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Averages of central L-values of Hilbert modular
forms with an application to subconvexity
Brooke Feigon∗
Institute for Advanced Study
Einstein Drive
Princeton, NJ 08450
David Whitehouse†
Institute for Advanced Study
Einstein Drive
Princeton, NJ 08450
June 26, 2009
Abstract
We use the relative trace formula to obtain exact formulas for central
values of certain twisted quadratic base change L-functions averaged over
Hilbert modular forms of a fixed weight and level. We apply these formulas
to the subconvexity problem for these L-functions. We also establish an
equidistribution result for the Hecke eigenvalues weighted by these L-
values.
Contents
1 Introduction 2
1.1 Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 About the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Waldspurger’s result . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Relative trace formula . . . . . . . . . . . . . . . . . . . . 6
1.2.3 The case of modular forms of weight 2 . . . . . . . . . . . 8
1.3 About the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 Summary of conditions . . . . . . . . . . . . . . . . . . . 10
1.3.2 Outline of the paper . . . . . . . . . . . . . . . . . . . . . 11
1.4 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Notation 12
2.1 Normalization of measures . . . . . . . . . . . . . . . . . . . . . . 13
∗Email address: bfeigon@math.toronto.edu
†Email address: dw@math.mit.edu
1

3 Spectral side of the trace formula 14
3.1 The test function . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Local preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Global calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Geometric side of the trace formula 22
4.1 Irregular cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Regular cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.1 Exact calculations . . . . . . . . . . . . . . . . . . . . . . 29
4.2.2 Bounds on I(ξ,1v) . . . . . . . . . . . . . . . . . . . . . . 32
4.3 Global calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5 A measure on the Hecke algebra 39
5.1 An application of the Plancherel formula . . . . . . . . . . . . . . 40
5.2 The distribution I˜ . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6 Main results 43
6.1 Average L-values . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.2 Subconvexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.3 Classical reformulation . . . . . . . . . . . . . . . . . . . . . . . . 53
1 Introduction
1.1 Statement of results
In this paper we use the relative trace formula, together with period formu-
las originating in work of Waldspurger [Wal85], to study central values of L-
functions associated to Hilbert modular forms. Let F be a totally real number
field and let Σ∞ denote the set of archimedean places of F . Given an ideal N of
OF and a tuple of positive integers k = (kv : v ∈ Σ∞) we let F(N, 2k) denote
the set of cuspidal automorphic representations of PGL(2,AF ) which are of ex-
act level N and holomorphic of weight 2k . We recall that each π ∈ F(N, 2k)
may be identified with a normalized holomorphic Hilbert modular newform of
level N, weight 2k and trivial nebentypus which is an eigenfunction for all the
Hecke operators.
Let E be a quadratic extension of F and for each π ∈ F(N, 2k) let πE denote
the base change of π to an automorphic representation of PGL(2,AE). Given a
unitary character Ω of the idele class group E×\A×E of E one may consider the
completed L-function L(s, πE⊗Ω) which satisfies a functional equation relating
s to 1 − s. We note that if σΩ denotes the induction of Ω to an automorphic
representation of GL(2,AF ) then,
L(s, πE ⊗ Ω) = L(s, π × σΩ).
The object of study in this paper is L(1/2, πE ⊗ Ω), the central value of this
L-function. In particular we prove an explicit formula for L(1/2, πE⊗Ω) as one
2

averages over π of a fixed weight and level. As an application of this formula
we establish subconvexity as π and Ω vary in a certain range and prove an
equidistribution result for the Hecke eigenvalues of such π.
Throughout this paper we make the following assumptions on the data in-
troduced above.
• E is an imaginary quadratic extension of F ,
• N is squarefree, each prime p dividing N is inert and unramified in E and
the number of primes dividing N has the same parity as the degree of the
extension [F : Q], and
• the character Ω is trivial when restricted to A×F , unramified at the places
of E above N, and at each archimedean place Ωv has weight mv < kv.
See Section 1.3.1 below for a discussion on the relevance and seriousness of these
assumptions.
The results of this paper are all derived from an exact formula for
∑
π∈F(N,2k)
L(1/2, πE ⊗ Ω)
L(1, π, Ad) fˆp(πp),
obtained via the relative trace formula. Here L(s, π,Ad) is the adjoint L-
function of π and fˆp denotes a Hecke operator at p ∤ N.
When N is large with respect to E, c(Ω) (the conductor of Ω) and fp our
formula simplifies considerably. The simplest version of it is given by,
Theorem 1.1. Assume that not all kv = 1 and N has absolute norm larger
than dE/F c(Ω)hF . Then,
2[F :Q]
|N|
(
2k− 2
k + m− 1
) ∑
π∈F(N,2k)
L(1/2, πE ⊗ Ω)
L(1, π, Ad) = 4|∆F |
3
2LS(Ω)(1, η)
where S(Ω) denotes the set of places of F above which Ω is ramified and η is
the quadratic idele class character of F associated to E.
For the full formula see Section 6 and for any undefined notation see Section
2.
Over Q we can express this formula more classically. We identify F(N, 2k)
with the set of normalized modular newforms of level N and weight 2k which
are Hecke eigenforms. Let E = Q(
√
−d) be an imaginary quadratic extension
of Q of discriminant −d. We take Ω as before and let gΩ denote the modular
form of weight 2|m|+ 1, level dc(Ω) and nebentypus χ−d associated to Ω. For
f ∈ F(N, 2k) we take the Rankin-Selberg L-function, L(s, f × gΩ) which has
functional equation relating s to 2k+2|m|+1−s. Using the relationship between
L(1, πf , Ad), where πf denotes the automorphic representation generated by f ,
and the square of the Petersson norm of f ,
(f, f) =
∫
Γ0(N)\H
|f(x+ iy)|y2k dx dyy2
3

