yang t - taylor expansion of eisenstein series - trans. amer. math. soc. 355 (2003) no.7 2663-2674

8/9/2019 Yang T - Taylor Expansion of Eisenstein Series - Trans. Amer. Math. Soc. 355 (2003) No.7 2663-2674

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TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 355, Number 7, Pages 26632674S 0002-9947(03)03194-5Article electronically published on February 27, 2003

TAYLOR EXPANSION OF AN EISENSTEIN SERIES

TONGHAI YANG

Abstract. In this paper, we give an explicit formula for the first two termsof the Taylor expansion of a classical Eisenstein series of weight 2k + 1 for0(q). Both the first term and the second term have interesting arithmetic

interpretations. We apply the result to compute the central derivative of someHecke L-functions.

0. IntroductionConsider the classical Eisenstein series

\ SL2(Z)Im()s,

which has a simple pole at s = 1. The well-known Kronecker limit formula givesa closed formula for the next term (the constant term) in terms of the Dedekind-function and has a lot of applications in number theory. It seems natural andworthwhile to study the same question for more general Eisenstein series. Forexample, consider the Eisenstein series

(0.1) E(, s) =

\0(q)(d)(c + d)2k1 Im()

s2k.

Here =

a bc d

, q is a fundamental discriminant of an imaginary quadratic field,

and = ( q ). This Eisenstein series was used in the celebrated work of Gross andZagier ([GZ, Chapter IV]) to compute the central derivative of cuspidal modularforms of weight 2k + 2. The Eisenstein series is holomorphic (as a function ofs) atthe symmetric center s = 0 with the leading term (constant term) given by a thetaseries via the Siegel-Weil formula. The analogue of the Kronecker limit formulawould be a closed formula for the central derivative at s = 0the main objectof this paper. This would give a direct proof of [GZ, Proposition 4.5]. Anotherapplication is to give a closed formula for the central derivative of a family of HeckeL-series associated to CM abelian varieties, which is very important in the arith-metic of CM abelian varieties in view of the Birch and Swinnerton-Dyer conjecture.This application will be given in section 4. We will also prove a transformation

equation for the tangent line of the Eisenstein series at the center, which should beof independent interest.

Received by the editors September 9, 2002.

2000 Mathematics Subject Classification. Primary 11G05, 11M20, 14H52.

Key words and phrases. Kronecker formula, central derivative, elliptic curves, Eisenstein series.

Partially supported by an AMS Centennial fellowship and NSF grant DMS-0070476.

c2003 American Mathematical Society

2663


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2664 TONGHAI YANG

To make the exposition simple, we assume that q > 3 is a prime congruent to 3modulo 4. Let k = Q(

q).Set

(0.2) (s, ) = s+12 s + 1

2

L(s, )

and

(0.3) E(, s) = qs+12 (s + 1, )E(, s).

It is well known that E(, s) is holomorphic.As in [GZ, Propositions 4.4 and 3.3], we define

(0.4) pk(t) =k

m=0

(km)(t)m

m!

and

(0.5) qk

(t) = 1

etu(u

1)kuk1du, t > 0.

We remark that pk(t) and qk(t) are two basic solutions of the differential equa-tions

(0.6) tC(t) + (1 + t)C(t) kC(t) = 0.Finally, let (n) be given by

(0.7) k(s) =

(n)ns.

Theorem 0.1. Let the notation be as above, and let h be the ideal class number ofk. Write = u + iv. Then

E(, 0) = vk(h + 2n>0

(n)pk(4nv)e(n))

and

E(, 0) +1

4

kj=1

1

jE(, 0)

=1

2vk

a0(v) 2

n>0

anpk(4nv)e(n) 2n


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TAYLOR EXPANSION OF AN EISENSTEIN SERIES 2665

derivative. The other one is incoherent, contributes nothing to the value, and itscentral derivative can be computed by the method of [KRY], where we dealt withthe case k = 0. This consists of sections 1 and 2.

In section 3, we study how the value and derivative behave under the Frickeinvolution 1/q and obtain the following functional equation. One inter-esting point about the equation is that it basically follows from the definition ofautomorphic forms (see (3.2)).

Theorem 0.2. The modular forms E(, 0) and E(, 0) satisfy the following func-tional equation:

E( 1q, 0)E( 1q , 0)

= i(

q)2k+1

1 0k

j=11j +

12 log q 1

E(, 0)E(, 0)

.

Finally, let be a canonical Hecke character of weight 1 ofk (see section 4 for thedefinition). It is associated to the CM elliptic curve A(q) studied by Gross ([Gro]).

