Arithmetic of the Yoshida Lift

144 Pages·2010·1.18 MB·English

Arithmetic of the Yoshida Lift
by
Johnson Xin Jia
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
(Mathematics)
in The University of Michigan
2010
Doctoral Committee:
Professor Stephen M. DeBacker, Co-Chair
Professor Christopher M. Skinner, Co-Chair
Professor Jeffrey C. Lagarias
Associate Professor Kevin J. Compton
Associate Professor Kartik Prasanna
To Mom and Dad
ii
Acknowledgements
I am in debt to my thesis advisor, Chris Skinner, for his steadfast support and for
suggesting this interesting problem. His keen insights and wealth of knowledge were
pivotal for overcoming numerous obstacles; his encouragements kept the work going
through pressing times.
I am equally in debt to Stephen DeBacker, whose generosity and kindness are
unparalleled. His often humorous but always incisive career advice and mathematical
suggestions have made me a better mathematician and a better person.
There are countless other individuals without whom this work would have been
simply impossible. Among them I would like to mention in particular Brian Conrad,
whose single-minded devotion to mathematics has profoundly shaped my mathemat-
ical temperaments.
Lastly I would like to thank the Mathematics Department at the University of
Michigan, as well as numerous other mathematical organizations, for the generous
support and the opportunities they provided. I like to especially thank the American
Institute of Mathematics for hosting the Workshop on Generalizing the Theta Cor-
respondence during the summer of 2008. It is while attending this workshop that I
learned many of the recent results used in this work.
iii
Contents
Dedication ii
Acknowledgements iii
Abstract vi
Introduction 1
0.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.2 An overview of the results and methods . . . . . . . . . . . . . . . . . . 5
0.3 A synopsis of the contents . . . . . . . . . . . . . . . . . . . . . . . . . . 9
0.4 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . 12
0.5 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Chapter 1. Quaternion Algebras 17
1.1 Structures of quaternion algebras . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Representations of PGL . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2
1.3 Eichler orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Representations of D× and D× . . . . . . . . . . . . . . . . . . . . . . . . 30
p
Chapter 2. Automorphic Forms on D× 36
2.1 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Automorphic representations on D× . . . . . . . . . . . . . . . . . . . . 37
2.3 Automorphic forms on D× . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Chapter 3. GSO(D) and GSp 50
4
3.1 From D× to GSO(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Some representation theory of GSO(D) . . . . . . . . . . . . . . . . . . 54
3.4 Automorphic forms on GSp . . . . . . . . . . . . . . . . . . . . . . . . . 60
4
3.5 Fourier coefficients and Bessel models . . . . . . . . . . . . . . . . . . . 69
iv
3.6 Arithmetic theory of Siegel modular forms . . . . . . . . . . . . . . . . 73
Chapter 4. The Yoshida Lift 78
4.1 Review of the Weil representation . . . . . . . . . . . . . . . . . . . . . . 78
4.2 The reductive dual pair (Sp ,O(D)) . . . . . . . . . . . . . . . . . . . . 80
4
4.3 Representation-theoretic aspects of the Yoshida lift . . . . . . . . . . . 85
4.4 Good theta kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.5 An arithmetic Yoshida lift . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Chapter 5. Fourier Coefficients 101
5.1 An integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2 Specializing to good choices . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3 Integrality of Fourier coefficients . . . . . . . . . . . . . . . . . . . . . . . 109
Chapter 6. Non-Vanishing 118
6.1 Bessel model of the Yoshida lift . . . . . . . . . . . . . . . . . . . . . . . 118
6.2 Equidistribution and non-vanishing . . . . . . . . . . . . . . . . . . . . . 122
References 134
v
Abstract
This thesis concerns the arithmetic properties of the Yoshida lift, Y, which is a scalar-
valued holomorphic Siegel modular form of degree 2 obtained as the theta lift of a
pair of automorphic forms f ,f on D×, where D is a definite quaternion algebra over
1 2
Q.