we can rewrite Theorem 1.1 in the following way: Assume Ω is not quadratic,
k > 1 and N > dc(Ω), then
(2k − 2)!u−d
√
dLS(Ω)(1, χ−d)
2π(4π)2k−1
∑
f∈F(N,2k)
Lfin(k + |m|+ 12 , f × gΩ)
(f, f) = h−d
where χ−d is the quadratic Dirichlet character of discriminant −d, h−d is the
class number of E and u−d = #O×E/{±1}. See Section 6.3 for further details.
We note that over Q and with Ω trivial an asymptotic version of Theorem
1.1 has been known for a while. First by Duke [Duk95] for prime level and
weight 2 and then by Iwaniec, Luo and Sarnak [IS00b], [ILS00] for squarefree
level and higher weight. An exact formula has been established by Michel and
Ramakrishnan in the case F = Q and Ω is a character of the ideal class group
of E. Their work uses Gross’ formula together with a geometric argument in
the weight two case, and the theta correspondence in higher weight; see [MR].
By an extension of Theorem 1.1 to include Hecke operators, we obtain a
result that includes a restriction at a prime p ∤ N. Let π = ⊗vπv ∈ F(N, 2k).
We let {αp, α−1p } denote the Satake parameters of πp and set ap(π) = αp +
α−1p . We recall that ap(π) ∈ [−2,+2] by Ramanujan’s conjecture; [Bla06].
The distribution of the ap(π) has been considered by Sarnak [Sar87] and Serre
[Ser97]. The spherical Plancherel measure on PGL(2, Fp) is given by
µp =
qp + 1
(q
1
2
p + q
− 12
p )2 − x2
√
4− x2
2π dx
on [−2, 2], here qp denotes the order of the residue field at p. Serre has proven
that when F = Q,
{ap(π) : π ∈ F(N, 2k)}
become equidistributed with respect to µp as N → ∞. We prove a variant of
this result where we include a weighting by Lp(1/2, πE ⊗ Ω).
Theorem 1.2. For any J ⊂ [−2,+2] we have
lim
N→∞
1
|N|
∑
π∈F(N,2k)
ap(π)∈J
Lp(1/2, πE ⊗ Ω)
Lp(1, π, Ad)
=4|∆F |
3
2
1
2[F :Q]
(
2k− 2
k + m− 1
)−1
LS(Ω)∪{p}(1, η)L(2, 1Fp)µp(J).
We note that in the case that F = Q and Ω is trivial, we recover the main
result of [RR05]. A similar result has been obtained by Royer [Roy00] for the
single L-value L(1/2, π) averaged over modular forms π of level N and weight
2.
We remark that one could consider the average by normalizing by |F(N, 2k )|
rather than |N|. In order to have a finite limit with this normalization we need
4

to add the technical restriction thatb |N|∏p|N(1 − 1|p|) ∼ |N|. For F = Q this
condition reduces to ϕ(N) ∼ N where ϕ is the Euler totient function. Using the
well known fact that |F(N, 2k)| ∼ 2k−112 ϕ(N) as N → ∞ we get the following
statement.
Corollary 1.3. Let F = Q and J ⊂ [−2,+2]. Then
lim
N→∞
1
|F(N, 2k)|
∑
π∈F(N,2k)
ap(π)∈J
Lp(1/2, πE ⊗ Ω)
Lp(1, π, Ad)
is equal to
24
2k − 1
(
2k − 2
k +m− 1
)−1
LS(Ω)∪{p}(1, η)L(2, 1Qp)µp(J)
where the limit is taken over squarefree N such that ϕ(N) ∼ N and each prime
dividing N is inert and unramified in E and does not divide c(Ω).
Finally we apply our work to the problem of subconvexity. Using a version of
Theorem 1.1 that is also valid for smaller N, combined with the non-negativity
of L(1/2, π×σΩ), established in [JC01], and an upper bound for L(1, π, Ad), we
get the following theorem.
Theorem 1.4. Fix a totally real number field F and a CM extension E of F .
Let N be a squarefree ideal in OF such that the number of primes dividing N
has the same parity as [F : Q] and such that each prime of F dividing N is inert
and unramified in E. Let Ω be a character of A×FE×\A×E which is unramified
above N and has weights at the archimedean places strictly less than k. Then
for any ǫ > 0,
Lfin(1/2, π × σΩ) ≪F,E,k,ǫ |N|1+ǫc(Ω)ǫ + |N|ǫc(Ω)
1
2+ǫ,
for all π ∈ F(N, 2k).
Hence for 0 ≤ t < 16 and ǫ > 0,
Lfin(1/2, π × σΩ) ≪F,E,k,ǫ (c(Ω)|N|)
1
2−t,
for π ∈ F(N, 2k) with N such that
c(Ω)
2t+ǫ
1−(2t+ǫ) ≤ |N| ≤ c(Ω)
1−(2t+ǫ)
1+2t+ǫ .
This result clearly beats the convexity bound Lfin(1/2, π×σΩ) ≪ǫ (dE/Qc(Ω)N)
1
2+ǫ
for all ǫ > 0. Similar results have been obtained in [MR], where Ω is a character
of the ideal class group of E and N and E vary. We remark that Michel and
Harcos ([HM06] and [Mic04]) have proven subconvexity in the level aspect for
Lfin(1/2, π1 × π2) where π1 and π2 are cusp forms on GL(2)/Q with π2 fixed.
We also mention the work of Cogdell, Piatetski-Shapiro and Sarnak [Cog03]
which proves subconvexity for the central value of a fixed Hilbert modular form
twisted by a ray class character.
5