When q 3 mod 8, S. Miller and the author proved recently that the centralderivative L(1, ) = 0 ([MY]). Since the central derivative encodes very importantinformation in the arithmetic of A(q), it is important to find a good formula for thecentral derivative. Standard calculation shows that the L-series L(s, ) is E(, 2s)evaluated at a CM cycle. So Theorem 0.1 gives an explicit formula for the centralderivative L(1, ) (Corollary 4.2).

1. Coherent and incoherent Eisenstein series

Let G = SL2 over Q, and let B = T N be the standard Borel subgroup, whereT is the standard maximal split torus of B and N is the unipotent radical of B.Their rational points are given by

T(Q) ={

m(a) = a 00 a1

: a Q}and

N(Q) = {n(b) =

1 b0 1

: b Q}.

Consider the global induced representation

I(s, ) = IndG(A)B(A)| |sA

of G(A), where A is the ring of adeles ofQ. By definition a section (s) I(s, )satisfies

(1.1) (n(b)m(a)g, s) = (a)|a|s+1(g, s)for a

A and b

A. Let K = SL2(Z) and let K

= SO(2)(R). Associated to a

standard section , which means that its restriction on KK is independent ofs,one defines the Eisenstein series

(1.2) E(g,s, ) =

B(Q)\G(Q)(g,s).

It is absolutely convergent for Re s > 1 and has a meromorphic continuation tothe whole complex s-plane. We consider three standard sections 0, in thispaper. For every prime p q, let p I(s, p) be the unique spherical section


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2666 TONGHAI YANG

such that p(x) = 1 for every x Kp = SL2(Zp). Let I(s, ) be the uniquesection of weight 2k + 1 in the sense that

(1.3) (gk, s) = (g, s)ei(2k+1)

for every k =

cos sin sin cos

K. For p = q, letJq =

a bcq d

SL2(Zq) : a,b,c,d Zq

be the Iwahori subgroup of Kq. Then q defines a character of Jq via

(1.4) q(

a bcq d

) = q(d).

As described in [KRY, section 2], the subspace of I(s, q) consisting of q eigenvec-tors ofJq is two-dimensional and is spanned by the cell functions of iq, determinedby

(1.5) iq(wj , s) = ij , where w0 = 1 and w1 = w =

0 11 0

.

We denote this subspace by W(Jq, q, s). A better basis for this subspace turns outto be given by

(1.6) q = 0q

1q 1q,

which are eigenfunctions of some intertwining operator (see Lemma 2.2). Set

(1.7) 0 = 0q

p=qp and

= q

p=qp.

Clearly, 0 = 12 (+ + ). For = u + iv with v > 0, let

(1.8) g = n(u)m(

v).

Then standard computation gives

Proposition 1.1. Let the notation be as above. Then

E(, s) = vk12 E(g, s, 0)

=1

2vk

12 (E(g, s, +) + E(g, s, )).

Here

E(g,s, ) = qs+12 (s + 1, )E(g,s, )

is the completion of the Eisenstein series E(g,s, ).

As we will see in Proposition 2.4, the Eisenstein series with behave almost aseven/odd functions respectively, and both have nice functional equations. This

is not a coincidence. Indeed, from the point of view of representation theory,+(g, 0) is a coherent section in I(0, ) in the sense that it comes from a global(two-dimensional) quadratic space, while (g, 0) is an incoherent section in I(0, ),coming from a collection of inconsistent local quadratic spaces. We refer to [Ku]for explanation of this terminology and for a general idea for computing the cen-tral derivative of incoherent Eisenstein series. Every section in I(0, ) is a linearcombination of coherent and incoherent sections; we just made it explicit in thiscase.


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2. Proof of Theorem 0.1

Let = p be the canonical additive character ofA viap(x) =

e2ix if p = ,e2i(x) if p = .

Here is the canonical map Qp Qp/Zp Q/Z. For a standard section =

p I(s, ) and d Q, one defines the local Whittaker function

(2.1) Wd,p(g,s, ) =

Qp

(wn(b)g, s)p(db)db.

Let

(2.2) Wd,p(g,s, ) = Lp(s + 1, )Wd,p(g,s, )

be its completion. We also set Mp(s) = W0,p(s) and Mp (s) = W0,p(s). So M

(s) =Mp (s) is a normalized intertwining operator from I(s, ) to I(s, ).In general, an Eisenstein series E(g,s, ) has a Fourier expansion

(2.3) E(g,s, ) =

d

Ed (g,s, )

with

(2.4) Ed (g,s, ) = qs+12

p

Wd,p(g,s, )

for d = 0 and(2.5) E0 (g,s, ) = q

s+12 (s + 1, )(g, s) + q

s+12 M(s)(g, s).