Specifically, wedefinearefinedversionoftheYoshidalift, Y, whichhasthespecial
property that it preserves p-integral structures and is not identically zero under mild
conditions. For p-integrality, we compute a formula for the Fourier coefficients aT of
Y byexploitinganinherentfreedominthedefinitionofY. TheformulaforaT inturn
allows us to compute the Bessel model of the Yoshida lift, and apply an argument
of Cornut–Vatsal to conclude that Y is non-zero. Furthermore, if we assume Artin’s
conjecture on primitive roots, then we show that Y is in fact not zero modulo p.
vi
Introduction
The present work is the first part of an ongoing project aimed at understanding and
affirming some deep connections between special values of L-functions and certain
Selmer groups. The main object we study here is a certain theta lift. Since the
subject could get a bit technical, we begin with some informal excerpts from the
elementary aspects of the subject to set the context for the non-experts.
0.0.1 Theta functions.
The simplest example of a theta function is the holomorphic function on the upper-
half plane H={z ∈C∶Im(z)>0} given by the series
θ(z)= ∑e2πın2z.
n∈Z
It is an automorphic form on for the congruence subgroup
⎧ ⎡ ⎤ ⎫
⎪⎪ ⎢⎢a b⎥⎥ ⎪⎪
Γ (4)=⎨γ =⎢ ⎥∈SL (Z)∶c∈4Z⎬
0 ⎪⎪ ⎢c d⎥ 2 ⎪⎪
⎩ ⎣ ⎦ ⎭
of weight 1, in that it satisfies the transformation law [Iwa97, §2 (2.73)]
2
θ(γz)=(∗)⋅(cz+d)1 ⋅θ(z)
2
⎡ ⎤
⎢ ⎥
⎢a b⎥
for all γ = ⎢ ⎥ ∈ Γ (4) and z ∈ H.∗ Also note that the Fourier coefficients a(m)
⎢c d⎥ 0
⎣ ⎦
of θ is 1 exactly when m ∈ Z2 and 0 otherwise, which so happens to be equal to the
number of ways that we can represent m by a square.
The point to take home is that theta functions, such as θ(z), are in automorphic
forms; moreover,theirFourierexpansionsarerelativelytractableandcarryinteresting
arithmetic information.
∗We have suppressed a constant ∗ dependent on c and d to keep the presentation elementary.
1
0.0.2 Automorphic forms.
Roughlyspeaking, automorphicformsarefunctionsthatsatisfystronginvariantprop-
erties under certain group actions. For us, the most exciting aspect of automorphic
forms is that they are expected to give rise to L-functions, and can be used to study
the analytic properties of these L-functions. Take θ(z) for example, one can show
that [GM04, §2.1 (2.10)]
s ∞ 1 1 1 1
π−s ⋅Γ( )⋅ζ(s)=∫ (ts−1+t1−s−1)( ⋅θ(it)− )dt− −
2 2 2 2 2 2 s 1−s
1
where Γ(s) is the usual Gamma function and ζ(s) = ∑∞ n−s is the Riemann zeta
n=1
function. This integral representation of ζ(s) together with Jacobi’s transformation
law
1 ı
θ(ıt)= √ ⋅θ( )
t t
allow us to establish a functional equation for ζ(s).
Infact, allL-functionsthathaveEulerproductfactorizationsandfunctionalequa-
tions should be the L-functions associated with automorphic forms. Since Galois
representations arising from geometry or otherwise give rise naturally to L-functions,
one expects automorphic forms to be associated with Galois representations. This is
one of the aims of the Langlands program.
0.0.3 Integrality.
In addition to their relations to zeta functions, the Fourier coefficients of automorphic
forms (most notably those on GL ) have shown to carry a lot of number-theoretic in-
2
formation. Forexample, letQ(m ,...,m )=∑r α z2 beapositive-definitequadratic
1 r i=1 i i
form with α ∈Z . We define a theta function
i >0
θ(z,Q)= ∑ e2πı⋅Q(m)z
m∈Zr
for z ∈H. It is an automorphic form for Γ (2N) of weight k = r [Iwa97, §11.3]. It has
0 2
a Fourier expansion
θ(z,Q)= ∑r(n,Q)e2πız
n∈Z
where r(n,Q) is the representation number of n by Q, that is, the number of ways
we can right n = Q(m) for some m ∈ Zr. In particular, we see that these Fourier
coefficients r(n,Q) are all integers.