1.2 About the proof
Before continuing we give some background about the tools used in the proofs
of the results contained in this paper.
1.2.1 Waldspurger’s result
An important result of Waldspurger [Wal85] relates L(1/2, πE ⊗ Ω) to period
integrals of automorphic forms over the torus E×. More precisely let X(π,E)
be the set of isomorphism classes of quaternion algebras D/F such that E →֒ D
and such that π comes from an automorphic representation πD of D× via the
Jacquet-Langlands correspondence. We note that X(π,E) is finite, since D
must be isomorphic to the matrix algebra at all places where π is unramified,
and non-empty, since M(2, F ) ∈ X(π,E).
Take D ∈ X(π,E) and let ϕ be an element in the space of πD. Thus, ϕ is a
function on D×\D×(AF ) and we can define period integrals by
PD(ϕ) =
∫
E×A×F \A
×
E
ϕ(t)Ω−1(t) dt,
with A×E viewed as a subgroup of D×(AF ) via the inclusion E →֒ D. In [Wal85,
Proposition 7], |PD(ϕ)|2 is expressed in terms of L(1/2, πE ⊗ Ω). For suitable
choices of measure Waldspurger proves that,
|PD(ϕ)|2
(ϕ,ϕ) =
L(1/2, πE ⊗ Ω)
2L(1, π, Ad)
∏
v
αv(Ev, ϕv,Ωv),
where αv(Ev, ϕv,Ωv) are local period integrals which are equal to 1 for almost
all v. Subsequent refinements of Waldspurger’s result, [Gro87], [Zha01], [Xue06],
[Pop06], [MW09], have sought to compute these local period integrals so as to
provide an exact formula relating the L-value and the period integral. Under
certain additional ramification conditions these results take the form,
|PD(ϕπ)|2
(ϕπ , ϕπ)
= C(E, π,Ω)L(1/2, πE ⊗ Ω), (1)
where D is a suitable element of X(π,E) and ϕπ ∈ πD is a test vector defined
by Gross and Prasad [GP91]. The constant C(E, π,Ω) is of the form,
C(E, π,Ω) = 1L(1, π, Ad)
√
∆F
2
√
c(Ω)∆E
∏
v
Cv(E, π,Ω)
where the product is taken over the places v of F which are “bad” for π or Ω
and Cv(E, π,Ω) consists of certain local L-factors.
1.2.2 Relative trace formula
Having fixed the extension E/F let D/F be a quaternion algebra such that
E →֒ D. We set G equal to the group PD× over F and let T be the torus in G
such that T (F ) is equal to the image of E× in G(F ).
6

Let f ∈ C∞c (G(AF )). Then we have the map
R(f) : L2(G(F )\G(AF )) → L2(G(F )\G(AF ))
given by
(R(f)ϕ)(x) =
∫
G(AF )
f(y)ϕ(xy) dy.
R(f) is an integral operator with kernel
Kf (x, y) =
∑
γ∈G(F )
f(x−1γy).
When D is a division algebra, as a representation of G(AF ),
L2(G(F )\G(AF )) =
⊕
π∈A(G)
π
with the sum taken over the set of irreducible automorphic representationsA(G)
of G(AF ). Since R(f) preserves each of the spaces π, the kernel has a spectral
expansion
Kf(x, y) =
∑
π∈A(G)
∑
ϕ∈B(π)
(R(f)ϕ)(x)ϕ(y),
where B(π) denotes an orthonormal basis of the space of π.
Let I(f) be the distribution defined by
I(f) =
∫
T (F )\T (AF )
∫
T (F )\T (AF )
Kf(t1, t2)Ω(t−11 t2) dt1 dt2.
The spectral expansion for Kf (x, y) gives,
I(f) =
∑
π∈A(G)
∑
ϕ∈B(π)
∫
T (F )\T (AF )
(R(f)ϕ)(t1)Ω−1(t1) dt1
∫
T (F )\T (AF )
ϕ(t2)Ω−1(t2) dt2.
From the geometric expansion for the kernel, and after interchanging summation
and integration,
I(f) =
∑
γ∈T (F )\G(F )/T (F )
vol(Tγ(F )\Tγ(AF ))
∫
Tγ(AF )\(T (AF )×T (AF ))
f(t−11 γt2)Ω(t−11 t2) dt1 dt2,
(2)
where, for γ ∈ G(F ),
Tγ =
{
(t1, t2) ∈ T × T : t−11 γt2 = γ
}
.
We now return to the setting of the previous section. Having fixed N we let
D be the quaternion algebra over F which is ramified at the primes dividing N
and all the infinite places of F . Since the extension E/F is inert at all the places
7

