A First Course in Modular Forms

447 Pages·2005·2.93 MB·English

by  Fred Diamond, Jerry Shurman (auth.)

228
Graduate Texts in Mathematics
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S. Axler F.W. Gehring K.A. Ribet
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(continued after index)
Fred Diamond
Jerry Shurman
A First Course
in Modular Forms
FredDiamond JerryShurman
DepartmentofMathematics DepartmentofMathematics
BrandeisUniversity ReedCollege
Waltham,MA02454 Portland,OR97202
USA USA
[email protected] [email protected]
EditorialBoard
S.Axler F.W.Gehring K.A.Ribet
MathematicsDepartment MathematicsDepartment MathematicsDepartment
SanFranciscoState EastHall UniversityofCalifornia,
University UniversityofMichigan Berkeley
SanFrancisco,CA94132 AnnArbor,MI48109 Berkeley,CA94720-3840
USA USA USA
[email protected] [email protected] [email protected]
MathematicsSubjectClassification(2000):25001,11019
LibraryofCongressCataloging-in-PublicationData
Diamond,Fred.
Afirstcourseinmodularforms/FredDiamondandJerryShurman.
p. cm.—(Graduatetextsinmathematics;228)
Includesbibliographicalreferencesandindex.
ISBN0-387-23229-X
1.Forms,Modular. I.Shurman,JerryMichael. II.Title. III.Series.
QA243.D47 2005
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Preface
This book explains a result called the Modularity Theorem:
All rational elliptic curves arise from modular forms.
Taniyamafirstsuggestedinthe1950’sthatastatementalongtheselinesmight
betrue,andapreciseconjecturewasformulatedbyShimura.ApaperofWeil
[Wei67] provides strong theoretical evidence for the conjecture. The theorem
was proved for a large class of elliptic curves by Wiles [Wil95] with a key
ingredient supplied by joint work with Taylor [TW95], completing the proof
of Fermat’s Last Theorem after some 350 years. The Modularity Theorem
was proved completely by Breuil, Conrad, Taylor, and the first author of this
book [BCDT01]. Different forms of it are stated here in Chapters 2, 6, 7, 8,
and 9.
To describe the theorem very simply for now, first consider a situation
from elementary number theory. Take a quadratic equation
Q:x2 =d, d∈Z, d(cid:2)=0,
and for each prime number p define an integer a (Q),
p
(cid:1) (cid:2)
the number of solutions x of equation Q
a (Q)= −1.
p
working modulo p
The values a (Q) extend multiplicatively to values a (Q) for all positive in-
p n
tegers n, meaning that a (Q)=a (Q)a (Q) for all m and n.
mn m n
Since by definition a (Q) is the Legendre symbol (d/p) for all p > 2, one
p
statement of the Quadratic Reciprocity Theorem is that a (Q) depends only
p
on the value of p modulo 4|d|. This can be reinterpreted as a statement that
the sequence of solution-counts {a (Q), a (Q), a (Q),...} arises as a system
2 3 5
of eigenvalues on a finite-dimensional complex vector space associated to the
equation Q. Let N = 4|d|, let G = (Z/NZ)∗ be the multiplicative group of
viii Preface
integer residue classes modulo N, and let V be the vector space of complex-
N
valued functions on G,
V ={f :G−→C}.
N
For each prime p define a linear operator T on V ,
p N
(cid:3)
f(pn) if p(cid:1)N,
T :V −→V , (T f)(n)=
p N N p 0 if p|N,
where the product pn ∈ G uses the reduction of p modulo N. Consider a
particular function f =f in V ,
Q N
f :G−→C, f(n)=a (Q) for n∈G.
n
ThisiswelldefinedbyQuadraticReciprocityasstatedabove.Itisimmediate
that f is an eigenvector for the operators T ,
p
(cid:3)
f(pn)=a (Q)=a (Q)a (Q) if p(cid:1)N,
(T f)(n)= pn p n
p 0 if p|N
=a (Q)f(n) in all cases.
p
That is, T f = a (Q)f for all p. This shows that the sequence {a (Q)} is a
p p p
system of eigenvalues as claimed.
The Modularity Theorem can be viewed as giving an analogous result.
