Table Of ContentSpringer Monographs in Mathematics
Goro Shimura
EEElementary Dirichlet Series
and Modular Forms
Goro Shimura
Department of Mathematics
Princeton University
Princeton, New Jersey 08544-1000
[email protected]
ISBN: 978-0-387-72473-7 eISBN: 978-0-387-72474-4
Library ofCongress Control Number: 2007931368
Mathematics Subject Classification (2000):11F11, 11F67, 11G05, 11G10, 11G15, 11R29, 11M06
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PREFACE
A book on any mathematical subject above textbook level is not of much
value unless it contains new ideas and new perspectives. Also, the author may
be encouraged to include new results, provided that they help the reader gain
newinsightsandarepresentedalongwithknownoldresultsinaclearexposition.
ItiswiththisphilosophythatIwritethisvolume. Thetwosubjects,Dirichlet
seriesandmodularforms,aretraditional,butItreattheminbothorthodoxand
unorthodox ways. However, I try to make the book accessible to those who are
not familiar with such topics, by including plenty of expository material. More
specific descriptions of the contents will be given in the Introduction.
To some extent, this book has a supplementary nature to my previous book
Introduction to the Arithmetic Theory of Automorphic Functions, published by
PrincetonUniversity Pressin1971, thoughIdonotwritethepresentbookwith
that intent. While the 1971 book grew out of my lectures in various places, the
essentialpointsofthisnewbookhaveneverbeenpresentedpubliclyorprivately.
I hope that it will draw an audience as large as that of the previous book.
Princeton
March 2007 Goro Shimura
v
TABLE OF CONTENTS
Preface v
Introduction 1
Chapter I. Preliminaries on Modular Forms and
Dirichlet Series 5
1. Basic symbols and the definition of modular forms 5
2. Elementary Fourier analysis 13
3. The functional equation of a Dirichlet series 19
Chapter II. Critical Values of Dirichlet L-functions 25
4. The values of elementary Dirichlet series at integers 25
5. The class number of a cyclotomic field 39
6. Some more formulas for L(k, χ) 45
Chapter III. The Case of Imaginary Quadratic Fields
and Nearly Holomorphic Modular Forms 53
7. Dirichletseriesassociatedwithanimaginaryquadratic
field 53
8. Nearly holomorphic modular forms 55
Chapter IV. Eisenstein Series 59
9. Fourier expansion of Eisenstein series 59
10. Polynomial relations between Eisenstein series 66
11. Recurrence formulas for the critical values of certain
Dirichlet series 75
Chapter V. Critical Values of Dirichlet Series Asso-
ciated with Imaginary Quadratic Fields 79
12. The singular values of nearly holomorphic forms 79
13. The critical values of L-functions of an imaginary qua-
dratic field 84
14. Thezetafunctionofamemberofaone-parameterfam-
ily of elliptic curves 96
Chapter VI. Supplementary Results 113
15. Isomorphism classes of abelian varieties with complex
multiplication 113
vii
viii TABLE OF CONTENTS
15A. The general case 113
15B. The case of elliptic curves 117
16. Holomorphic differential operators on the upper half
plane 120
Appendix 127
A1. Integration and differentiation under the integral sign 127
A2. Fourier series with parameters 130
A3. The confluent hypergeometric function 131
A4. The Weierstrass ℘-function 136
A5. The action of G on modular forms 139
A+
References 145
Index 147
INTRODUCTION
There are two types of Dirichlet series that we discuss in this book:
(cid:1)
(1) Dr (s)= nr|n|−r−s,
a,N
0=(cid:1) n∈a(cid:1)+NZ
(2) Lr(s; α, b)= ξ−r|ξ|r−2s.
0=(cid:1) ξ∈α+b
Here s is a complex variable as usual, r is 0 or 1 for Dr and 0 < r ∈ Z for
a,N
the latter series; a ∈ Z and 0 < N ∈ Z. To define the series of (2) we take an
imaginary quadratic field K embedded in C and take also an element α of K
and a Z-lattice b in K. One of our principal problems is to investigate the na-
ture of the values of these series at certain integer values of s. As a preliminary
step, we discuss their analytic continuation and functional equaltions. We ob-
tain Dirichlet L-functions and certain Hecke L-functions of K as suitable linear
combinations of these series, and so the values of such L-functions are included
in our objects of study. As will be explained below, these series are directly and
indirectly related to elliptic modular forms, and the exposition of such functions
in that context forms a substantial portion of this volume. Thus, as we said in
the preface, the main objective of this book is to present some new ideas, new
results, and new perspectives, along with old ones in this area covering certain
aspects of the theory of modular forms and Dirichlet series.