of ramification for D there exists an embedding E →֒ D. For each π ∈ F(N, 2k)
we let ϕπ be an element in the space of πD as in (1) above. We can choose a
test function f = fN,k in C∞c (G(AF )), such that the operator R(f) projects
L2(G(F )\G(AF )) onto the span of the set {ϕπ ∈ πD : π ∈ F(N, 2k)}. (This
isn’t quite true if all kv = 1, in this case the function fN,k may also pick out
some 1-dimensional representations as well.) Hence for this function,
I(fN,k ) =
∑
π∈F(N,2k)
|
∫
T (F )\T (AF ) ϕπ(t)Ω
−1(t) dt|2
(ϕπ, ϕπ)
.
Applying the identity (1) for each π ∈ F(N, 2k) we get,
I(fN,k ) = C(E,Ω, k ,N)
∑
π∈F(N,2k)
L(1/2, πE ⊗ Ω)
L(1, π, Ad) ,
with C(E,Ω, k ,N) an explicit constant. From the geometric expansion for
I(fN,k ), (2), we obtain a closed expression for this average in terms of orbital
integrals of the function fN,k over double cosets. Furthermore one can show that
only finitely many T (F ) double cosets contribute to the sum. The results of this
paper then stem from calculations of these geometric terms. When the level N
is large we show that only the identity double coset contributes which gives an
exact formula for the average (Theorem 6.1), when the character Ω is every-
where unramified we can compute all the necessary orbital integrals (Theorem
6.5) and when the character Ω ramifies we are able to bound the terms on the
geometric side of the trace formula which we can use towards the subconvexity
problem for these L-values (Theorem 6.8).
1.2.3 The case of modular forms of weight 2
When F = Q the results of this paper are rephrased classically in Section 6.3.
Here we illustrate our methods for modular forms of weight 2 and prime level
in more classical language.
We fix an imaginary quadratic extension E = Q(
√
−d) and take N to be
a prime which is inert and unramified in E. We take Ω to be a character of
Pic(E), the ideal class group of E. Let D be the quaternion algebra over Q
which is ramified at N and ∞. We fix an embedding E →֒ D and take R to be
a maximal order in D such that R∩E = OE . Let X denote the finite set of left
equivalence classes of right R-ideals; see [Gro87, Section 1]. Given an ideal a in
E we may form the right R-ideal aR in D. In this way we get a well defined map
ι : Pic(E) → X . Let F (X) denote the space of complex valued functions on
X which can be endowed with a natural inner product and an action of Hecke
operators.
The Jacquet-Langlands correspondence, in this case, gives a Hecke-equivarient
isomorphism JL : F (X) ∼−→M2(N) of F (X) with the space of modular forms
of weight 2 and level N ; see [Gro87, Section 5]. We set S(X) = JL−1(S2(N))
8

where S2(N) denotes the space of cusp forms of level N and weight 2. The
orthogonal complement of S(X) in F (X) is the space of constant functions.
We identify F(N, 2) with the set of normalized eigenforms in S2(N). For
f ∈ F(N, 2) let f ′ ∈ F (X) be such that JL(f ′) = f . Then the period relation
for the central L-value can be expressed as,
L(1, f × gΩ)
(f, f) = C(N,E)
∣∣∣
∑
y∈Pic(E) f ′(ι(y))Ω(y)
∣∣∣
2
(f ′, f ′) ,
where gΩ is the θ-series associated to Ω and L(1, f × gΩ) denotes the central
value of the Rankin-Selberg L-function of f with gΩ. Here C(N,E) is an explicit
constant depending only on N and E. Thus,
∑
f∈F(N,2)
L(1, f × gΩ)
(f, f) = C(N,E)
∑
f∈F(N,2)
∣∣∣
∑
y∈Pic(E) f ′(ι(y))Ω(y)
∣∣∣
2
(f ′, f ′) .
Since the orthogonal complement of S(X) is spanned by the constant functions
we see that if B is any orthonormal basis of F (X) then
∑
f∈F(N,2)
L(1, f × gΩ)
(f, f) +C(N,E)δΩ
#Pic(E)2
(1,1) = C(N,E)
∑
h∈B
∣∣∣∣∣∣
∑
y∈Pic(E)
h(ι(y))Ω(y)
∣∣∣∣∣∣
2
,
where 1 denotes the function which is identically 1 on X , and
δΩ =
{
1, if Ω is trivial;
0, otherwise.
For each x ∈ X let δx ∈ F (X) be the function defined by
δx(y) =
{
1, if y = x;
0, otherwise.
Now suppose we take for B the set {δx/‖δx‖ : x ∈ X}. Then,
∑
h∈B
∣∣∣∣∣∣
∑
y∈Pic(E)
h(ι(x))Ω(x)
∣∣∣∣∣∣
2
=
∑
x∈X
∑
y1∈Pic(E)
∑
y2∈Pic(E)
δx(ι(y1))δx(ι(y2))
‖δx‖2
Ω(y1y−12 )
=
∑
y1∈Pic(E)
∑
y2∈Pic(E)
(δx(ι(y1))δx(ι(y2))
‖δx‖2
)
Ω(y1y−12 ).
For y1, y2 ∈ Pic(E) clearly,
∑
x∈X
δx(ι(y1))δx(ι(y2)) =
{
1, if ι(y1) = ι(y2);
0, otherwise.
9

