SUBCONVEXITY BOUNDS FOR AUTOMORPHIC L-FUNCTIONS
ON GL2
A DISSERTATION
SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL
OF THE UNIVERSITY OF MINNESOTA
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BY
PR
EV
IE
Delia Daria Letang
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Paul Garrett, advisor
May 2009
UMI Number: 3358635
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c Delia Daria Letang, May 2009
ACKNOWLEDGEMENTS
I would like to express my sincere thanks to Professor Paul Garrett - my advisor,
teacher, and mentor. Thank you for meeting me week after week, answering my countless
questions, providing solid advice, and always setting me on the right path. Because of
you, I am a better mathematician and a better person. I also thank Professor Adrian
Diaconu who unfortunately could not be here for my final oral exam. Thank you for
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clearing up some of the muddling issues and for helping to make this thesis a reality.
Special thanks to Professors Dihua Jiang, Yongdae Kim, and Richard McGehee for
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consenting to serve on my examination committee and for your comments on this thesis.
I would not have completed this dissertation without the support and love from my
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husband. Babs, thank you so much for the many days and nights of taking care of our
sons even when I knew you were burning the candle at both ends. Thank you for all
that you did to help me to complete the PhD program. Many thanks to my mother, my
brother and sister-in-law Nathaniel and Wendy, my sister Sandra and nephew Rommell
for visiting me and taking care of my family. Thanks to the rest of my family and my
friends for your encouragement and support. A special thank you to Chuck and Peggy
MacCarthy for introducing me to the University of Minnesota, helping me to make
a smooth transition from the Caribbean to Minnesota, and for being such influential
factors in my life.
Finally, I would like to thank my Savior and Lord Jesus Christ for seeing me through.
Truly, “I can do all things through Him who strengthens me”.
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DEDICATION
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This thesis is dedicated to my husband Earlsworth and my two sons, Earl and Sam.
ii
ABSTRACT
Asymptotics for integral moments of automorphic L-functions are highly non-trivial
to obtain, but have serious implications. Suitable asymptotics for integral moments
of L-functions would prove the Lindelöf Hypothesis. Conjectures for moments of Lfunctions were initiated by Hardy and Littlewood in 1918. Subconvexity bounds in a
given aspect have geometric and number-theoretic applications and are sufficient for
providing solutions to many problems.
In this thesis, we develop asymptotics for the second integral moments of families of
automorphic L-functions for GL2 over an arbitrary number field. These L-functions are
twisted by idele class characters χ. The weight functions are derived from archimedean
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data as well as data associated with a finite prime at which χ has arbitrary ramification.
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We break convexity at this non-archimedean place.
iii
Contents
1 INTRODUCTION
1
Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2.1
The Euler-Riemann-zeta function . . . . . . . . . . . . . . . . . .
3
1.2.2
The Riemann Hypothesis . . . . . . . . . . . . . . . . . . . . . .
4
1.2.3
The Prime Number Theorem . . . . . . . . . . . . . . . . . . . .
4
1.2.4
Lindelöf Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . .
6
Development of the Theory of L-functions . . . . . . . . . . . . . . . . .
7
1.3.1
Riemann’s method . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.3.2
Hecke on GL1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.3.3
Weil’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.3.4
Maass Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.3.5
Tate’s thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.3.6
Jacquet-Langlands method . . . . . . . . . . . . . . . . . . . . .
13
1.3.7
GLn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.4
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1.3
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1.1
The Main result
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 THE MOMENT EXPANSION
15
17
2.1
Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2
Unwinding to an Euler Product . . . . . . . . . . . . . . . . . . . . . . .
17
2.3
The non-decoupled integrals . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.4
Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
iv
3 SPECTRAL DECOMPOSITION OF THE POINCARÉ SERIES
29
3.1
Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.2
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.3
The Cuspidal Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.4
The continuous part . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
4 PRELIMINARIES TO SUBCONVEXITY
46
4.1
Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
4.2
Meromorphic continuation of Z(w′ ) . . . . . . . . . . . . . . . . . . . . .
47
4.3
Polynomial growth of Z(w′ ) . . . . . . . . . . . . . . . . . . . . . . . . .
50
57
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5 SUBCONVEXITY BOUNDS
Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
5.2
Trivial bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
5.2.1
The completed L-function . . . . . . . . . . . . . . . . . . . . . .
58
5.2.2
The local epsilon factors . . . . . . . . . . . . . . . . . . . . . . .
60
5.2.3
Ratio of gamma functions . . . . . . . . . . . . . . . . . . . . . .
62
5.2.4
The functional equation . . . . . . . . . . . . . . . . . . . . . . .
