When dimL = 2l and L is of signature

(2l− 2, 2), Φl∞ also comes from S(V∞) where V = L⊗R. In this case, the first author (see [18], see also

Section 10) has used the first derivative of the Eisenstein series to compute certain integral of Borcherds

modular forms which are of interest in Arakelov theory.


When l is an even integer, L = Hr, r > 0, and µ = 0, the Eisenstein series (6.1) are the classical

Eisenstein series discussed in Section 3.

In this section, we assume l ∈ 12 + Z, and κ = ±1 such that (2.10) holds, i.e.,

l1 :=1

2

(l − κ

2

)∈ Z, (6.2)

First we take L = Z with quadratic form Q(x) = κx2. Then L∗/L = 12Z/Z is of order two with µ = 0, 12 .

Theorem 6.1. Assume that l ∈ 12 + Z>0 and κ = ±1 satisfy (6.2), and let L = Z with quadratic form

κx2. Let µ = 0, 12 ∈ L∗/L. Then

E(τ, s,Φl,µ) =∑

m∈Q(µ)+Z

Am(v, s; l, µ) qm,

where Am(v, s) = Am(v, s; l, µ) is given as follows for m ∈ Q(µ) + Z. Let

α =1

2(1 + s+ l), β =

1

2(1 + s− l)

be as in Proposition 2.3 and let c1(l) =√2π(−1)l1 .

(i) If m > 0, then

Am(v, s) = c1(l)vβ(2πm)s

L(s+ 12 , χκm)b(κm, s+ 1

2 )

ζ(2s+ 1)

Ψ−l(s, 4πmv)

Γ(α).

Here b(κm, s) =∏bp(κm, s) is given in (2.16), and χκm is the quadratic Dirichlet character associated

to Q(√κm), as in the notation section.

(ii) If m < 0, then

Am(v, s) = c1(l)vβ(2π|m|)s

L(s+ 12 , χκm)b(κm, s+ 1

2 )

ζ(2s+ 1)

Ψl(s, 4π|m|v)Γ(β)e4π|m|v .

(iii) When µ = 12 , the constant term is A0(v, s) = 0.

(iv) When µ = 0, the constant term is

A0(v, s) = v1−l+s

2 + v1−l−s

2 c1(l)2−sΓ(s)ζ(2s)

ζ(2s+ 1)Γ(α)Γ(β).

Proof. We sketch the proof of the case m > 0 and leave the other cases to the reader. By (2.6),

Propositions 2.3 and 5.1, one has

Em(τ, s,Φl,µ) =Wm,∞(τ, s,Φl∞)∏p<∞

γ(Vp)|2κ|12pWp(s,m, µ)

=√2π(−i)l

( ∏p<∞

γ(Vp)

)(2πm)svβ

L(s+ 12 , χκm)b(κm, s)

ζ(2s+ 1)

Ψ−l(s, 4πmv)

Γ(α)qm

× char(Q(µ) + Z)(m).

So it suffices to verify the identity

(−i)l∏p<∞

γ(Vp) = (−1)l1 .

This follows from the general fact that∏p6∞

γ(Vp) = 1 and γ(V∞) = (−i)κ2 .

The following corollary follows from Theorem 6.1 and the functional equations of L-function, bp-

function (see (2.15)), and Ψn (see (2.21)).


Corollary 6.2. Set

Gl(s) = Λ(2s+ 1)

l1−1∏j=0

(1 +

κ

2+ 2j + s

).

Then the normalized Eisenstein series

E(τ, s,Φl,µ) = Gl(s)E(τ, s,Φl,µ)

has the simple functional equation

E(τ, s,Φl,µ) = E(τ,−s,Φl,µ).

Taking l = 12 and s = l − 1 = −1

2 in Theorem 6.1, we get immediately

Proposition 6.3. For µ = 0, 12 E(τ,− 12 ,Φ

12 ,µ) =

∑n∈µ+Z q

n2

is the classical theta functions.

Proof. When m > 0, χκm is a real quadratic character and thus

L(0, χκm) =

{0, if d = 1,

ζ(0), if d = 1.

So, for m ∈ Q(µ) + Z, Theorem 6.1 gives

Am

(v,−1

2

)=

√2π(2πm)−

12L(0, χd) b(−m, 0)

1

Γ(12 )=

{0, if d = 1,

2, if d = 1.

When m < 0, 1Γ(β) = 0 at s = −1

2 , and thus Am(v,−12 ) = 0. Finally when m = 0, the same reasoning

gives

A0

(v,−1

2

)=

1, if µ = 0,

0, if µ =1

2.

2

By this proposition, the next result can be viewed as giving the ‘derivative’ of theta functions. We

need some additional notation.

Let H(N) be the Hurwitz class number of integral binary quadratic forms of discriminant −N for an

integer N > 0. It is well-known that (see [15, p. 69])

H(N) = L(0, χ−N )∑n|c

n∏p|n

(1− χ−N (p)p−1),

where −N = dc2 with d being the fundamental discriminant of Q(√−N). One usually defines H(0) =

ζ(0). We extend the definition of H(N) to N < 0 by letting

H(N) = L′(0, χ−N )∑n|c

n∏p|n

(1− χ−N (p)p−1).

In particular, when −N is a fundamental discriminant, i.e., when c = 1, one has

H(N) =

2h(N)

w(N), if N > 0,

h(N) log ϵ(N)

w(N), if N < 0,

− 1

12, if N = 0.

Here h(N), w(N), and ϵ(N) are the class number, the number of roots of unity, and the fundamental

unit (of value larger than 1) of Q(√−N).


Proposition 6.4. Let the notation be as above, and µ = 0, 12 . Then,

E′(τ,−1

2,Φ

12 ,µ) =

(1

2log v − π

3

√v

)δ0,µ +

∑n∈µ+Zn>0

an(v) qn2

− 2∑

m∈µ2+Zm>0

m not a square

H(−4m)√m

qm

−∑

m∈−µ2+Zm>0

H(4m)√m

q−m∫ ∞

1

e−4πmvr(r − 1)−12 dr,

where

an(v) = γ + log 2 + 2 log π + 2∑p|2n

(pkp − 1

pkp(p− 1)− kp

)log p+ J

(− 1

2, 4πn2v

)with kp = ordp(2n).

When l = 32 and κ = −1, Theorem 6.1 gives the well-known Zagier’s Eisenstein series and its ‘deriva-

tive’.