The ideal N of F is assumed to be squarefree and such that each prime
divisor of N is inert and unramified in E. This is, for the most part, a technical
assumption so that we avoid the problem of having to deal with oldforms. This
condition can most likely be removed with some extra work. We also assume
that the number of prime divisors of N has the same parity as [F : Q], this is
a necessary condition as it ensures that the sign of the functional equation of
L(s, πE ⊗ Ω) is +1.
We place certain restrictions on the character Ω as well. We assume that
Ω is unramified above N. This assumption is needed in the work of Gross
and Prasad [GP91] which provides the test vector ϕπ which appears in (1).
We also assume, at the archimedean places, that the weights of Ω are strictly
smaller than the weights k of the Hilbert modular forms. This ensures that
the quaternion algebra D which appears in (1) is ramified at the archimedean
places of F . Without this assumption the function fN,k would not be compactly
supported and one would not obtain a finite closed formula for the average L-
values. In this case it is likely that one could work out an asymptotic version
of the formulas in this paper. This would require some additional archimedean
calculations similar to [RR05, Section 2].
Finally it would also be interesting to consider the case of Maass forms. It
would seem that again with some additional archimedean calculations this could
be carried out.
1.3.2 Outline of the paper
This paper is set up as follows. We begin by introducing our notation and
defining our measures. In the following section we pick our test functions and
compute their spectral expansion in the relative trace formula. In the fourth
section we explicitly compute the geometric side of the relative trace formula
for these test functions. While, a priori, there are an infinite number of orbital
integrals that appear in the geometric expansion, we show that for our test
functions only a finite number of terms are nonzero. This is what allows us to
get an exact, rather than asymptotic, formula.
In the fifth section we compute a distribution on the Hecke algebra which
appears in the geometric expansion of the relative trace formula. In the last
section we combine the spectral and geometric calculations to get our main
formulas. We also establish our application to subconvexity and rewrite our
formulae over Q in classical language.
The original motivation for our work came from trying to understand [RR05],
which applies the relative trace formula of [Jac86] to average values of base
change L-functions. We found that by integrating over nonsplit rather than
split tori, and applying [MW09], we obtain more general and exact results.
Furthermore, by working on quaternion algebras we avoid having to deal with
contributions from oldforms.
11

1.4 Acknowledgements
We wish to thank Peter Sarnak for his valuable suggestions and encouragement.
We thank Dinakar Ramakrishnan for his useful comments, as well as furnishing
us with early versions of his work with Philippe Michel [MR]. We also thank
Dipendra Prasad for helpful conversations and the Institute for Advanced Study
for a stimulating mathematical environment. Finally, we thank the referees for
their helpful comments.
The first author was supported by NSF grant DMS-0111298 and the Natural
Sciences and Engineering Research Council of Canada. The second author was
supported by NSF grants DMS-0111298 and DMS-0758197. Any opinions, find-
ings and conclusions or recommendations expressed in this material are those
of the authors and do not necessarily reflect the views of the National Science
Foundation.
2 Notation
Let F be a totally real number field of discriminant ∆F and class number hF .
For a finite place v of F , ̟v denotes a uniformizing parameter in Fv and qv
denotes the cardinality of the residue class field at v. The ring of integers in Fv
is denoted by OFv , the units by UFv and for n > 0 we set UnFv = 1 + ̟nvOFv .
For an ideal a of OF , we denote by |a| the absolute norm of a. We denote by
Σ∞ the set of infinite places of F . We take N to be a squarefree ideal in OF
such that the number of primes dividing N has the same parity as [F : Q]. For
each place v ∈ Σ∞ we fix an integer kv ≥ 1. We denote by F(N, 2k) the set
of cuspidal automorphic representations of PGL(2,AF ) of level N and weight
2k = (2kv).
We now take a CM extension E/F such that each prime of F dividing N
is inert and unramified in E. We let DE/F be the different of E/F , dE/F be
the discriminant of E/F and dE/F = |dE/F |. We denote by η the quadratic
character of F×\A×F associated to E/F by class field theory and by N the
norm map from E to F . For a place v of F we denote by Ev = E ⊗F Fv, by
OEv the integral closure of OFv in Ev and by UEv = O×Ev . We denote the action
of the non-trivial element in Gal(E/F ) on α ∈ Ev by α¯.
We take a unitary character Ω : E×\A×E → C× such that Ω|A×F is trivial.
We assume that Ω is unramified above N and that at each place v ∈ Σ∞, Ωv
has the form
z 7→
(z
z¯
)mv
with |mv| strictly less than kv. We set m = (mv). At each finite place v of F
we denote by n(Ωv) the smallest non-negative integer such that Ω is trivial on
(OF +̟n(Ωv)v OE)×. We denote by c(Ω) the norm of the conductor of Ω in F
and by c(Ω) the absolute norm of c(Ω). We note that c(Ω) = ∏v<∞ q
2n(Ωv)
v .
We use S(Ω) to denote the set of places of F above which Ω is ramified.
12