64
5.2.5
Applying the Phragmen-Lindelöf Principle . . . . . . . . . . . . .
64
Subconvexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
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5.1
BIBLIOGRAPHY
v
76
Chapter 1
1.1
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INTRODUCTION
Prologue
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The Riemann Hypothesis and its extension to general L-functions (the Grand Riemann
Hypothesis) are among the most important open problems in mathematics. These are
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central because of consequences for the distribution of primes, and because analogous
applications exist for broad classes of L-functions.
One consequence of the Riemann Hypothesis which would yield many of its corollaries is the Lindelöf Hypothesis, which states
1
ζ( + it) ≪ |t|ǫ for all ǫ > 0
2
Although unproven, some progress has been made in obtaining approximations to the
Lindelöf Hypothesis. To begin with, Riemann’s functional equation for the zeta function,
Stirling’s approximation for the Gamma function, and the Phragmen-Lindelöf principle
were used to obtain the convexity or trivial bound for the zeta function:
1
1
|ζ( + it)| ≪ |t| 4 +ǫ
2
Any improvement over
1
4
in this upper bound “breaks convexity”, and, not only could
be felt as progress toward Lindelöf, but also has far-reaching applications. Moreover, in
recent years, L-functions have played a central role in number theory and one issue is
1
estimating the size of L-functions inside the critical strip, and certainly on the critical
line. Recently, [Molteni 2000] proved the trivial bound for GLn−1 × GLn L-functions.
Various authors have obtained subconvexity bounds in different aspects. For example, [Weyl 1921] gave a subconvex bound
1
1
|ζ( + it)| ≪ |t| 6 +ǫ
2
[Burgess 1962] broke convexity in the conductor aspect for Dirichlet L-functions over
Q. Subconvexity bounds were also obtained for GL2 L-functions in [Good 1982, 1986],
[Meurman 1987] and [Duke-Friedlander-Iwaniec 1993, 1994, 2001]. In recent years, subconvexity results were obtained by several authors including Kowalski, Michel, Van-
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derkam and Venkatesh (see [Kowalski-Michel-Vanderkam 2002] and [Michel-Venkatesh
2006]).
Asymptotics for integral moments of automorphic L-functions have stimulated much
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work, since a sufficiently good estimate on all moments of ζ( 21 + it) would prove Lindelöf. Conjectures for moments of the Riemann-zeta function on the critical line were
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first introduced by Hardy and Littlewood in [Hardy-Littlewood 1918], who obtained the
second moment of the Riemann-zeta function
Z T
1
|ζ( + it)|2 dt ∼ T log T
2
0
In 1926, Ingham [Ingham 1926] obtained the fourth moment
Z T
1
1
|ζ( + it)|4 dt ∼
· T (log T )4
2
2
2π
0
Various authors have predicted other moments. However, until recently, all of these results concerned integral moments of automorphic L-functions over Q, or over quadratic
extensions of Q, and not over an arbitrary number field. In 2006, Diaconu and Goldfeld [Diaconu-Goldfeld 2006a, 2006b] reconsidered the cases of groundfield Q or complex quadratic extensions. Then Diaconu and Garrett [Diaconu-Garrett 2008] obtained
asymptotics with error-term for second integral moments of GL2 automorphic L-functions
over an arbitrary number field. Diaconu and Garrett obtained these moments by a spectral decomposition using the representation theory of adele groups GL1 and GL2 . Here
2
the L-functions were twisted by all unramified idele class characters χ. That is, in the
relevant spectral identity, Diaconu-Garrett obtained asymptotics with power-saving in
the error term, for
χ
∞
1
|L( + it, f ⊗ χ)|2 Mχ (t) dt
2
−∞
XZ
where Mχ (t) are smooth weights, and used these asymptotics with error-term to break
convexity in the t-aspect.
My research takes Diaconu-Garrett’s ideas in a new direction. I will also focus on the
second weighted moment of a GL2 automorphic L-function over an arbitrary number
field, but I will change the data associated with a fixed non-archimedean place v1 . I will
allow χ to have increasing ramification at v1 (Diaconu-Garrett only treated unramified
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χ). Thus the weights in the moment expansion will be obtained from the archimedean
data as well as the data associated with the ramification at the finite prime v1 . I will
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then obtain asymptotics for that second moment expansion and will break convexity in
1.2
1.2.1
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the χ-depth-aspect at the non-archimedean place v1 .