Proposition 6.5. Let the notation be as above. For µ = 0, 12 ,

E

(τ,

1

2,Φ

32 ,µ

)= δ0,µ − 12

∑m∈−µ2+Z>0

H(4m)qm − 3

2π√v

∑n∈µ+Z

∫ ∞

1

e−4πn2vrr−32 drq−n

2

.

In particular,

ζ(−1)

(E

(4τ,

1

2,Φ

32 ,0

)+ E

(4τ,

1

2,Φ

32 ,

12

))= − 1

12+

∞∑m=1

H(m)qm +1

16π√v

∑n∈Z

∫ ∞

1

e−4πn2vrr−32 drq−n

2

is Zagier’s Eisenstein series in [34].

Proof. This follows from Theorem 6.1 by a simple calculation. We only check the special case

m = −n2 ∈ −µ2 + Z. In this case, −4m = (2n)2, d = 1 and L(s, χ−m) = ζ(s). So, by Theorem 6.1, one

has

Em(τ, s,Φ32 ,µ) = − 2π√

2vβ(2πn2)s

b(−m, s+ 12 )

ζ(2s+ 1)Ψ 3

2(s, 4πn2v)

ζ(s+ 12 )

Γ(β)q−n

2

.

Therefore,

Em

(τ,

1

2,Φ

32 ,µ

)= − 2π√

2(2πn2)

12b(−m, 1)ζ(2)

Ψ 32

(1

2, 4πn2v

)1

2q−n

2

= − 3

π√v

∫ ∞

1

e−4πn2vrr−32 dr q−n

2

.

Similarly, one can check that

E0(τ,1

2,Φ

32 ,µ) =

(1− 3

π√v

)δ0,µ.

Notice that

1 =1

2

∫ ∞

1

r−32 dr.

One sees that

δ0,µ +∑

n∈µ+Z,n>0

∫ ∞

1


2

=1

2

∑n∈µ+Z

∫ ∞

1


2

.


Now the first part of the proposition is clear. For the relation with Zagier’s Eisenstein series, it suffices

to note that H(m) = 0 unless m ≡ 0,−1 (mod 4). 2

Theorem 6.6. Let the notation be as above. Write −m = dc2.

(i) The derivative of the constant term is

E′0

(τ,

1

2,Φ

32 ,µ

)= δ0,µ

(1

2log v − 3

π√v

(1

2log v − γ − log 8π +

ζ ′(−1)

ζ(−1)

)).

(ii) When m > 0 and m ∈ −µ2 + Z,

E′m

(τ,

1

2,Φ

32 ,µ

)= −12H(4m) qm

(1 + 2

ζ ′(−1)

ζ(−1)− 1

2γ − 1

2log π − log d

− L′(0, χ−m)

L(0, χ−m)−∑p|c

(b′p(−m, 0)bp(−m, 0)

− 2kp(c) log p

)+

1

2J

(1

2, 4πmv

)).

Here kp(c) = ordpc.

(iii) When m < 0, m ∈ −µ2 + Z, and −m is not a square,

E′m

(τ,

1

2,Φ

32 ,µ

)= − 3

π√v

H(4m)√|m|

∫ ∞

1

e4πmvrr−32 drqm.

(iv) When m = −c2 ∈ −µ2 + Z,

E′−n2(τ,

12 ,Φ

32 ,µ)

E−n2(τ, 12 ,Φ32 ,µ)

=1

2log v + 2− 2 log 2 +

1

2log 2π − 1

2γ + 2

ζ ′(−1)

ζ(−1)−∑p|2c

b′p(−m, 0)bp(−m, 0)

+Ψ′

32

Ψ 32

.

For an integer r > 1 and positive integer N , write (−1)rN = dc2 such that d is the fundamental

discriminant of Q(√(−1)rN). Cohen defined

H(r,N) = L(1− r, χ−N )∑n|c

n2r−1∏p|n

(1− χ(−1)rN (p)p−r). (6.3)

He further let H(r, 0) = ζ(1− 2r). Notice that H(1, N) = H(N). He proved that [11, Theorem 3.1]

Hr(τ) =∑N>0

H(r,N) qN (6.4)

is a modular form of weight r + 12 for Γ0(4) for r > 2—the so-called Cohen’s Eisenstein series. This

Eisenstein series can be derived from Theorem 6.1 as follows.

Proposition 6.7. Let the notation be as in Theorem 6.1 with l = r + 12 and κ = (−1)r. Then

Hr(τ) = ζ(1− 2r)

(E

(τ, r − 1

2,Φr+

12 ,0

)+ E

(τ, r − 1

2,Φr+

12 ,

12

)).

Proof. This follows from a simple calculation with Theorem 6.1. 2

7 An incoherent Eisenstein series of weight 1

Let k = Q(√D) be an imaginary quadratic field with fundamental discriminant D, ring of integer Ok,

and different ∂. Let χ be the quadratic Dirichlet character associated to k. Let a be a fractional ideal

and let L = a with integral quadratic form Q(x) = −N(x)N(a) . Then L∗ = ∂−1a. Let V = L ⊗Z Q, so that

V∞ has signature (0, 2). Notice that Φ1∞(s) is associated to a quadratic space over R of signature (2, 0).

Thus, the Eisenstein series E(τ, s,Φ1,µ) of weight 1 associated to µ ∈ L∗/L is an incoherent Eisenstein

series in the sense of [17], and is automatically zero at s = 0. Its derivative at s = 0 is very important and


was calculated and used in [22] for µ = 0 and in [30] for general µ. It is also used in [8]. The analogous

series for a CM number field was used in [9].

Let

Λ(s, χ) = |D| s2π− s+12 Γ

(s+ 1

2

)L(s, χ) (7.1)

be the complete L-function of χ.

Proposition 7.1. Let the notation be as above. For m ∈ Q(µ) + Z, define o(D) to be the number of

primes p|D such that ordp(mD) > 0. Let

ρ(m, s) =∏p-D

σ−s,p(m,χ),

and

ρD(m, s) =∏p|D

1

2(1 + χp(−mN(a))|mD|sp).

Then

E(τ, s,Φ1,µ) =∑

m∈Q(µ)+Z

Am(v, s, µ) qm

where Am(v, s, µ) is given as follows.

(i) When m > 0,

Λ(s+ 1, χ)Am(v, s, µ) = −2o(m)+1(2m√|D|πv)sρ(m, s)ρD(m, s)Ψ−1(s, 4πm, v).

(ii) When m < 0,

Λ(s+ 1, χ)Am(v, s, µ) = −2o(m)s(2m√|D|πv)sρ(m, s)ρD(m, s)Ψ1(s, 4πmv)e

−4π|m|v.