The local Whittaker integrals are computed in the next three lemmas.

Lemma 2.1 ([KRY, Lemma 2.4]). For a finite prime number p = q, one hasWd,p(1, s, p) = 0 unless ordp d 0. In such a case, one has

Wd,p(1, s, p) =ordp d

r=0

(p(p)ps)r

and

Mp (s)(s) = Lp(s, )p(s).Here p is the unique spherical section defined in section 1. In particular,

Wd,p(1, 0, p) = p(d),

where p(d) = (pordp d) for p < .

Lemma 2.2. For p = q, one has

Wd,q(w0, s,

)Wd,q(w1, s,

)

=

(1 q(d)qs(ordq d+1))

1q1q

if ordq d 0,

(1 q(d))

0

q1

if ordq d = 1,

0 otherwise


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2668 TONGHAI YANG

and

Mq (s)q =

1

qq .

Proof. The first formula follows from [KRY, (3.26)-(3.29)]. For the second formula,notice that Mq (s) is an intertwining operator between eigenspaces W(Jq, q, s) andW(Jq, q, s) of Jp. So

Mq (s)q = a

+q + bq

for some constants a and b. Plugging in g = w0 and w1, and applying the firstformula, one gets the desired formula.

Lemma 2.3. Let = be the local section in I(s, ) defined by (1.3).(1)

Wd,(g, s, ) = 2iv1+s2

s2 e(du)

k

j=0j s2j + s2

(2v,d, s2 + k + 1,s2 k)

( s2 ).

Here

(g,h,,) =

xh>0

egx(x + h)1(x h)1dxis Shimuras eta function for g > 0, h R, and Re and Re sufficiently large[Sh].

(2) For d > 0, one has

Wd,(g, 0, ) = 2iv12pk(4dv)e(d),

where pk is defined by (0.4).(3) For d < 0 , one has Wd,(g, 0, ) = 0, and

Wd,(g, 0, ) = iv12 qk(4dv)e(d),

where qk is given by (0.5).(4) M(s)(s) = ik

j=0js/2j+s/2 L(s, )(s).

Proof. The proof is the same as that of [KRY, Proposition 2.6] and is left to thereader.

Proposition 2.4. One has the functional equation as s goes to s:

(2.6)k

j=0

(j s2

)E(g, s, ) = k

j=0

(j +s

2)E(g,s, ).

Proof. By Lemmas 2.1-2.3, one has

(2.7) M(s)(g, s) = q 12k

j=0

j s2

j + s2(s, )(g, s).

Now the proposition follows from the functional equations

qs2 (s, ) = q

s2 (s, )

andE(g,s, ) = E(g, s, M(s)).

Here M(s) = M(s)(s + 1, )1 is the unnormalized intertwining operator fromI(s, ) to I(s, ).


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Theorem 2.5. One has

(2.8) v12 E(g, 0, +) = 2(hq + 2

n>0

(n)pk(4nv)e(n))

and

v12 E(g, 0, )(2.9)

= hq(log qv + 2(1, )(1, )

+k

j=1

1

j) 2

n>0

anpk(4nv)e(n)

2n 0 is an integer, the lemmas and (2.10) imply

Ed (g, 0, +) = q

12

pq

p(d)1 + q(d)q 2iv

12pk(4dv)e(d)

= 4v12 (d)pk(4dv)e(d).(2.11)

The same lemmas also imply

E0 (g, 0, +

) = q1

2 (1, )+

(g, 0) + q1

2 M(0)+

(g, 0)

= hv12 + (0, )v

12

= 2hv12 .

This proves (2.8).As for (2.9), we again check term by term, and it is clear from the lemmas that

E

d (g, 0, ) = 0 unless d is an integer, which we assume from now on.

When d < 0, Wd,(g, 0, ) = 0 by Lemma 2.3(3), and so (using Lemmas

2.1-2.3 and (2.10))

Ed (g, 0, ) = q

12 Wd,(g, 0, )W

d,q(1, 0,

q )

pqWd,p(1, 0, p)

= 2v12

qk(4dv)e(d)(1 q(d)) pq

p(d)

= 2v 12 (d)qk(4dv)e(d),as desired.

When d > 0 and q(d) = 1, one has Wd,q(1, 0,

) = 0 and

Wd,q(1, 0, ) =

1q (ordq d + 1)log q.