Background
The Euler-Riemann-zeta function
The Euler-Riemann-zeta function ζ(s) is a function of the complex variable s = σ + it
∞
X
1
ζ(s) =
(for ℜ(s) > 1)
ns
n=1
The zeta function ζ(s) has a meromorphic continuation to the entire complex plane with
a simple pole at s = 1 with residue 1. It has an Euler product
ζ(s) =
Y
p
1
1 − p−s
Riemann’s functional equation for the zeta function is
s
s
ξ(s) = ξ(1 − s) (where ξ(s) = π − 2 Γ( )ζ(s))
2
3
The functional equation, Stirling’s approximation for the gamma function, and the
Phragmen-Lindelöf principle, can be used to prove the existence of the convexity bound
for ζ(s).
The Riemann-zeta function has trivial zeros at the negative even integers s = −2, −4, −6, . . .
and its non-trivial zeros are of the form s = σ + it for 0 < σ < 1.
1.2.2
The Riemann Hypothesis
The Riemann Hypothesis was formulated by Bernhard Riemann in 1859, speculating
that all non-trivial zeros of ζ(s) lie on the critical line ℜ(s) = 21 . A more general state1
2.
general L-functions have real part equal to
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ment, called the Grand Riemann Hypothesis, conjectures that the non-trivial zeros of
The Riemann Hypothesis remains un-
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reachable today although we understand much more about L-functions. Riemann knew
that the non-trivial zeros lie within 0 ≤ ℜ(s) ≤ 1. In 1896, Hadamard and de la ValléePoussin proved that no zero lies on the line ℜ(s) = 1. This fact, together with other
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properties of the zeta function, showed that all non-trivial zeros must lie within the
critical strip 0 < ℜ(s) < 1. This was the key step in the proof of the Prime Number
Theorem. Recent work by van de Lune [Lune-Riele-Winter 1986] has shown that the
first ten billion zeros are on the critical line. Odlyzko [Odlyzko 1989] has also calculated
millions of zeros on the critical line.
1.2.3
The Prime Number Theorem
The Prime Number Theorem states that
π(x) ∼
x
log x
i.e.
lim π(x) ·
x→∞
log x
=1
x
Although the first recorded result about π(x) was about 300 B.C. when Euclid or his
predecessors proved that there is an infinite number of primes, the problem of the dis4
tribution of prime numbers received attention for the first time from Fermat in the
seventeenth century, and from Euler, Gauss and Legendre in the eighteenth century.
P1
Euler used the product expansion of ζ(s) and the divergence of
n to prove the diverP log p
P1
gence of
p and
p giving a quantitative proof of the infinitude of primes. Gauss
discovered empirically that
π(x) ≈
x
log x
and Legendre speculated subsequently that
π(x) =
x
A log x + B
where A = 1 and B = −1.08366. In 1850, Tchebychef showed that
Tchebychef also showed that if limx→∞
π(x) log x
x
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7
π(x) log x
9
<
<
8
x
8
exists, then it must be 1. In 1859, Rie-
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mann proposed a new method for proving the Prime Number Theorem, using Euler’s
expression of ζ(s) as an Euler product. Riemann did not prove the Prime Number Theo-
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rem but he remarked that if certain properties of the zeta function could be established,
then the Prime Number Theorem would be proven. In 1896, the Prime Number Theorem was completely proven by Hadamard and de la Vallée Poussin; the key step in the
proof was showing that ζ(s) has no zeros on the line ℜ(s) = 1. When Hadamard and
de la Vallée Poussin proved the Prime Number Theorem, they showed that
√
π(x) = Li(x) + O(xe−a
log x
)
for some positive constant a, where
Li(x) =
Z
2
The error term
√
O(xe−a log x )
x
1
dt
log t
was dependent on what was known about the zero-free
region within the critical strip. As knowledge of the size of the region increases, the
error term decreases. In the extreme case that we know the Riemann Hypothesis, we
get the best error estimate: in 1901, von Koch showed that the Riemann Hypothesis is
equivalent to
1
π(x) = Li(x) + O(x 2 log x)
5
Other statements equivalent to the Riemann Hypothesis are
1
π(x) − Li(x) = O(x 2 +ǫ ) for all ǫ > 0
and
π(x) =
x
1
+ O(xσ+ǫ ) for all ǫ > 0, σ ≥
log x
2
The Riemann Hypothesis would give the best possible error term in the Prime Number
Theorem.
1.2.4
Lindelöf Hypothesis
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The Lindelöf Hypothesis is a consequence of the Riemann Hypothesis, somewhat weaker
than the Riemann Hypothesis. It was formulated by E. Lindelöf in 1908 and conjectures
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estimates of the rate of growth of ζ(s) on the critical line. It speculates that
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1
ζ( + it) ≪ |t|ǫ (for all ǫ > 0)
2
i.e. ζ(s) grows rather slowly on the critical line. The Lindelöf Hypothesis for ζ(s) or
any L-function goes far beyond the trivial bound.