(iii) When m = 0, the constant term is

Λ(s+ 1, χ)Am(v, s, µ) = δ0,µ(v

s2Λ(−s, χ)− v−

s2Λ(s, χ)

).

Here δ0,µ = 1 or 0 depending on whether µ ∈ L or not.

Proof. This follows from Propositions 2.1, 2.3, and 5.2 and the fact that

(−i)∏p<∞

γ(Vp) = −∏p6∞

γ(Vp) = −1. 2

To see that E(τ, 0,Φ1,µ) = 0 and to calculate the central derivative, notice first that

ρ(m, 0) = ρ(|mD|) = |{b ⊂ Ok : N(b) = |mD|}|.

For m > 0, let Diff(m) be the set of primes p <∞ such that χp(−mN(a)) = −1. For m < 0, let Diff(m)

be the set of such finite primes together with ∞. It is easy to see that Diff(m) is a finite set of odd

cardinality. It is also easy to see that if p ∈ Diff(m) is finite and unramified in k, then p is inert in k,σ−s,p(m,χ) is zero at s = 0, and

ρ′(m, 0) =1

2(ordp(m) + 1)ρ(|mD|/p) log p.

If p ∈ Diff(m) is finite and ramified in k, then ρD(m, 0) = 0 and

ρ′D(m, 0) =1

2(ordp(mD))

∏q|D,q =p

1

2(1 + χq(−mN(a))) log p.

On the other hand, when m < 0 one has

Ψ1(0, 4π|m|v) = e−4π|m|v∫ ∞

1

e−4π|m|vrr−1 dr.


Now a simple calculation using Proposition 7.1 gives the following theorem, which is essentially [30,

Theorem 4.1]. It was also used in [8] and [25].

Theorem 7.2. Let the notation be as in Proposition 7.1. Then E(τ, 0,Φ1,µ) = 0. Assume that

m ∈ Q(µ) + Z. Then A′m(v, 0, µ) = 0 unless |Diff(m)| = 1. Assume Diff(m) = {p}. Then

(i) If m > 0 and p inert in k, then

Λ(0, χ)A′m(v, 0, µ) = −2o(m) (ordp(m) + 1) ρ(|mD|/p) log p,

and it is independent of v.

(ii) If m > 0 and p is ramified in k, then

Λ(0, χ)A′m(v, 0, µ) = −2o(m) ordp(mD) ρ(|mD|) log p,

and it is independent of v.

(iii) If m < 0, then

Λ(0, χ)A′m(v, 0, µ) = −2o(m)ρ(|mD|)

∫ ∞

1

e−4π|m|vrr−1 dr.

(iv)

A′0(v, 0, µ) = δ0,µ

(log v − 2

Λ′(0, χ)

Λ(0, χ)

).

8 Quaternion algebras—ternary forms

Let B be a quaternion algebra over Q and let D = D(B) be the product of finite primes at which B is

ramified, i.e., D(B) is the reduced discriminant of B. Let OB be a maximal order of B.

Let l > 3/2 be a half-integer and κ = ±1 such that (2.10) is satisfied, i.e., l1 = 12 (l −

κ2 ) is an integer.

Let

L = {x ∈ OB : trx = 0}, Q(x) = −κdetx

where detx is the reduced norm of x, and let V = L ⊗ Q. Then |L∗/L| = 2D2. In this section, we will

compute the Eisenstein series E(τ, s; l,D, µ) = E(τ, s,Φl,µ) for all µ ∈ L∗/L. When µ = 0, and l = 32 ,

the Eisenstein series is studied in detail in [23] and its derivative at s = 1/2 is shown to be the generating

functions of Faltings’s heights of certain arithmetic CM cycles. The hope of extending this result to all

µ ∈ L∗/L is one important motivation for the present paper.

Theorem 8.1. Let the notation be as above, and set α = 12 (1 + l + s), β = 1

2 (1 − l + s), and

C1(l,D) = (−1)l1+o(D) 2πD , where o(D) is the number of prime factors of D.

(i) When m > 0 and m ∈ Q(µ) + Z, the m-th Fourier coefficient is

Em(τ, s; l,D, µ) =C1(l,D)√

2

L( 12 + s, χκm)b(κm, s+ 12 ;D)

ζD(2s+ 1)(2πm)svβ

Ψ−l(s, 4πmv)

Γ(α)qm.

Here b(κm, s;D) =∏p bp(κm, s;D) and Ψn is given by (2.10).

(ii) When m < 0 and m ∈ Q(µ) + Z,

Em(τ, s; l,D, µ) =C1(l,D)√

2

L( 12 + s, χκm)b(κm, s+ 12 ;D)

ζD(2s+ 1)(2π|m|)svβΨl(s, 4π|m|v)

Γ(β)e4π|m|v qm.

(iii) The constant term is

E0(τ, s; l,D, µ) = δ0,µ

(vβ +

C1(l,D)√2

vβ−s2−sΓ(s)ζ(2s)

∏p|D(1− p1−2s)

Γ(α)Γ(β)ζD(2s+ 1)

).


Proof. It follows from (2.6), Propositions 2.3, and 5.2, and the fact that

(−i)l∏p<∞

γ(Vp) = (−1)l1+o(D),

since ∏p6∞

γ(Vp) = 1,

and

γ(V∞) =

{(−i)−κ

2 , if B∞ =M2(R),−(−i)−κ

2 , if B∞ is a division algebra.2

The functional equations for L-functions, bp, and Ψn, together with Theorem 8.1, yield:

Corollary 8.2. Set

G(s; l,D) = π−s− 12Γ

(s+

1

2

)ζD(2s+ 1)

l1−1∏j=0

(1 +

κ

2+ 2j + s

).

Then the normalized Eisenstein series

E(τ, s,Φl,µ) = G(s; l,D)E(τ, s,Φl,µ)

has the simple functional equation

E(τ, s,Φl,µ) = E(τ,−s,Φl,µ).

For computational purposes, the following variant, which is a consequence of Theorem 8.1, the func-

tional equation for L(s, χκm), and (2.18), is useful.

Theorem 8.3. Let the notation be as above.

(i) For m > 0 and m ∈ Q(µ) + Z,

Em(τ, s; l,D, µ) =vβ∏

p|D(1− p1+2s)

× Γ(β − s)

Γ(−s)L( 12 − s, χκm)b(κm, 12 − s;D)

ζ(−2s)Ψ−l(s, 4πmv) q

m.