Consider a cubic equation
E :y2 =4x3−g x−g , g ,g ∈Z, g3−27g2 (cid:2)=0.
2 3 2 3 2 3
Such equations define elliptic curves, objects central to this book. For each
prime number p define a number a (E) akin to a (Q) from before,
p p
(cid:1) (cid:2)
the number of solutions (x,y) of equation E
a (E)=p− .
p
working modulo p
One statement of Modularity is that again the sequence of solution-counts
{a (E)} arises as a system of eigenvalues. Understanding this requires some
p
vocabulary.
A modular form is a function on the complex upper half plane that sat-
isfies certain transformation conditions and holomorphy conditions. Let τ be
a variable in the upper half plane. Then a modular form necessarily has a
Fourier expansion,
(cid:4)∞
f(τ)= a (f)e2πinτ, a (f)∈C for all n.
n n
n=0
Each nonzero modular form has two associated integers k and N called its
weight and its level. The modular forms of any given weight and level form
Preface ix
a vector space. Linear operators called the Hecke operators, including an op-
erator T for each prime p, act on these vector spaces. An eigenform is a
p
modular form that is a simultaneous eigenvector for all the Hecke operators.
By analogy to the situation from elementary number theory, the Modular-
ity Theorem associates to the equation E an eigenform f = f in a vector
E
space V of weight 2 modular forms at a level N called the conductor of E.
N
The eigenvalues of f are its Fourier coefficients,
T (f)=a (f)f for all primes p,
p p
and a version of Modularity is that the Fourier coefficients give the solution-
counts,
a (f)=a (E) for all primes p. (0.1)
p p
Thatis,thesolution-countsofequationE areasystemofeigenvalues,likethe
solution-counts of equation Q, but this time they arise from modular forms.
This version of the Modularity Theorem will be stated in Chapter 8.
Chapter 1 gives the basic definitions and some first examples of modular
forms. It introduces elliptic curves in the context of the complex numbers,
where they are defined as tori and then related to equations like E but with
g ,g ∈ C. And it introduces modular curves, quotients of the upper half
2 3
planethatareinsomesensemorenaturaldomainsofmodularformsthanthe
upperhalfplaneitself.ComplexellipticcurvesarecompactRiemannsurfaces,
meaningtheyareindistinguishableinthesmallfromthecomplexplane.Chap-
ter2showsthatmodularcurvescanbemadeintocompactRiemannsurfaces
as well. It ends with the book’s first statement of the Modularity Theorem,
relating elliptic curves and modular curves as Riemann surfaces: If the com-
plex number j = 1728g3/(g3−27g2) is rational then the elliptic curve is the
2 2 3
holomorphic image of a modular curve. This is notated
X (N)−→E.
0
Much of what follows over the next six chapters is carried out with an eye
to going from this complex analytic version of Modularity to the arithmetic
version(0.1).Thusthisbook’saimisnottoproveModularitybuttostateits
different versions, showing some of the relations among them and how they
connect to different areas of mathematics.
Modularformsmakeupfinite-dimensionalvectorspaces.Tocomputetheir
dimensions Chapter 3 further studies modular curves as Riemann surfaces.
Two complementary types of modular forms are Eisenstein series and cusp
forms. Chapter 4 discusses Eisenstein series and computes their Fourier ex-
pansions.Intheprocessitintroducesideasthatwillbeusedlaterinthebook,
especially the idea of an L-function,
(cid:4)∞
a
L(s)= n.
ns
n=1
x Preface
Here s is a complex variable restricted to some right half plane to make the
series converge, and the coefficients a can arise from different contexts. For
n
instance, they can be the Fourier coefficients a (f) of a modular form. Chap-
n
ter 5 shows that if f is a Hecke eigenform of weight 2 and level N then its
L-function has an Euler factorization
(cid:5)
L(s,f)= (1−a (f)p−s+1 (p)p1−2s)−1.
p N
p
The product is taken over primes p, and 1 (p) is 1 when p (cid:1) N (true for all
N
but finitely many p) but is 0 when p|N.
Chapter 6 introduces the Jacobian of a modular curve, analogous to a
complex elliptic curve in that both are complex tori and thus have Abelian
group structure. Another version of the Modularity Theorem says that every
complexellipticcurvewitharationalj-valueistheholomorphichomomorphic
image of a Jacobian,
J (N)−→E.