We define (
2k − 2
k + m − 1
)
=
∏
v∈Σ∞
(
2kv − 2
kv +mv − 1
)
.
Having fixed N, we let D be the quaternion algebra defined over F which
is ramified precisely at the infinite places of F and the places dividing N. We
fix an embedding E →֒ D. We let Z denote the center of D and denote by G
the group Z\D× viewed as an algebraic group defined over F . We denote by
ND : D× → F× the reduced norm. At each finite place v of F we fix a maximal
order Rv in Dv such that Rv ∩Ev = OFv +̟n(Ωv)v OEv . We note that Rv is well
defined up to E×v conjugacy.
All L-functions are completed.
2.1 Normalization of measures
We fix an additive character ψ of F\AF which, for the sake of convenience, we
take to be ψ = ψ0 ◦ trF/Q where ψ0 denotes the standard character on Q\AQ.
For a place v of F we take the additive Haar measure dx on Fv which is self-dual
with respect to ψv. On F×v we take the measure
d×x = L(1, 1Fv)
dx
|x|v
.
We define measures on Ev and E×v similarly with respect to the additive char-
acter ψ ◦ trE/F . We note that with these choices of measures we have
∏
v<∞
vol(UFv , d×xv) = |∆F |−
1
2 ,
and similarly for E. We also have
vol(F×v \E×v ) = vol(R×\C×) = 2
for v ∈ Σ∞.
For the group D we recall that there exists an ǫ ∈ F×, well defined in
F×/NE×, such that D is isomorphic to
{(
α ǫβ
β¯ α¯
)
: α, β ∈ E
}
.
For a place v of F we take the Tamagawa measure dgv on D×v which is given by
dgv = L(1, 1Fv)|ǫ|v
dαv dβv
|αvα¯v − ǫβvβ¯v|2v
.
We note that this measure only depends on the choice of ǫ modulo NE×. Fur-
thermore with this definition
vol(R×v ) = L(2, 1Fv)−1 vol(UFv , d×xv)4
13

for non-archimedean v ∤ N since the isomorphism of D×v with GL(2, Fv) pre-
serves Tamagawa measures. We have
vol(R×v ) =
L(2, 1Fv)−1
qv − 1
vol(UFv , d×xv)4
for v | N by a straightforward calculation. We also note that
vol(G(Fv)) = 4π2
for v ∈ Σ∞.
Globally we take the product of these local measures and give discrete sub-
groups the counting measures. In this way we get
vol(A×FE×\A×E) = 2L(1, η)
and
vol(G(F )\G(AF )) = 2.
3 Spectral side of the trace formula
Having fixed the quaternion algebra D we have taken G to be the algebraic
group defined over F with G(F ) = D×/F×. Since D is anisotropic the quotient
G(F )\G(AF ) is compact and hence, as a representation of G(AF ),
L2(G(F )\G(AF )) =
⊕̂
π′∈A(G)
Vπ′ .
Here the sum is taken over the irreducible automorphic representations A(G)
of G(AF ) and for each π′ ∈ A(G), Vπ′ denotes the space of π′. The Jacquet-
Langlands correspondence [JL70] yields an injection JL : A(G) →֒ A(PGL(2)),
whereA(PGL(2)) denotes the set of automorphic representations of PGL(2,AF ).
The set A(G) can be decomposed as,
A(G) = Acusp(G) ∐Ares(G),
where
Acusp(G) = {π′ ∈ A(G) : JL(π′) is cuspidal}
and
Ares(G) =
{
δ ◦ND : δ : F×\A×F → {±1}
}
.
The compatibility between the local and global Jacquet-Langlands corre-
spondence yields the following.
Fact 3.1. The image of Acusp(G) under JL is equal to the set of cuspidal au-
tomorphic representations π = ⊗vπv of PGL(2,AF ) such that πv is a discrete
series representation of PGL(2, Fv) at all places v where D ramifies. In partic-
ular F(N, 2k) is contained in the image of JL and for π′ = ⊗vπ′v ∈ Acusp(G)
we have JL(π′) ∈ F(N, 2k) if and only if,
14