(ii) For m < 0 and m ∈ Q(µ) + Z,

Em(τ, s; l,D, µ) =vβ∏

p|D(1− p1+2s)

Γ( 12 (1 + l − s))Γ(12 (32 − l + s))

Γ(−s)Γ(β + 1)

×L( 12 − s, χκm)b(κm, 12 − s;D)

ζ(−2s)

Ψl(s, 4π|m|v)e4π|m|v qm.

Corollary 8.4. The values at s = l − 1 of the Eisenstein series E(τ, s; l,D, µ) as µ runs over L∗/L

are the components of a holomorphic vector-valued with rational Fourier coefficients form for SL2(Z) of

weight l. Explicitly,

E(τ, s; l,D, µ) = δ0,µ +1∏

p|D(1− p2l−1)ζ(2− 2l)

∑m>0

m∈Q(µ)+Z

H(l,m;D) qm.

Here

H(l,m;D) = L

(3

2− l, χκm

)b

(κm,

3

2− l;D

)(8.1)

is the analogue of Cohen’s general class number H(r,N) in (6.3). Indeed, H(l,m, ; 1) = H(l − 12 , 4m).


Theorem 8.5. Assume that l > 52 , and µ ∈ L∗/L.

(i) For m > 0 and m ∈ Q(µ) + Z,

E′m(τ, l − 1; l,D, µ)

Em(τ, l − 1; l,D, µ)=− 1

2log π|d|+ 1

2

Γ′(1− l)

Γ(1− l)+ 2

ζ ′(2− 2l)

ζ(2− 2l)−L′( 32 − l, χκm)

L( 32 − l, χκm)

+∑p|cD

[2p2l−1

1− p2l−1kp(D)− kp(c)−

1

log p

b′p(κm,32 − l;D)

bp(κm,32 − l;D)

]log p

+1

2J(l − 1, 4πmv).

Here c, d are given by 4κm = dc2 (see (2.15)), kp(c) = ordpc, and J(n, t) is given in Lemma 2.2.


E′m(τ, l − 1; l,D, µ) = − (4πv)1−l

Γ(1− l)ζ(2− 2l)|m|1−lH(l,m;D)

∫ ∞

1

e4πmvrr−ldrqm.

Here

H(l,m;D) = L′(3

2− l, χκm

)b

(κm,

3

2− l;D

)is the analogue of (8.1) when L( 32 − l, χκm) = 0.

(iii) Finally, the constant term is given by

E′0(τ, l − 1; l,D, µ) =

1

2log v − (−1)l1+o(D)2

52−lv1−l

D

ζ ′(3− 2l)

ζ(2− 2l)

∏p|D

1− p3−2l

1− p1−2l.

Proof. We first claim that, for m = 0, L( 32 − l, χκm) = 0 if and only if m < 0. Indeed, 32 − l =

3−κ2 −2l1

is a negative integer. First assume that m < 0. In this case 32 − l is odd if and only if κ = 1, which

is equivalent to that Q(√κm) is imaginary. On the other hand, it is well known, by the functional

equation, that L(s, χκm) vanishes at odd negative integers when Q(√κm) is imaginary and vanishes at

even negative integers when Q(√κm) is real. So one has always L( 32 − l, χκm) = 0 if m < 0. The same

argument shows that L( 32 − l, χκm) = 0 when m > 0. Now both (i) and (ii) follow from Theorem 8.3

and Lemma 2.2. For (iii), notice that 1Γ(β) vanishes at s = l − 1 and has derivative 1

2 at s = l − 1. So

Theorem 8.1 gives

E′0(τ, l − 1; l,D, µ) =

1

2log v +

(−1)l1+o(D)232−lv1−lπ

D

Γ(l − 1)ζ(2l − 2)

Γ(l)ζ(2l − 1)

∏p|D

1− p3−2l

1− p1−2l.

Now the functional equation for the zeta function gives

Γ(l − 1)

ζ(2l − 2)Γ(l)ζ(2l − 1) = − 2

π

ζ ′(3− 2l)

ζ(2− 2l).

This proves (iii). 2

The case where l = 32 , o(D) > 1 is even, and µ = 0, is dealt with in [23, Theorem 8.8], and can be

extended to the case l = 32 , o(D) > 1 and any µ without any difficulty. Here is the result in our notation.

It can also easily be derived from Theorem 8.3.

Theorem 8.6. Assume that l = 32 , o(D) > 1, and µ ∈ L∗/L. Assume further that µ ∈ Q(µ) + Z.

(i) When m > 0 and there is no prime p|D for which χκm(p) = 1,

E′m(τ, 12 ;

32 , D, µ)

Em(τ, 12 ;32 , D, µ)

=− 1

2log π|d|+ 1

2

Γ′(− 12 )

Γ(−12 )

+ 2ζ ′(−1)

ζ(−1)− L′(0, χκm)

L(0, χκm)

+∑p|cD

[2p2

1− p2kp(D)− kp(c)−

1

log p

b′p(κm, 0;D)

bp(κm, 0;D)

]log p+

1

2J

(1

2, 4πmv

).


Here 4κm = dc2 as in (2.15) and kp(c) = ordpc.

(ii) When m > 0 and there is a unique prime p|D for which χκm(p) = 1,

E′m

(τ,

1

2;3

2, D, µ

)= − 24∏

q|D(1− q)δ

(d,D

p

)H0(m,D)

1 + p− pkp

1 + plog p,

where

δ(d,D) =∏q|D

(1− χκm(q)), and H0(m,D) =1

2L(0, χκm)

∏q-D

bq(κm, 0;D),

are given by [23, (8.19)–(8.20)].

(iii) When m < 0 and m ∈ Q(µ) + Z,

E′m

(τ,

1

2;3

2, D, µ

)= − 3

π√v

H( 32 ,m;D)√|m|

∫ ∞

1

e4πmvrr−32 dr qm.

(iv) Finally, when µ = 0, the constant term is given by E′0(τ,

12 ;

32 , D, µ) =

12 log v.

The case l = 32 and D = 1 is the same as Zagier’s Eisenstein series and its derivative as discussed

in Proposition 6.5 and Theorem 6.6. Finally, when l = 32 , and D = p is a prime number, i.e., B is the

quaternion algebra ramified exactly at p and ∞, Theorem 8.6 still holds with two modifications as follows.

1. When −m = n2 is a square, in this case, L(s, χκm) = ζ(s) does not vanish at s = 0, but bp(n, s;D)

does and

b′p(κm, 0;D) = (1 + p− pkp) log p.

So in this case

E′−n2

(τ,

1

2;3

2, D, µ

)= −1 + p− pkp

1− p23

π√v

∫ ∞

1

e4πmvrr−32 dr q−n

2

.