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2670 TONGHAI YANG

The same computation using Lemmas 2.1-2.3 and (2.10) yields

Ed (g, 0, ) =

2v

12pk(4dv)e(d)(ordq d + 1)(d)log q

= 2v 12 anpk(4dv)e(d),(2.12)since an = (ordq d + 1)(d)log q in this case.

When d > 0 and q(d) = 1, there is a prime l|d such that Wd,l(1, 0, l) =l(d) = 0 by (2.10). In this case,

Wd,l(1, 0, l) =1

2(ordl d + 1) log l.

The same calulation yields

Ed (g, 0, ) = 2v 12 anpk(4dv)e(d),

as desired.Finally, when d = 0, one has by the same lemmas,

(2.13) E0 (g, s, ) = 1k

j=1(j +s2 )

(G(s) G(s))

with

(2.14) G(s) = (qv)1+s2 (1 + s, )

kj=1

(j +s

2).

So

(2.15) E0 (g, 0, ) =

2G(0)k!

= hv12

log(qv) + 2 (1, )

(1, )+

kj=1

1

j

.

This finishes the proof of (2.9).

Proof of Theorem 0.1. One has by Proposition 2.4,

(2.16) E(, 0) =1

2vk

12 E(g, 0, +)

and

(2.17) E(, 0) =1

2vk

12

E(g, 0, ) 1

2

kj=1

1

jE(g, 0, +)

.

Now Theorem 0.1 easily follows from Propositions 1.1 and 2.4 and Theorem 2.5.

3. Proof of Theorem 0.2

By Proposition 1.1 and Formulas (2.16) and (2.17), Theorem 0.2 is equivalent

to the identity

(3.1)

||

2k+1E(g 1q

, 0, +)

E(g 1q

, 0, )

= i

1 012 log q 1

E(g, 0, +)E(g, 0, )

.

To prove (3.1), one observes the following trivial but fundamental identity andcomputes both sides:

(3.2) E(w1 gq, s, ) = E(wfgq, s, ).


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Here wf and w are the images of w =

0 11 0

in G(Af) and G(R) respectively.

The left-hand side of this identity is given by

Lemma 3.1.

E(w1 g, s, ) =

||

2k+1E(g 1

, s, ).

Proof. Write w1 g = g 1

k; then ei = ||/. So one has, for any G(Q),

(w1 g, s) = ||

2k+1(g 1

, s).

Plugging this into the definition of the Eisenstein series, one gets the lemma.

For the right-hand side of (3.2), one has

Lemma 3.2.E(wfgq, 0, +)E(wfgq, 0, )

= i

1 012 log q 1

E(g, 0, +)E(g, 0, )

.

Proof. We verify these identities by comparing the Fourier coefficients Edq

(wfgq,

s, ) with Ed (g, s, ). We may assume that d is an integer by Lemmas 2.1-2.3.

Straightforward calculation using the same lemmas yields, for any integer d,

(3.3) Wdq

,p(wfgq, s, ) = Fp (d)W

d,p(g, s,

)

with

(3.4) Fp (d) =

1 if p q,q1s2 if p =

,

1q 1q(d)qsr1q(d)qs(r+1) if p = q.Here r = ordq d. We will verify the derivative part and leave the value part to thereader. First assume d = 0. It follows from (3.3) that

Edq

(wfgq, 0, ) = iEd (g, 0,

)

1 if q(d) = 1,1 1ordq d+1 if q(d) = 1.

When q(d) = 1, one has by (2.11) and (2.12),

Ed (g, 0, ) = Ed (g, 0, +)

ordq d + 1

2log q.

So

(3.5) Edq

(wfgq, 0, ) = iEd (g, 0, ) + i2log qEd (g, 0, +),

as desired. When q(d) = 1 we have Ed (g, 0, +) = 0, and (3.5) still holds.It remains to check the constant term. Recall (2.13)-(2.15). Direct calculation

using Lemmas 2.1-2.3 also gives

(3.6) E0 (wfgq, s, ) = ik

j=1(j +s2 )

(qs2 G(s) qs2 G(s)).


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2672 TONGHAI YANG

Therefore,

E0

(wfgq, 0, ) = i

2G(0)

k!+ i

2G(0)

k!

1

2log q

= iE0 (g, 0, ) +

i

2log qE0 (gq, 0,

+),(3.7)

as expected, too.