To be more specific, let us first consider the Dirichlet L-function
(cid:1)∞
L(s, χ)= χ(n)n−s
n=1
with a primitive Dirichlet character χ modulo a positive integer N. It is well
known that if k is a positive integer and χ(−1)=(−1)k, then
(cid:1)N
2k!G(χ)
(3) L(k, χ)=− χ(a)B (a/N),
(2πi)k k
a=1
where B is the Bernoulli polynomial of degree k and G(χ) is the Gauss sum of
k
χ. If k = 1 in particular, the last sum for χ such that χ(−1) = −1 becomes
(cid:2)
N χ(a)a/N,whichremindsusofanotherwellknownresultaboutthesecond
a=1
factor of the class number of a cyclotomic field, which is written h /h . Here
K F
1
2 INTRODUCTION
h resp. h is the class number of K resp. F; K is an imaginary subfield of
K F
Q(ζ) with a primitive mth root of unity ζ for some positive integer m and
F is the maximal real subfield of K. There is a classical formula for h /h ,
(cid:3) (cid:2) K F
which is easy factors times a product χ(a)a, where X is a certain set
χ∈X a
of primitive Dirichlet characters χ such that χ(−1)=−1.
No alternative formulas have pr√eviously been presented except when K is an
imaginary quadratic field K = Q( −d). Given such a K, take a real character
χ of conductor d that corresponds to K. Then it is well known that
√
(cid:1)q
w d w
(4) K L(1, χ)=h = (cid:4) K (cid:5) χ(a), q =[(d−1)/2],
2π K 2 2−χ(2)
a=1
where w is the number of roots of unity in K.
K
Now we will prove as one of the main results of this book that there are new
formulas for L(k, χ). The most basic one is
(k−1)!G(χ) 1 (cid:1)q
(5) (2πi)k L(k, χ)= 2(cid:4)2k−χ(2)(cid:5) χ(a)E1,k−1(2a/d),
a=1
where q =[(d−1)/2]andE1,k−1(t)istheEulerpolynomialofdegree k−1.This
clearly includes (4) (without h ) as a special case, as E (t)=1. This formula
K 1,0
is better than (3) at least from the computational viewpoint, as E1,k−1(t) is a
polynomial in t of degree k−1, whereas B (t) is of degree k. We will present
k
many more new formulas for L(k, χ) in Sections 4 and 6. As applications, we
will prove some new formulas for the quotient h /h .
K F
Toavoidexcessivedetails,westateitinthisintroductiononlywhenK =Q(ζ)
with m=2r >4:
(cid:7) (cid:8)
h (cid:6)r (cid:6) (cid:1)ds
K =2γ χ(a) , d =2s−2−1, γ =r−1−2r−2.
h s
F s=3 χ∈Ys a=1
Here Y is the set of primitive Dirichlet characters χ of conductor 2s such that
s (cid:2)
χ(−1)=−1. Notice that we have ds χ(a), which is of far “smaller size” than
(cid:2) a=1
the sum χ(a)a in the classical formula. A similar but somewhat different
a
formula can be obtained in the case m=(cid:8)r with an odd prime (cid:8).
The latter part of the book concerns the critical values of the series of type
(2). Inthiscaseweevaluateitat k/2 withaninteger k suchthat 2−r ≤k ≤r
and r−k ∈2Z. Then we can show that:
(6) There is a constant γ which depends only on K (that is, independent of
α, b, r,and k)suchthatLr(k/2; α, b)is π(r+k)/2γr timesanalgebraicnumber.
This was proved in one of the author’s papers. Though we will give a proof
of this fact in this book, it is merely the starting point. Indeed one of our
main problems is to find a suitable γ so that the algebraic number can be
INTRODUCTION 3
computed. Before going into this problem, we note that the constant γ can
be giv(cid:9)en as (cid:10)ϕ(τ) with a(cid:11)modular form ϕ of weight 1 and τ ∈ K ∩H, where
H = z ∈C(cid:10)Im(z)>0 .
If r = k (cid:5)= 2, the value Lr(r/2; α, b) can be given as πrh(τ) with a holo-
morphic modular form h of weight r. Thus our task is to find (h/ϕr)(τ).
This can be achieved as follows. We fix a congruence subgroup Γ of SL (Z)
2
to which both ϕ and h belong, and assume that we can find two modular
forms f and g that generate the algebra of all modular forms of all nonneg-
ative weights w(cid:4)ith respect to Γ. The(cid:5)n h = P(f, g) with a polynomial P, and
(h/ϕr)(τ)=P (f/ϕk)(τ), (g/ϕ(cid:5))(τ) , where k resp. (cid:8) is the weight of f resp.
g. However there are two essential questions:
(I) How can we find P?
(II) In the general case in which r (cid:5)=k or k =2, Lr(k/2; α, b) can be given
as π(r+k)/2p(τ) with some nonholomorphic modular form (which we call nearly
holomorphic) p of weight r. Then, how can we handle (p/ϕr)(τ)?