2. The constant term for µ = 0 is

E′0

(τ,

1

2;3

2, D, 0

)=

1

2log v +

6

π√v

1

p− p−1.

9 Quaternion algebras—quaternary forms

In this section, let B be a quaternion algebra over Q with reduced discriminant D = D(B), and let x 7→ x

be the main involution in B. Let R be an Eichler order of B of level N with (N,D) = 1. Let V = B

with the quadratic form Q(x) = detx, the reduced norm of x, and let L = R. Then

L∗ = {y ∈ B : tr(xy) ∈ Z for all x ∈ R}

and |L∗/L| = D2N2. For µ ∈ L∗/L, let ϕµ = char(µ + L) =∏p<∞ char(µ + Lp). For a positive even

integer l, we are interested in computing the Eisenstein series E(τ, s; l, µ,R) = E(τ, s,Φl,µ). For simplicity,

we will assume that N is square free. For p such that for µ ∈ Rp, let

Bp(s;m,µ,R) = (1−X)−1

1−Xkp+1, if p - DN,1− pX + pXkp+1 −Xkp+2, if p|D,1− 2X + pX − pXkp+1 +Xkp+2, if p|N.

(9.1)

Here kp = ordpm and X = p−s. For p with µ /∈ Rp, we set Bp(s;m,µ,R) = 1. It is easy to check that

Bp(s;m,µ,R) satisfies the following simple functional equation for p - N and kp > −1

Bp(s;m,µ,R) = |mD|sBp(−s;m,µ,R). (9.2)

Theorem 9.1. Let α = 12 (1 + l + s), and β = 1

2 (1− l + s). Then the Fourier expansion of

E(τ, s; l, µ,R) =∑

m∈Q(µ)+Z

Am(τ, s; l, µ,R) qm


is given as follows.

(i) For m > 0 and m ∈ Q(µ) + Z,Am(τ, s; l, µ,R) = C+(s, l,D)msvβB(m, s;µ,R)Ψ−l(s, 4πmv),

where

C+(s, l,D) = (−1)o(D)+ l2

(2π)s+1

DNζDN (s+ 1)Γ(α)

= (−1)o(D)

( ∏06j< l

2

s− 2j

s+ 2j + 1

)π

12 2s+1

Γ( l−s2 )ζ(−s)∏p|DN (p− p−s)

.


Am(τ, s; l, µ,R) = C−(s, l,D)|m|svβB(m, s;µ,R)Ψl(s, 4π|m|v)Γ(β)e4π|m|v ,

where

C−(s, l,D) = (−1)o(D)+ l2

(2π)s+1

DNζDN (s+ 1)= (−1)o(D)+ l

2

√π2s+1Γ( s+1

2 )

Γ(− s2 )ζ(−s)

∏p|DN (p− p−s)

.

(iii) For Q(µ) ∈ Z, the constant term is

A0(τ, s; l, µ,R) = vβδ0,µ + (−1)o(D)+ l2 vβ−s

2π

DN

×∏p|D

1− p1−s

1− p−1−s

∏p|N,µ∈Rp

1 + p1−s − 2p−s

1− p−1−s

∏p|N,µ/∈Rp

1− p−s

1− p−1−s2−sΓ(s)ζ(s)

Γ(α)Γ(β)ζ(s+ 1).

(iv) For Q(µ) /∈ Z, the constant term is 0.

Proof. First, for m = 0, we have by (4.4), and Propositions 5.7 and 5.8,

Wm,p(s, ϕµ) = γ(Vp)|DN |p char(Q(µ) + Zp)(m)Bp(m, s;µ,R)

ζp(s+ 1)−1, if p - DN,−1, if p|D,1, if p|N.

One can check that∏p<∞ γ(Vp) = γ(V∞) = (−1)o(D). Thus, for m = 0,

Em(τ, s; l, µ,R) =(−1)o(D)

DN

B(m, s;µ,R)

ζDN (s+ 1)Wm,∞(τ, s; Φl∞)char(Q(µ) + Z)(m).

Now applying Proposition 2.2, one proves claims (i) and (ii) with the first formula for C±(s, l,D). The

second formula for C± follows from the first one, the functional equation of ζ(s) and the basic properties

of Γ(s). To see (iii) and (iv), recall from Proposition 5.7 and Corollary 5.9 that

W0,p(s, ϕµ) = γ(Vp)|DN |pζp(s)

ζp(s+ 1)

1, if p - DN,

− 1− p1−s

1− p−1−s , if p|D,µ ∈ Rp,

1 + p1−s − 2p−s

1− p−1−s , if p|N,µ ∈ Rp,

1− p−s

1− p−1−s , if p|N,µ /∈ Rp, Q(µ) ∈ Zp,

0, otherwise.

We notice that, for p|D, µ ∈ Rp if and only if Q(µ) ∈ Zp. We also recall from Proposition 2.2 that

W0,∞(τ, s,Φl∞) = 2π(−1)l2 vβ−s

2−sΓ(s)

Γ(α)Γ(β).


Now (iii) and (iv) follow from the formula

E0(τ, s; l, µ,R) = vβδ0,,µ +W0,∞(τ, s,Φl∞)∏p<∞

W0,p(s, ϕµ). 2

Corollary 9.2. Let the notation be as above.

(i) When l > 2 or D > 1, the special value at s = l− 1 is a holomorphic modular form of weight l with

rational Fourier coefficients and is given by

E(τ, l − 1; l, µ,R) = δ0,µ + C+(l,D)∑

m∈Q(µ)+Z,m>0

ml−1B(m, l − 1;µ,R) qm,

where

C+(l,D) = (−1)o(D)

( ∏06j< l

2

l − 2j − 1

l + 2j

)1

DN∏p|DN (1− p−l)

2l

ζ(1− l).

(ii) When l = 2 and D = 1, so that B =M2(Q),

E(τ, 1; 2, µ,R) = δ0,µ − 1

ζ(−1)∏p|N (p− p−1)

∑m∈Q(µ)+Z,m>0

mB(m, 1;µ,R) qm − 3

πvδ(µ,N),

where

δ(µ,N) =

0, if Q(µ) /∈ Z,2|{p|N,µ∈Rp}|

∏p|N

(p+ 1)−1, if Q(µ) ∈ Z.