0
Modularity refines to say that the elliptic curve is in fact the image of a
quotientofaJacobian,theAbelianvarietyassociatedtoaweight2eigenform,
A −→E.
f
This version of Modularity associates a cusp form f to the elliptic curve E.
Chapter 7 brings algebraic geometry into the picture and moves toward
number theory by shifting the environment from the complex numbers to the
rational numbers. Every complex elliptic curve with rational j-invariant can
be associated to the solution set of an equation E with g ,g ∈ Q. Modular
2 3
curves, Jacobians, and Abelian varieties are similarly associated to solution
sets of systems of polynomial equations over Q, algebraic objects in contrast
to the earlier complex analytic ones. The formulations of Modularity already
in play rephrase algebraically to statements about objects and maps defined
by polynomials over Q,
X (N) −→E, J (N) −→E, A −→E.
0 alg 0 alg f,alg
We discuss only the first of these in detail since X (N) is a curve while
0 alg
J (N) and A are higher-dimensional objects beyond the scope of this
0 alg f,alg
book. These algebraic versions of Modularity have applications to number
theory,forexampleconstructingrationalpointsonellipticcurvesusingpoints
called Heegner points on modular curves.
Chapter 8 develops the Eichler–Shimura relation, describing the Hecke
operator T in characteristic p. This relation and the versions of Modularity
p
alreadystatedhelptoestablishtwomoreversionsoftheModularityTheorem.
One is the arithmetic version that a (f)=a (E) for all p, as above. For the
p p
other, define the Hasse–Weil L-function of an elliptic curve E in terms of the
solution-counts a (E) and a positive integer N called the conductor of E,
p
Preface xi
(cid:5)
L(s,E)= (1−a (E)p−s+1 (p)p1−2s)−1.
p N
p
Comparing this to the Euler product form of L(s,f) above gives a version of
Modularity equivalent to the arithmetic one: The L-function of the modular
form is the L-function of the elliptic curve,
L(s,f)=L(s,E).
Asafunctionofthecomplexvariables,bothL-functionsareinitiallydefined
only on a right half plane, but Chapter 5 shows that L(s,f) extends ana-
lytically to all of C. By Modularity the same now holds for L(s,E). This is
important because we want to understand E as an Abelian group, and the
conjecture of Birch and Swinnerton-Dyer is that the analytically continued
L(s,E) contains information about the group’s structure.
Chapter 9 introduces (cid:2)-adic Galois representations, certain homomor-
phisms of Galois groups into matrix groups. Such representations are asso-
ciated to elliptic curves and to modular forms, incorporating the ideas from
Chapters 6 through 8 into a framework with rich additional algebraic struc-
ture. The corresponding version of the Modularity Theorem is: Every Galois
representation associated to an elliptic curve over Q arises from a Galois
representation associated to a modular form,
ρ ∼ρ .
f,(cid:3) E,(cid:3)
ThisistheversionofModularitythatwasproved.Thebookendsbydiscussing
twobroaderconjecturesthatGaloisrepresentationsarisefrommodularforms.
Manygoodbooksonmodularformsalreadyexist,buttheycanbedaunt-
ing for a beginner. Although some of the difficulty lies in the material itself,
the authors believe that a more expansive narrative with exercises will help
students into the subject. We also believe that algebraic aspects of modu-
lar forms, necessary to understand their role in number theory, can be made
accessible to students without previous background in algebraic number the-
ory and algebraic geometry. In the last four chapters we have tried to do so
by introducing elements of these subjects as necessary but not letting them
take over the text. We gratefully acknowledge our debt to the other books,
especially to Shimura [Shi73].
Theminimalprerequisitesareundergraduatesemestercoursesinlinearal-
gebra, modern algebra, real analysis, complex analysis, and elementary num-
ber theory. Topics such as holomorphic and meromorphic functions, congru-
ences, Euler’s totient function, the Chinese Remainder Theorem, basics of
general point set topology, and the structure theorem for modules over a
principal ideal domain are used freely from the beginning, and the Spectral
Theorem of linear algebra is cited in Chapter 5. A few facts about represen-
tations and tensor products are also cited in Chapter 5, and Galois theory is