Problem (II) can be reduced to Problem (I) and Problem (II) for simpler
(cid:2)
p. In the easiest case we can express p as a polynomial [r/2]Eah , where
a=0 2 a
h is a holomorphic modular form of weight r −2a and E is a well known
a 2
nonholomorphic Eisenstein series of weight 2:
(cid:12) (cid:13)
(cid:1)∞ (cid:1)
1 1
E (z)= − + d e(nz).
2 8πy 24
n=1 0<d|n
Then the problem can be reduced to (E /ϕ2)(τ) and (h /ϕr−2a)(τ). The latter
2 a
quantity is handled by P for h . As for E , we have to deal with it in a special
a 2
way. Foragiven τ wewillfindaspecialholomorphicmodularform q ofweight
2 such that (E /q)(τ) can be explicitly given.
2
Inthiswaythevalueofnonholomorphicfunctionscanbereducedtothecaseof
holomorphicmodularforms,andtoProblem(I).Wealsohavetofind(f/ϕk)(τ)
and(g/ϕ(cid:5))(τ),whichisnontrivial,butourideaistoreduceinfinitelymanyvalues
to finitely many values. In general, there is no clear-cut answer to (I). However,
we can produce two types of recurrence formulas for Eisenstein series, which
seem to be new and by which the problem about h/ϕr of an arbitrary weight
r can be reduced to the case of smaller r. (See (10.8) and (10.15c).) Without
stating it, we content ourselves by mentioning its application to Lr(k/2; α, b).
Assuming that α∈/ b, 0<k ≤r, and r−k ∈2Z, put n=(r−k)/2, and
Ln(α, b)=(−1)k(2πi)−k−nΓ(k+n)Lr(k/2; α, b)
k (cid:14)
2(2i)nIm(τ)n(DnE )(τ) if k =2,
− 2 2
0 if k (cid:5)=2,
where Dn is a differential operator of the type mentioned above. Then we have
2
a recurrence formula
4 INTRODUCTION
(cid:12) (cid:13) (cid:12) (cid:13)
(cid:1)t (cid:1)n
Ln (α, b)=12 t n Lj (α, b)·Ln−j (α, b)
t+5 i j i+3 t−i+2
i=0 j=0
for 0 ≤ t ∈ Z and 0 ≤ n ∈ Z. Thus the values of Ln(α, b) for k > 4 can be
k
reduced inductively to those for 2≤k ≤4. If α∈b, there is another recurrence
formula which reduces Ln (α, b) for 2k ≥ 8 to the cases with 2k = 4 and
2k
2k =6.
(cid:2)
We can form a Hecke L-function L(s, λ) = λ(a)N(a)−s, with a Hecke
a
ideal character λ of K such that
λ(αr)=α−r|α|r if α∈K× and α−1∈c,
where c is an integral ideal of K. Since this is a finite linear combination of
series of type (2), statement (6) holds for L(k/2, λ) in place of Lr(k/2; α, b). In
Section 13, we will present many examples of numerical values of L(k/2, λ).
When r =1,thefunctionL(s, λ)iscloselyconnectedwithanellipticcurveC
definedoveranalgebraicnumberfield h withcomplexmultiplication inK.Ina
certaincase,itisindeedthezetafunctionofC over h.Wewillstudythisaspect
in Section 14, and compare L(1/2, λ) with a period of a holomorphic 1-form on
C, when C is a member of a one-parameter family {Cz}z∈H of elliptic curves.
However,withoutgoingintodetailsofthistheory,letusendthisintroduction
by briefly mentioning some other noteworthy features of the book.
(A) A discussion of irregular cusps of a congruence subgroup of SL (Z) in
2
§1.11 and Theorem 1.13.
(B) The functional equation of the Eisenstein series
(cid:1)
EN(z, s; p, q)=Im(z)s (mz+n)−k|mz+n|−2s
k
(m,n)
under s(cid:7)→1−k−s (Theorem 9.7). Here (z, s)∈H ×C, 0<N <Z, 0≤k ∈
Z, (p, q)∈Z2, and (m, n) runs over Z2 under the condition 0(cid:5)=(m, n)≡(p, q)
(mod NZ2).
(C) The explicit Fourier expansion of EN(z, 1−k; p, q) given in (9.14).
k
(D) In Section 15, we discuss isomorphism classes of abelian varieties, elliptic
curves in particular, with complex multiplication defined over a number field
with the same zeta function.
(cid:9) (E(cid:11))InSection16wepresentanewclassofholomorphicdifferentialoperators
Ap ∞ . The operator Ap sends an automorphic form of weight k to that of
k p=2 k
weight kp+2p, and every operator of the same nature can be reduced to this
class.