The Lindelöf Hypothesis is equivalent to the statement that
Z T
1
Ik (T ) =
|ζ( + it)|k dt ∼ T 1+ǫ (for k = 2, 4, 6, . . .)
2
−T
Thus, estimates of higher moments may eventually prove Lindelöf. Unfortunately, very
little is known about higher moments; no asymptotic formulas are presently known for
k ≥ 6. However, some work on formulating conjectures for the 2kth moment has been
done through the conjectural relationships between L-functions and Random Matrix
Theory. [Conrey-Ghosh 1984] conjectured that
Z T
1
ak
2
|ζ( + it)|2k dt ∼ gk 2 T logk T
2
k
!
0
where
∞ Y
1 k2 X k + m − 1 −m
ak =
(1 − )
p
p
m
p
m=0
6
and gk is an integer which could not be predicted. However, Keating and Snaith
[Keating-Snaith 2000] believe that
gk = k2 !
k−1
Y
j=0
j!
(k + j)!
Diaconu, Goldfeld and Hoffstein also showed that conjectures about the meromorphic
continuation and polar divisors of multiple Dirichlet series imply the Conrey-Ghosh
conjecture [Diaconu-Goldfeld-Hoffstein 2001]. It is not clear that random matrix theory
offers any mechanism to prove anything about moments, but it is an interesting heuristic.
Development of the Theory of L-functions
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1.3
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An L-function of an automorphic form f is a Dirichlet series
L(s, f ) =
∞
X
an
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n=1
ns
The theory of L-functions of modular forms originated in 1859 in Riemann’s paper
on ζ(s). Since then, there were other proofs of the analytic continuation of many Lfunctions by Hecke, Iwasawa, Tate, Rankin, Selberg, Langlands, Jacquet, Godement,
Piatetski-Shapiro, Bump, Shalika, Garrett, Rallis, Jiang, Ginzburg and others. Dirichlet
generalized the zeta function by introducing the L-functions
L(s, χ) =
∞
X
χ(n)
n=1
ns
=
Y
(1 − χ(p)p−s )−1
(χ a Dirichlet character)
p
but as functions of a real variable. Hecke’s theory [Gunning 1962] linked modular forms
and Dirichlet series with functional equations and showed the connection between Euler
products and Hecke operators. Tate and Iwasawa ([Tate 1950], [Iwasawa 1992]), in what
is often known as Tate’s thesis, gave an intelligible proof of the meromorphic continuation and functional equation of GL1 L-functions. Subsequently, Jacquet and Langlands
rewrote Hecke’s and Maass’ treatment of GL2 L-functions of automorphic representations. Their approach followed Hecke and the local-global techniques of Tate-Iwasawa.
7
In 1972, Godement and Jacquet [Godement-Jacquet 1972] treated standard L-functions
for GLn in a style resembling Tate-Iwasawa theory. In 1979, Jacquet, Piatetski-Shapiro
and Shalika [Jacquet-PS-Shalika 1979] extended Hecke’s construction of L-functions of
automorphic forms on GLn , using Fourier-Whittaker expansions on GLn .
1.3.1
Riemann’s method
Riemann’s method for proving the meromorphic continuation of ζ(s) and for deriving
the functional equation applied the Poisson summation formula to the Gaussian
2
f (x) = e−πx t , t > 0
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so obtaining the Jacobi identity
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1
−1
Θ(it) = √ Θ( )
it
t
Riemann then obtained an integral representation
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s
s
ξ(s) = π − 2 Γ( )ζ(s)
2
Z ∞
Θ(iy) − 1 s dy
=
y2
2
y
0
from which the analytic properties of the zeta function follow.
1.3.2
Hecke on GL1
Let k be a number field, o its ring of integers, p a prime ideal in o, and χ a Hecke
character (or Größencharakter). Hecke’s L-function for χ is defined as:
L(s, χ) =
Y
p6∈S
(1 − χ(p)N p−s )−1 (for a finite set S of bad primes)
where N p is the number of elements in the finite field o/p. Hecke proved the analytic
continuation and derived the functional equation of L(s, χ) by using a theta function
much in the style of Riemann.