2
The integrality of these Fourier coefficients is not only interesting in its own right,
but it is also important in constructing Galois representations. In fact, it is one of
the (abide minor) ingredients used in Wiles’s proof of Fermat’s Last Theorem.
0.0.4 Theta lifts.
Let χ∶(Z/NZ)× →C1 be a Dirichlet character of conductor N. It turns out that χ is
naturally an automorphic form for the group SO ≃S1. We define a theta lift of χ to
2
the group GL by
2
n
θ (z)= ∑ χ(n)⋅nν ⋅e2πın2z
χ
i=−n
where ν = 0 if χ(−1) = 1 and 1 otherwise. It is an automorphic form for the group
Γ (4) of weight ν+1. So we have lifted the automorphic form χ on SO to the group
0 2 2
GL .
2
This simple example is actually quite representative of the general phenomenon.
The theta lift which we study, Y, called the Yoshida lift, can be expressed as a sum
of a similar shape
r 1
Y (Z)=∑ ∑ f●(β )⋅⟨P˜●(x),f●(α )⟩ ⋅e2πıtr(TxZ),∗
e 2 i k 1 i 2k,0
i=1 i x∈h−1⋅X ∩XT
i f Q
where Y is the holomorphic Siegel modular form of degree 2 associated to Y. Note
this expression also give a Fourier expansion
Y (Z)= ∑ a(T)⋅e2πıtr(TZ)
T∈T
of Y where the index T runs over the set of semi-definite symmetric 2×2 matrices
with entries in Q, and Z ∈H .
2
The goal of this thesis is to study the arithmetic properties of these Fourier co-
efficients a(T), and also to show that the Yoshida lift Y is not trivial, that is, not
identically zero (or zero modulo a prime) as a function.
0.1 Motivations
As we mentioned at the beginning, the current work is not just an end in itself. In
fact, we see it really as a stepping stone for achieving our goal of exhibiting evidence
∗ Here the h ’s are elements in an orthogonal group, e is a positive integer, and h−1⋅X ∩XT
i i i f Q
is a finite collection of integral maps for each i. Also P˜● is a vector-valued harmonic polynomial
k
which we will define precisely.
3
for some deep conjectures in number theory. Since casting this work in this larger
context helps one tobetter appreciate its relevance and utility, let us outline the vista.
0.1.1 The classical Yoshida lift.
To explain this, let D be a definite quaternion algebra over Q. In [Yos80], Yoshida
studied the theta lift taking a pair of automorphic forms, f , i = 1,2 on D× to a
i
holomorphic automorphic form Y on GSp . To do this, he first showed that the
4
product f ⊗f is an automorphic form on the orthogonal similitude group GSO(D)
1 2
of D. This is an immediate consequence of the fact that GSO(D) is essentially
D××D×.
Then the theory of Weil representations give rise to a collections of automorphic
forms, Θ , on a subgroup of GSO(D)×GSp . They are naturally indexed by a choice
ϕ 4
of a Bruhat–Schwartz function ϕ. The Yoshida lift
Y =jΘ ,f ⊗f o
ϕ 1 2 GSO(D)
is essentially the Petersson inner product of Θ with f ⊗f over the group GSO(D).
ϕ 1 2
Moreover, Yoshida computed the Satake parameters of Y in terms of those of f
i
at the unramified places. This allows us to study the L-functions associated with Y
in terms of the L-functions associated with the f ’s.
i
0.1.2 Elements in a Selmer group.
By a standard construction, we can associate to Y a holomorphic Siegel modular
form Y on the Siegel upper-half space H of degree 2. Under favorable conditions,
2
Y occurs in the cohomology of H , then we can associate to Y a p-adic Galois
2
representation [Tay93], [Wei05]
ρ ∶Gal(Q¯/Q)→GSp (Q ).
Y 4 p
What makes ρ interesting is that it is expected to be semi-simple but not irreducible
Y
[Art04, pg. 78]. By constructing a congruence
Y ≡F (mod p)
between Y and a stable Siegel modular form F whose Galois representation is irre-
ducible, one can then (in theory) follow a well-known procedure dating back to the
4