4. L-series

Recall that q is a prime congruent to 3 modulo 4 and k = Q(q) is the

associated imaginary quadratic field. Recall also ([Roh]) that a canonical Heckecharacter ofk of weight 2k + 1 is a Hecke character satisfying

(1) The conductor of isqOk.

(2) (A) = (A) for an ideal A relatively prime toqOk.

(3) (

Ok) =

2k+1.

In this section, we will give an explicit formula for the central derivative of itsL-function, which has deep arithmetic implications as mentioned in the introduc-tion. We refer to [Gro] for the arithmetics of elliptic curves associated to theseHecke characters (see also [MY] and [Ya] and the reference there for more recentdevelopments). For each ideal class C ofk, we can define the partial L-series by

(4.1) L(s,,C) =

BC, integral(B)(NB)s.

Of course, L(s, ) =

C CL(k) L(s,,C). The following proposition is standard.

Proposition 4.1. LetA C be a primitive ideal of k relatively prime to 2q, andwrite

(4.2) A = [a,

b +

q

2 ], with a > 0, b 0 mod q.Let A =

b+q

2aq . Then

L(s + k + 1, , C ) =(A)

(NA)2k+1(2

q)skL(2s + 1, )E(A, 2s).

Set

(4.3) k() = h + 2n>0

(n)pk(4nv)e(n)

and

(4.4) k() = a0(v) 2n>0anpk(4nv)e(n) 2n


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2674 TONGHAI YANG

Acknowledgement

This work was inspired by joint work with Steve Kudla and Michael Rapoport.

The author thanks them for the inspiration. He thanks Rene Schoof for numericallyverifying the formulae in Corollary 4.2, which corrects a mistake in an earlier versionof this paper. Finally, he thanks Dick Gross, Steve Miller, and David Rohrlich forstimulating discussions.

References

[Gro] B. Gross, Arithmetic on elliptic curves with complex multiplication, Lecture Notes inMath., no. 776, Springer-Verlag, Berlin, 1980. MR 81f:10041

[GZ] B. Gross and D. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84(1986), 225-320. MR 87j:11057

[Ku] S. Kudla, Central derivatives of Eisenstein series and height pairings, Ann. Math. 146(1997), 545-646. MR 99j:11047

[KRY] S. Kudla, M. Rapoport, and T.H. Yang, On the derivative of an Eisenstein series of

weight one, Internat. Math. Res. Notices 7 (1999), 347-385. MR 2000b:11057

[MY] S. D. Miller and T. H. Yang, Non-vanishing of the central derivative of canonical HeckeL-functions, Math. Res. Letters 7 (2000), 263-277. MR 2001i:11058

[Roh] D. Rohrlich, Root numbers of Hecke L-functions of CM fields, Amer. J. Math. 104(1982), 517-543. MR 83j:12011

[Sh] G. Shimura, Confluent hypergeometric functions on tube domains, Math. Ann. 260(1982), 269-302. MR 84f:32040

[Ya] T.H. Yang, On CM abelian varieties over imaginary quadratic fields, preprint.

Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

E-mail address: [email protected]
http://www.ams.org/mathscinet-getitem?mr=81f:10041http://www.ams.org/mathscinet-getitem?mr=81f:10041http://www.ams.org/mathscinet-getitem?mr=87j:11057http://www.ams.org/mathscinet-getitem?mr=87j:11057http://www.ams.org/mathscinet-getitem?mr=99j:11047http://www.ams.org/mathscinet-getitem?mr=99j:11047http://www.ams.org/mathscinet-getitem?mr=2000b:11057http://www.ams.org/mathscinet-getitem?mr=2000b:11057http://www.ams.org/mathscinet-getitem?mr=2001i:11058http://www.ams.org/mathscinet-getitem?mr=2001i:11058http://www.ams.org/mathscinet-getitem?mr=83j:12011http://www.ams.org/mathscinet-getitem?mr=83j:12011http://www.ams.org/mathscinet-getitem?mr=84f:32040http://www.ams.org/mathscinet-getitem?mr=84f:32040http://www.ams.org/mathscinet-getitem?mr=84f:32040http://www.ams.org/mathscinet-getitem?mr=83j:12011http://www.ams.org/mathscinet-getitem?mr=2001i:11058http://www.ams.org/mathscinet-getitem?mr=2000b:11057http://www.ams.org/mathscinet-getitem?mr=99j:11047http://www.ams.org/mathscinet-getitem?mr=87j:11057http://www.ams.org/mathscinet-getitem?mr=81f:10041

yang t - taylor expansion of eisenstein series - trans. amer. math. soc. 355 (2003) no.7 2663-2674

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