Proof. Noting that C+(l,D) = C+(1, l, D) and Ψ−l(l − 1, z) = 1, the case m > 0 follows directly from

Theorem 9.1(i). When m < 0, Γ(β)−1 = 0 at s = l − 1, and thus Em(τ, l − 1; l, µ,R) = 0. For the same

reason one has for l > 2, E0(τ, l − 1; l, µ,R) = δ0,µ. When l = 2, ζ(s) has a simple pole at s = l − 1 = 1,

cancelling the zero of Γ(β)−1. If D > 1, the term∏p|D

1− p1−s

1− p−1−s

is zero at s = 1 and we still have E0(τ, l − 1; l, µ,R) = δ0,µ. If D = 1, then a direct computation using

(iii) and (iv) gives the extra term − 3πv δ(µ,N) for the constant term. 2

Corollary 9.3. (i) When m > 0 and m ∈ Q(µ) + Z,

E′m(τ, l − 1; l, µ,R)

Em(τ, l − 1; l, µ,R)=∑

06j< l2

(1

l − 2j − 1− 1

l + 2j

)− 1

2(γ + log(4π)) +

ζ ′(1− l)

ζ(1− l)

+∑

p|mDN

Kp log p+1

2J(l − 1, 4πmv).

Here γ is the Euler’s constant, kp = ordpm, J(n, z) is given in Lemma 2.2, and Kp is given as follows.

When µ ∈ Rp, Kp is equal to

kp +1

p(l−1)(kp+1) − 1− 1

pl−1 − 1, if p - DN,

kp −pl + pl−1 − 2

(pl−1 − 1)(pl − 1)+p2−l − (kp + 1)p(1−l)(kp+1)+1 − (kp + 2)p(1−l)(kp+2)

1− p2−l + p(1−l)(kp+1)+1 − p(1−l)(kp+2), if p|D,

kp −pl + pl−1 − 2

(pl−1 − 1)(pl − 1)+

2− p2−l − (kp + 1)p(1−l)(kp+1)+1 + (kp + 2)p(1−l)(kp+2)

1− 2p1−l + p2−l − p(1−l)(kp+1)+1 + p(1−l)(kp+2), if p|N.

When µ /∈ Rp, Kp =12kp−

1pl−1

. Finally J(n, t) is given in Lemma 2.2, and the value Em(τ, l− 1; l, µ,R)

is given by Corollary 9.2.


(ii) When m < 0 and m ∈ Q(µ) + Z,

E′m(τ, l − 1; l, µ,R) = C−(l,D) (2πv)1−lB(m, l − 1;µ,R)

∫ ∞

1

e4πmvrr−ldr qm

with

C−(l,D) =

√πΓ( l2 )(−1)o(D)+ l

2

Γ(1−l2 )ζ(1− l)DN∏p|DN (1− p−l)

Corollary 9.4. When l > 2, the derivative of the constant term at s = l − 1 is

E′0(τ, l − 1; l, µ,R) =

1

2δ0,µ log v +

(−1)o(D)+ l2√πΓ( l2 )

DN(l − 1)Γ(1−l2 )(2πv)l−1

ζ(l − 1)

ζ(1− l)

×∏p|D

1− p2−l

1− p−l

∏p|N,µ∈Rp

1 + p2−l − 2p1−l

1− p−l

∏p|N,µ/∈Rp

1− p1−l

1− p−l

=1

2δ0,µ log v + (2πv)1−lC−(l,D)

ζ(l − 1)

l − 1

×∏p|D

(1− p2−l)∏

p|N,µ∈Rp

(1 + p2−l − 2p1−l)∏

p|N,µ/∈Rp

(1− p1−l),

if Q(µ) ∈ Z. Otherwise, the derivative is zero.

Corollary 9.5. Suppose that l = 2.

(i) If o(D) > 1, then E′0(τ, 1; 2, µ,R) =

12 δ0,µ log v.

(ii) If o(D) = 1 so that D is a prime number, then

E′0(τ, 1; 2, µ,R) =

1

2δ0,µ log v +

3

πvδ(µ,N)

D

D2 − 1,

where δ(µ,N) is given in Corollary 9.2.

(iii) If D = 1, then

E′0(τ, 1; 2, µ,R) =

1

2δ0,µ log v − 3

πvδ(µ,N)

(− 1

2log v +

1

2− log(4π) +

ζ ′(−1)

ζ(−1)

)− 3

πvδ(µ,N)

( ∑p|N,µ/∈Rp

p

p2 − 1log p−

∑p|N,µ∈Rp

p

2(p+ 1)log p

).

Proof. (sketch) Since 1− p1−s vanishes at s = 1, (i) and (ii) are clear. When D = 1 and l = 2, one has

E0(τ, s; 2, µ,R) = vβδ0,µ − vβ−s2−sΓ(s)

Γ( s+32 )ζ(s+ 1)

ζ(s)

Γ( s−12 )

× 2π

N

∏p|N,µ/∈Rp

1− p−s

1− p−1−s

∏p|N,µ∈Rp

1 + p1−s − 2p−s

1− p−1−s .

Thus,

E0(τ, 1; 2, µ,R) = δ0,µ − 3

πvδ(µ,N) and E′

0(τ, 1; 2, µ,R) =1

2δ0,µ log v − 3

πvδ(µ,N)(A1 +A2),

where

A1 = −1

2log v − log 2 +

Γ′(1)

Γ(1)− 1

2

Γ′(2)

Γ(2)− ζ ′(2)

ζ(2)+

d

dslog

ζ(s)

Γ( s−12 )

∣∣∣∣s=1

,

and

A2 =∑

p|N,µ/∈Rp

(p−1

1− p−1− p−2

1− p−2

)log p+

∑p|N,µ∈Rp

(−1 + 2p−1

2− 2p−1− p−2

1− p−2

)log p


=∑

p|N,µ/∈Rp

p

p2 − 1log p−

∑p|N,µ∈Rp

p

2(p+ 1)log p.

To compute A1, recall the basic formulas

Γ′(1)

Γ(1)= −γ, Γ′(2)

Γ(2)= 1− γ, and

Γ′(− 12 )

Γ(− 12 )

= 2− γ − 2 log 2.

These formulas and the functional equation of ζ(s) give

ζ ′(2)

ζ(2)= log 2π − 1 + γ − ζ ′(−1)

ζ(−1).

Finally, one has

ζ(s)

Γ( s−12 )

=1

2

(1 + (γ − 1

2

Γ′( 1+s2 )

Γ( 1+s2 )2)(s− 1) + · · ·

),

so thatd

dslog

ζ(s)

Γ( s−12 )

∣∣∣∣s=1

=3

2γ,

and hence A1 = −12 log v − log 4π + 1

2 + ζ′(−1)ζ(−1) . This proves (iii). 2

Using (9.3), and (2.10), one can easily obtain from Theorem 9.1 the following simple functional equation

when N = 1, i.e., R is a maximal order of B, which we write as a theorem without proof.