so that I(f) = Icusp(f) + Ires(f). In the same we also write
Kf(x, y) = Kf,cusp(x, y) +Kf,res(x, y).
Our goal in this section is to choose a suitable test function f ∈ C∞c (G(AF ))
such that R(f) kills all π′ ∈ Acusp(G) such that JL(π′) 6∈ F(N, 2k ). We will
then compute I(f) and use a precise version of Waldspurger’s formula [MW09,
Theorem 4.1] to relate Icusp(f) to the L-values being considered in this paper.
3.1 The test function
We fix a finite prime p ∤ N of F . We will consider test functions
f =
∏
v
fv ∈ C∞c (G(AF ))
defined as follows.
At a finite place v 6= p we take fv = 1v, the characteristic function of ZvR×v .
At v = p we allow fp to be any element in the Hecke algebra H(G(Fp), ZpR×p ).
Let v ∈ Σ∞. We fix an isomorphism D×(Fv) ∼= GL(2,C) which gives
an irreducible 2-dimensional representation V of D×(Fv). We set π′2kv =
Sym2kv−2 V ⊗det−kv+1, which descends to a well defined representation ofG(Fv)
corresponding, via the local Jacquet-Langlands correspondence, to the weight
2kv discrete series on PGL(2,R). We note that dim π′2kv = 2kv − 1. Let 〈 , 〉
be a G(Fv) invariant inner product of π′2kv . As is well known, since |mv| < kv
the subspace of π′2kv ,
π′2kv (Ωv) =
{
w ∈ π′2kv : π′2kv (t)w = Ωv(t)w for all t ∈ E×v
}
is 1-dimensional. We fix a unit vectorwv ∈ π′2kv (Ωv) and define fv ∈ C∞c (G(Fv))
by
fv(g) = 〈π′2kv (g)wv, wv〉.
Given fp ∈ H(G(Fp), ZpR×p ) we denote by fˆp the function on unramified
representations πp defined by
fˆp(πp) =
1
vol(Zp\ZpR×p )
Tr πp(fp).
We also view fˆp as a function on [−2, 2] in the usual way.
3.2 Local preliminaries
Before continuing onto the calculation of I(f) for f as in Section 3.1 we record
some local preliminaries. For an irreducible admissible representation σ ofG(Fv)
acting on the space Vσ and fv ∈ C∞c (G(Fv)) we let,
σ(fv) : Vσ → Vσ : w 7→
∫
G(Fv)
fv(gv)σ(gv)w dgv.
We record the following basic results for use later.
16

Lemma 3.2. Let v be a finite place of F not dividing N and let fv ∈ C∞c (G(Fv))
be as in Section 3.1. Let σ be an irreducible unitarizable representation of G(Fv).
Then σ(fv) kills the orthogonal complement of σR
×
v in Vσ and for w ∈ σR
×
v ,
σ(fv)w = vol(Zv\ZvR×v )fˆv(σ)w.
Furthermore σR×v has dimension at most one.
Proof. For v ∤ N, G(Fv) ∼= PGL(2, Fv) and this result is well known.
Lemma 3.3. Let v be a finite place of F dividing N and let fv ∈ C∞c (G(Fv)) be
as in Section 3.1. Let σ be an irreducible unitarizable representation of G(Fv).
Then σ(fv) kills the orthogonal complement of σR
×
v in Vσ and for w ∈ σR
×
v ,
σ(fv)w = vol(Zv\ZvR×v )w.
Furthermore if σR×v 6= 0 then σ = δ ◦NDv for an unramified character δ of F×v .
Proof. The first part of the lemma is clear. It remains to prove the last state-
ment. In this case we use the exact sequence,
1 // R×v // D×v
NDv
// F×v /UFv // 1 .
From which it follows that if σR×v 6= 0 then σ = δ ◦ NDv for a character δ of
F×v /UFv .
Lemma 3.4. Let v be an infinite place of F and let fv ∈ C∞c (G(Fv)) be as in
Section 3.1. Let σ be an irreducible unitarizable representation of G(Fv). Then
σ(fv) kills the space of σ unless σ ∼= π′2kv . Furthermore for σ = π′2kv the map
π′2kv (fv) kills the orthogonal complement of π′2kv (Ωv) and for w ∈ π′2kv (Ωv),
π′2kv (fv)w =
vol(G(Fv))
2kv − 1
w.
Proof. Since fv is a matrix coefficient of π′2kv it follows that σ(fv) kills the space
of σ unless σ ∼= π′2kv . On the other hand if 0 6= w ∈ π′2kv (Ωv) then
fv(g) =
〈π′2kv (g)w,w〉
〈w,w〉 .
Since π′2kv (Ωv) is spanned by w it follows that π′2kv (fv) kills the orthogonal
complement of π′2kv (Ωv) in π′2kv . Finally,
〈π′2kv (fv)w,w〉 =
∫
G(Fv)
fv(g)〈π′2kv (g)w,w〉 dg
=
∫
G(Fv)
|〈π′2kv (g)w,w〉|2
〈w,w〉 dg
=
vol(G(Fv))
dimπ′2kv
〈w,w〉.
The last part of the Lemma now follows.
17