Hecke also developed the theory of L-functions of modular forms for congruence
subgroups of SL2 (Z). In 1937, he introduced a ring of operators acting on modular
8
forms systemizing Mordell’s example of Ramanujan’s ∆. Hecke’s idea was to obtain
“good” modular forms by finding an algebra acting on spaces of modular forms so
that the modular forms are diagonalizable. The eigenfunctions would then inherit the
structural properties of the algebra [Gunning 1962]. Thus an L-function was described as
“nice” (that is, has meromorphic continuation, Euler product and satisfies a functional
equation) if it is a Mellin transform of an eigenfunction of Hecke operators. That is, if
f is a normalized Hecke eigenfunction, or a holomorphic form of weight k for SL2 (Z),
then the L-function L(s, f ) has an Euler product
L(s, f ) =
Y
(1 − ap p−s + χ(p)pk−1−2s )−1
p
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Hecke also proved a converse theorem:
Suppose that |an | = O(nr ) for some r so that L(s) =
P
n an n
−s
converges absolutely
satisfies the functional equation
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when ℜ(s) is large. If L(s) has analytic continuation, is bounded in vertical strips and
k
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Λ(s, f ) = (−1) 2 Λ(k − s)
then f (z) =
P
n
an q n is a cusp form of weight k and level 1. If L(f, s) has an Euler
product, then f is a Hecke eigenform.
1.3.3
Weil’s method
Weil completed Hecke’s theory by characterizing modular forms for congruence subgroups such as Γ0 (N ). Weil’s breakthrough was to consider twists of an L-function by
a Dirichlet character , namely
L(f ⊗ χ, s) = L(χ, s) =
X an χ(n)
n
ns
Weil proved that if an L-function L(s) is absolutely convergent, and for sufficiently many
χ
Λ(f ⊗ s, χ) = (2π)−s Γ(s)L(f ⊗ s, χ)
9
satisfies a functional equation, and is bounded in vertical strips, then the function
f (z) =
∞
X
an e2πinz
n=1
belongs to the space of modular forms for Γ0 (N ) and satisfies a holomorphy condition
at the cusps.
1.3.4
Maass Forms
In 1949, Maass introduced non-holomorphic modular forms now called Maass forms or
waveforms. A Maass form is a non-constant eigenfunction of the Laplacian
∂2
∂2
+
)
∂x2 ∂y 2
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∆ = y2(
in L2 (Γ\H), where Γ is a discrete subgroup of SL2 (R) and H is the upper half plane. A
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Maass form f for SL2 (Z) satisfies the following properties:
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• f (γz) = f (z) for all γ ∈ SL2 (Z), z ∈ H
• f is an eigenfunction of ∆ with eigenvalue s(s − 1) for some s ∈ C
• f has moderate growth in a Siegel set
Apart from Eisenstein series and special waveforms, Maass forms lack explicit construction and identification. Selberg used the trace formula to show they exist for
SL2 (Z). An example of a Maass form is the non-holomorphic Eisenstein series
X
c,d∈Z2 −(0,0)
ys
|cz + d|2s
Removing a superfluous factor of ζ(2s), define
Es (z) =
X
c,d∈Z, gcd(c,d)=1
ys
|cz + d|2s
Since
ℑ(γz) =
y
(where γ = ( ac db ) ∈ Γ = SL2 (Z))
|cz + d|2
10
then
Es (z) =
X
N ∩Γ\Γ
ℑ(γz)s
The simplest Rankin-Selberg integral (Rankin 1939 and Selberg 1940) is
hf · Es , gi
where f and g are cusp forms. This integral converges for all s ∈ C.
hf · Es , gi =
Z
y s f (z) g(z) y 2k
P \H
dx dy
y2
= (4π)−(s+2k−1) Γ(s + 2k − 1)
X an bn
ns+2k−1
n≥1
f (z) =
X
an e2πinz , g(z) =
n>0
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where
X
n>0
bn e2πinz , P = ( ∗0 ∗∗ )
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Rankin and Selberg showed further that Es (z) has a meromorphic continuation to the
whole s-plane and satisfies the functional equation
where
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Es (z) = φ(s) E1−s (z)
1
φ(s) = π 2
Γ(s − 12 ) ζ(2s − 1)
ξ(2s − 1)
·
=
Γ(s)
ζ(2s)
ξ(2s)
Hecke’s method was extended by Maass who obtained the meromorphic continuation
and functional equations of L-functions of Maass forms for Γ0 (N ).
1.3.5
Tate’s thesis
Tate-Iwasawa method reproved the meromorphic continuation and functional equations of Hecke’s L-functions attached to his Größencharakters by using harmonic analysis on the adeles. This identifies the Hecke character χ as a continuous character
χ : J → C× trivial on k× . This establishes the theory of automorphic representations
and L-functions of GL1 in the adelic setting.
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