Theorem 9.6. Let R = OB be a maximal order of B. Then

E(τ, s, l;µ,OB) =

(D

π

) s+12

Γ

(s+ 1

2

)ζ(s+ 1)

( ∏06j< l

2

(1 + s+ 2j)

)E(τ, s; l, µ,OB)

satisfies the functional equation E(τ, s, l;µ,OB) = E(τ,−s, l;µ,OB).

It would be interesting to work out the (matrix) functional for E(τ, s; l, µ,R) for a general Eichler

order.

10 Integrals of Borcherds forms

The derivatives of Eisenstein series computed in the previous sections are useful in computing period

integrals of Borcherds’ modular forms, see, for example, [6, 18, 35]. We first briefly review the result

of [18], which is one of the main original motivations for this paper.

Let V be a non-degenerate quadratic space over Q of signature (n, 2), let L be an even lattice in V ,

and let L∗ be its dual. Let G = GSpin(V ), and let D be the Hermitian symmetric domain of the oriented

negative 2-planes in VR. Let K be a compact open subgroup of G(Q) which fixes L and acts trivially on

L∗/L. Then there is a Shimura variety XK = Sh(G,D)K over Q such that

XK(C) = G(Q)\(D×G(Q)/K

).

Let SL = ⊕Cϕµ be the space of Schwartz functions on V = V ⊗Q Q which are L-invariant and are

supported on L∗. For simplicity, we assume that n is even, and refer to [18] for the general case. A

weakly holomorphic modular form of weight k valued in SL is defined to be a holomorphic function

F : H → SL with the following properties:

1. For every γ = ( a bc d ) ∈ SL2(Z), F (γτ) = (cτ + d)kω−1V (γ)F (τ), where on the right-hand side we

identify γ with its image in SL2(Z).2. There is a polynomial PF (τ) ∈ SL[q

−1] such that F (τ)− PF (τ) = o(e−ϵv) as v = Im(τ) → ∞ for

some fixed constant ϵ > 0.


Such a modular form has a Fourier expansion

F (τ) =∑

µ∈L∗/L

∑m

cµ(m) qm ϕµ (10.1)

such that cµ(m) = 0 when −m > 0 is sufficiently large.

For a weakly holomorphic modular form of weight 1− n2 with cµ(m) ∈ Z whenever m < 0, Borcherds

constructed a meromorphic modular form Ψ(F ) on XK with some amazing properties, via a regularized

theta lifting, [2], see also [18] for a brief review. It is of interest in Arakelov theory to compute the integral

κ(F ) = − 1

Vol(XK)

∫X(C)

log ∥Ψ(z, F )∥2dµ(z)

where µ(z) is a normalized G(R)-invariant volume form on D, and ∥Ψ(z, F )∥ is a suitably normalized

Peterson metric of Ψ(z, F ) (see [18]).

Let E(τ, s; Φn2 +1,µ) be the Eisenstein series on SL(2) associated with Φ

n2 +1∞ ⊗ ϕµ, and let

κµ(m) =

1

2C δ0,µ, if m = 0,

limv→∞

E′m(τ,

n

2;Φ

n2 +1,µ) q−m, if m > 0.

Here C = log(2π) − Γ′(1) = log(2π) + γ. Then the first author proved the following theorem, which

basically says that κµ(m) is the kernel function for the integral (10.1).

Theorem 10.1 [18, Theorem II]. Let F be a weakly holomorphic modular form of weight 1− n2 valued

in SL such that all cµ(m) are integers for m 6 0. Then

κ(F ) =∑µ

∑m>0

cµ(−m)κµ(m).

Remark 10.2. In [4], Bruinier and Funke extended Borcherds’ construction to the so-called harmonic

weak Maass forms to obtain automorphic Green functions Φ(z, F ) on XK for certain special divisors

Z(F ). When F is weakly holomorphic, one has Φ(z, F ) = − log ∥Ψ(z, F )∥2. It thus makes sense to define

in the general case

κ(F ) =1

Vol(XK)

∫XK(C)

Φ(z, F ) dµ(z).

Zhao proved in his thesis [35] that κ(F )−∑µ

∑m>0 cµ(−m)κµ(m) is the derivative of a certain L-series.

We end this paper with an explicit calculation of κ(F ) in an interesting example of Borcherds. Let

V = Q12 with the quadratic form Q(x) = 12 (∑10i=1 x

2i − x211 − x212). Then the lattice considered in [2,

Example 13.7] is L = {x = (xi) ∈ Z12 :∑xi ∈ 2Z}. It is the maximal even lattice in the odd

unimodular lattice Z12 ⊂ V . It is also a sublattice of the even unimodular lattice L1 generated by L

and ρ = 12 (1, 1, . . . , 1) ∈ V. Let ei ∈ V be the element which has the i-th coordinate 1 and all the other

coordinates 0. Then L∗/L is of order 4 and a set of coset representatives is given by 0, ρ, ρ− e1, and e1.

The first three have norm in Z and the last one has norm 12 . For every p = 2, Lp is unimodular, and for

p = 2, we have

Lemma 10.3. The 2-adic lattice L2 = L⊗ Z2 is Z2-equivalent to Z122 with quadratic form

Q(x) = x1x2 + x3x4 + x5x6 + 2x7x8 + (x29 + x9x10 + x210) + (x211 + x11x12 + x212).

Proof. Set

f1 = −e10 − e12, f2 = e9 + e12, f3 = e9 + e10 + e11 + e12, f4 = −e8 − e11,

f5 = −e7 + e8 + e9 + e10 + e11 + e12, f6 = e6 − e7, f7 = e4 − e5, f8 = e4 − e3.


Then the vectors f1, . . . , f8 generate a unimodular Z-sublattice M of L of rank 8, whose Gram matrix is

diag(J1, J1, J2, J2) with J1 = ( 0 11 0 ), J2 = ( 2 1

1 2 ). So L =M ⊕M⊥ where

M⊥ =

{x = (xi) ∈ L : x3 = x4 = x5, x6 = x7 = · · · = x12,

∑xi ∈ 2Z

}is Z-equivalent to M1 = {x = (xi) ∈ Z4 :

∑xi ∈ 2Z} with quadratic form Q1(x1, x2, x3, x4) = x21 + x22 +

3x23 +3x24, since Q(e3 + e4 + e5) = 3 and Q(e6 + e7 + · · ·+ e12) = 3. Now we have to work over Z2. Take

g1 = e1 − e2 and g2 = e2 − e3 ∈ M1, then M2 = Z2g1 + Z2g2 is a unimodular Z2-sublattice of M1 ⊗ Z2

with

Q1(xg1 + yg2) = x2 − xy + 2y2

since Q1 ≡ (x + 2y)(x + 5y) (mod 8) there are ϵ1 ≡ −2 (mod 8) and ϵ2 ≡ −5 (mod 8) such that

Q1(xg1+yg2) = (x− ϵ1y)(x− ϵ2y) over Z2. That is, (M2, Q1) is Z2-equivalent to the hyperplane Z22 with

quadratic form xy. Now M1 ⊗ Z2 =M2 ⊕M⊥2 with

M⊥2 =

{x ∈ Z4

2 : x1 = x2 = 3x3,∑

xi ∈ 2Z2

}.

