MATH G9905 RESEARCH SEMINAR IN NUMBER THEORY
(FALL 2014)
LECTURE 11 (NOVEMBER 25, 2014)
MICHAEL WOODBURY
AN ADELIC KUZNETSOV TRACE FORMULA FOR GL(4)
NOTES TAKEN BY PAK-HIN LEE
Abstract. An important tool in analytic number theory for GL(2)-type questions is Kuznetsov’s
trace formula. Recently, in work of Blomer and of Goldfeld/Kontorovich, generalizations
of this to GL(3) have been given which are useful for number theoretic applications. In
my talks I will discuss joint work with Dorian Goldfeld in which we further generalize the
said GL(3) results to GL(4). I will discuss some of the new features and complications
which arise for GL(4) as well as applications to low lying zeros of L-functions and a vertical
Sato–Tate theorem.
1. GL(2)
To set things up, we will start by talking about GL(2) in classical language. Let {ϕj } be
a basis of Hecke Maass forms of full level (for simplicity). Each ϕj is an eigenfunction for
the Laplacian operator
∆ϕj = λj ϕj
and the Hecke operator
Tp ϕj = aj (p)ϕj
with a suitable normalization. For z = x + iy ∈ H,
X
√
ϕj (z) =
e2πinx yKitj (2π|n|y)aj (n).
n6=0
where λj = 41 + t2j .
Kuznetsov proved for “nice” test functions h,
√
Z
∞
∞
X
X
aj (m)aj (n)
δn,m ∞
s(n, m, c)
4π nm
h(tj ) + E = 2
r tanh(πr)h(r) dr +
× Jh
cosh(πt
)
π
c
c
j
−∞
c=1
j=1
where E is the Eisenstein contribution, s(n, m, c) is the Kloosterman sum and Jh is the
Kloosterman integral.
Applications of this include:
Last updated: November 25, 2014. Please send corrections and comments to [email protected].
1
• Kuznetsov used this to give bounds for the Kloosterman sums
X s(n, m, c)
1
X 6 + .
c
c≤X
• Important ingredient in proving GL(2)-type subconvexity results.
• Can be used to study {ϕj (n)}.
• Low lying zeros.
2. Applications (for GL(n))
We introduce the vertical Sato–Tate theorem. Define the Sato–Tate measure
 r
x2
1
1
−
if |x| ≤ 2,
(2)
dµ∞ = π
4

0
otherwise,
and the Plancherel measure
dµ(2)
p
√
(p + 1) 4 − x2
= 2π((p1/2 + p−1/2 )2 − x2 )

0


(2)
if |x| ≤ 2,
otherwise.
(2)
Note that as p → ∞, we have dµp → dµ∞ .
Theorem 1 (Sarnak, 1987). The sequence of eigenvalues a1 (p), a2 (p), a3 (p), · · · is equidis(2)
tributed with respect to dµp .
−1/4−ν 2
Theorem 2 (Bruggeman). Let hT (ν) = e T , and f : R → R be continuous. Then
P∞
Z
j=0 f (aj (p))hT (tj )
P∞
→ f dµ(2)
p
j=0 hT (tj )
as T → ∞.
Both of these results used the Selberg trace formula. In addition, Bruggeman proved that
using the Kuznetsov trace formula, one can get a formula for the weighted sum
P∞ f (aj (p))hT (tj )
Z
j=0
Lj
→ f dµ(2)
P∞ h (tj )
∞,
T
j=0
Lj
where Lj = L(1, Ad ϕj ).
Fan Zhou generalized these. His proof is unconditional whenever one has an asymptotic
character formula, which is known for GL(3).
3. Adelic KTF for GL(n)
Write A for the adeles of Q, and v will denote a place of Q. We have the Iwasawa
decomposition
G = GLn = U T K
2
where U is the upper triangular unipotent matrices, T is the diagonal matrices, and K is
the maximal compact. Denote
[G] = GLn (Q)\ GLn (A)/Z(Q).
For a = (a1 , a2 , · · · , an−1 ) ∈ Qn−1 , we can define the character on U


1 x1 ∗
∗
 . . . . . .

∗ 
 = e2πi(x1 a1 +···+xn−1 an−1 ) .

ψa,∞ 


1 xn−1 
1
We set
O
ψa,v : U (Q)\U (A) → C× .
N
N
We have a unitary representation H =
Hv : T (A) → σ =
v σv , where σv is a
finite-dimensional representation of Kv . We also have a function || · ||νtor : T (A) → C for
ν = (ν1 , · · · , νn ) ∈ Cn . Define the Poincaré series
X
P a (g, ν) =
ψ0 (γg)H(γg)||γg||νtor .
ψa =
γ∈Z(Q)U (Q)\G(Q)
Note H(utk) = σ(k)−1 H(t).
The trace formula is in principle very simple: we take two Poincaré series and compute
their inner product in two different ways. For the first way, we want to calculate
Z
a
b
0
hP (·, ν), P (·, ν )i[G] =
hP a (g, ν), P b (g, ν 0 )iσ dg.
[G]
At almost all places, the integrand is just P a P b . Plugging in and unfolding, we get that this
is equal to
X
X
YZ
hKlv (tv , ν, a, b, w, τ ), Hv (tv )i dt.
Tv
w∈W τ ∈Z(Q)\T (Q) v
This is a Kloosterman integral for each place. The second way uses the spectral expansion
X
P b (g, ν) =
hP b (·, ν), ϕj iϕj + rest of spectrum.
j
Our goal is to prove a formula of the type
X Aj (a)Aj (b)
λj ≤T
Lj
= cδa,b T + O(T 4 )
where 4 < .
From now on we will consider GL(4). Let
p#
T,R (α1 , α2 , α3 )
=e
2
2
2
α2
1 +α2 +α3 +α4
2T
Y
1≤j6=k≤3
Γ
2 + R + αj − αk
4
where α4 = α1 + α2 + α3 . This is the Lebedev–Whittaker transform of
ZZZ
Y
1
pT,R (y) =
p#
T,R
αj −αk Wα (y) dα
Γ
Re(α)=0
2
3
Next time we will bound the inner product using this.
4