Finally, M⊥2 has a basis g3 = (3, 3, 1, 1) and g4 = (0, 0, 0, 2) with respect to which we have

Q1(xg3 + yg4) = 2(6x2 + 3xy + y2) = 2(x− β1y)(x− β2y),

and this is Z2-equivalent to the hyperplane Z22 with quadratic form 2xy. Here βi ∈ Z2 satisfy β1 ≡ 2

(mod 8), β2 ≡ 3 (mod 8). This proves the lemma. 2

It is easy to check using the definition that

γ(Vp) = 1 (10.2)

for all p, including ∞. Now applying Theorem 4.4, we have

Proposition 10.4. Let µ = 0, ρ, ρ− e1, e1 ∈ L∗/L. Then

Wm,2(s, ϕµ) =1

2

char

(1

2+ Z2

)(m), if µ = e1,

char(Z2)(m), if µ = ρ, ρ− e1,

char(Z2)(m)1− 2 ·Xk2+1 +Xk2+2

1−X, if µ = 0.

Here X = 2−s and k2 = ordpm. In particular,

W0,2(s, ϕµ) =1

2

0, if Q(µ) /∈ Z2,

1, if Q(µ) ∈ Z2, µ /∈ L2,

ζ2(s), if µ ∈ L2.

Proposition 10.5. Let the notation be as above.

(i) If Q(µ) ∈ Z, m > 0, and m ∈ Z,

κµ(m) = Aµ(m)

(ζ ′(−5)

ζ(−5)+

137

120+

3

2γ − 1

2log π +

62

63log 2

+1

2logm+

∑p=2

(kp + 1

p5(kp+1) − 1− 1

p5 − 1

)log p+

W ′m,2(5, ϕµ)

Wm,2(5, ϕµ)

),

where

Aµ(m) = −28σ5(m)

31 · 25k2

25(k2+1) − 1, if µ /∈ L,

25(k2+2) − 63

25(k2+2) − 32, if µ ∈ L,


and

W ′m,2(5, ϕµ)

W ′m,2(5, ϕµ)

=

0, if µ /∈ L,(63k2 + 62

25(k2+2) − 63− 1

31

)log 2, if µ ∈ L.

(ii) If Q(µ) ∈ 12 + Z, i.e., µ = e1, then for m > 0 and m ∈ 1

2 + Z,

κµ(m) =− 8σ(2m)

(ζ ′(−5)

ζ(−5)+

137

120+

3

2γ − 1

2log π +

61

126log 2 +

1

2log(2m)

+∑p =2

(kp + 1

p5(kp+1) − 1− 1

p5 − 1

)log p

)

Proof. By Theorem 2.4, where S = {2} in this case, and Proposition 10.4, one has

q−mEm(τ, 5,Φ6,µ) =Wm,2(5, ϕµ)m5∏p =2

σ−5,p(m)(−2πi)5

Γ(6)ζ(6)(1− 2−6)

= −29Wm,2(5, ϕµ)m5∏p =2

σ−5,p(m).

When Q(µ) ∈ Z, one has

q−mEm(τ, 5,Φ6,µ) = −29 σ5(m)Wm,2(5, ϕµ)1− 2−5

1− 2−5(k2+1)= Aµ(m).

When Q(µ) ∈ 12 + Z, k2 = −1 and thus

q−mEm(τ, 5,Φ6,µ) = −291

2m5∏p=2

σ−5,p(m) = −8σ5(2m).

On the other hand, by Proposition 2.8, one has

E′m(τ, 5,Φ6,µ)

Em(τ, 5,Φ6,µ)=

1

2log(πm)− 1

2

Γ′(6)

Γ(6)+

1

2J(5, 4πmv)− ζ ′(6)

ζ(6)− 1

63log 2

+W ′m,2(5, ϕµ)

Wm,2(5, ϕµ)+∑p =2

(kp + 1

p5(kp+1) − 1− 1

p5 − 1

)log p.

The functional equation for ζ(s) gives

ζ ′(6)

ζ(6)= log π − 137

60− γ − log 2− ζ ′(−5)

ζ(−5)and

Γ′(6)

Γ(6)=

137

60− γ.

Putting them together, one proves the proposition. The formula forW ′

m,2(5,ϕµ)

Wm,2(5,ϕµ)follows directly from

Proposition 10.3. 2

Now let F =∑µ fµϕµ be the modular form of weight −4 for SL2(Z) valued in SL given by Borcherds [2,

Example 13.7] with

f0(τ) = 8η(2τ)8/η(τ)16 = 8 + 128q + 1152q2 + · · · ,fρ(τ) = fρ−e1(τ) = −8η(2τ)8/η(τ)16 = −8− 128q − 1152q2 + · · · ,

fe1(τ) = 8η(2τ)8/η(τ)16 + η(τ/2)8/η(τ)16 = q−12 + 36q

12 + 402q

32 + · · · .

Let Ψ(F ) be the corresponding Borcherds modular form for GSpin(V ). Then Theorem 10.1 gives

κ(F ) = 4(γ + log(2π)) + κe1

(1

2

)Applying Proposition 10.4, we obtain

κ(F ) = 4

(−ζ

′(−5)

ζ(−5)− 137

60− 2γ + 2 log π − 2

63log 2

). (10.3)


Acknowledgements T.Y. Yang is partially supported by Natural Science Foundation of USA (Grant No.

DMS-0855901) and National Natural Science Foundation of China (Grant No. 10628103). We would like to

dedicate this work to Professor Yuan Wang, who has made such important contributions to number theory and

Chinese mathematics. The second author thanks the AMSS, the Morningside Center of Mathematics, and the

Tsinghua University Mathematical Science Center for providing excellent working condition during his summer

visits to these institutes. We thank S. Ehlen and an anonymous referee for their careful reading and comments.

We thank Ye Tian for interesting discussion.

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