As a temporary expedient we set,
λ(N, k ) =
∏
v<∞
vol(Zv\R×v Zv)
∏
v∈Σ∞
vol(G(Fv))
2kv − 1
=
1
|∆F |
3
2L(2, 1F )
∏
v|N
1
qv − 1
∏
v∈Σ∞
4π
2kv − 1
.
3.3 Global calculations
Having chosen f ∈ C∞c (G(AF )) in Section 3.1 we now set about computing
I(f) = Icusp(f) + Ires(f).
We begin with the calculation of Ires(f) which, as we shall see, is often zero.
We define Xun(F ) to be the set of unramified characters χ : F×\A×F → {±1}.
Lemma 3.5. For f ∈ C∞c (G(AF )) as in Section 3.1, Kf,res(x, y) ≡ 0 unless
2kv = 2 for all v ∈ Σ∞, in which case
Kf,res(x, y) =
λ(N, k)
vol(G(F )\G(AF ))
∑
δ∈Xun(F )
δ(ND(xy−1))fˆp(δp ◦ND).
Proof. We recall that
Ares(G) =
{
δ ◦ND : δ : F×\A×F → {±1}
}
.
We fix a character δ : F×\A×F → {±1}. By definition we have
(R(f)(δ ◦ND))(x) =
∫
G(AF )
f(g)δ(ND(xg)) dg
=
∏
v
∫
G(Fv)
fv(gv)δv(NDv (xgv)) dgv.
We now set about computing the local integrals. Suppose first that v is a non-
archimedean place. If v 6= p then fv is the characteristic function of ZvR×v .
Hence,
∫
G(Fv)
fv(gv)δv(NDv (xgv)) dgv = δv(NDv (x))
∫
Zv\ZvR×v
δv(NDv (gv)) dgv.
Since the norm map NDv : R×v → UFv is surjective,
∫
Zv\ZvR×v
δv(NDv (xgv)) dgv =
{
0, if δv is ramified;
δv(NDv (x)) vol(Zv\ZvR×v ), if δv is unramified.
18

If v = p, then
∫
G(Fv)
fp(gp)δp(NDp(xgp)) dgp = δp(NDp(x))
∫
G(Fp)
fp(gp)δp(NDp(gp)) dgp
= δp(NDp(x))Tr(δp ◦NDp)(fp).
Clearly, since fp is bi-R×p -invariant we have Tr(δp ◦ NDp)(fp) = 0 unless δp is
unramified, in which case
∫
G(Fv)
fp(gp)δp(NDp(xgp)) dgp = δp(NDp(x)) vol(ZpR×p \R×p )fˆp(δp ◦NDp).
Finally, let v be an archimedean place of F . In this case we have Fv ∼= R
and, under this isomorphism, NDv (D×v ) = R+. Hence since δv is quadratic
δv(NDv (gv)) = 1 for all gv ∈ D×v . Thus for v ∈ Σ∞,
∫
G(Fv)
fv(gv)δv(NDv (xgv)) dgv =
∫
G(Fv)
〈π′2kv (g)wv, wv〉 dgv,
by definition of fv. Clearly this integral is zero unless π′2kv is the trivial repre-
sentation which is the case if and only if 2kv = 2. Thus for v ∈ Σ∞,
∫
G(Fv)
fv(gv)δv(NDv (xgv)) dgv =
{
0, if 2kv > 2;
vol(G(Fv)), if 2kv = 2.
Putting these local calculations together shows that (R(f)(δ ◦ ND))(x) is
zero unless δ is everywhere unramified and kv = 1 for all v ∈ Σ∞ in which case
(R(f)(δ◦ND))(x) = δ(ND(x))fˆp(δp◦NDp)
∏
v<∞
vol(Zv\ZvR×v )
∏
v∈Σ∞
vol(G(Fv)).
Finally to finish the proof it remains to observe that
Kf,res(x, y) =
∑
δ
(R(f)(δ ◦ND))(x)(δ ◦ND)(y)
vol(G(F )\G(AF ))
.
We can now compute Ires(f).
Lemma 3.6. For f ∈ C∞c (G(AF )) as in Section 3.1,
Ires(f) = C(k,Ω, fp)λ(N, k)
vol(A×FE×\A×E)2
vol(G(F )\G(AF ))
,
where C(k,Ω, fp) = 0 unless kv = 1 for all v ∈ Σ∞ and Ω is of the form
Ω = δ ◦NE/F for an everywhere unramified character δ of F×\A×F of order at
most 2. In this latter case we have
C(k,Ω, fp) =
{
fˆp(δp ◦ND) + fˆp(ηpδp ◦ND), if E/F is unramified everywhere;
fˆp(δp ◦ND), otherwise.
19

Proof. By definition
Ires(f) =
∫
E×A×F \A
×
E
∫
E×A×F \A
×
E
Kf,res(t1, t2)Ω(t−11 t2) dt1 dt2.
From Lemma 3.5 we see that Ires(f) = 0 unless kv = 1 for all v ∈ Σ∞, in which
case,
Ires(f) =
λ(N, k )
vol(G(F )\G(AF ))
∑
δ∈Xun(F )
fˆp(δp◦NDp)
∣∣∣∣∣
∫
E×A×F \A
×
E
δ(ND(t))Ω(t−1) dt
∣∣∣∣∣
2
.
Considering E× as a subgroup of D× we note that ND|E× : E× → F× is equal
to NE/F : E× → F×. Hence,
∣∣∣∣∣
∫
E×A×F \A
×
E
δ(ND(t))Ω(t−1) dt
∣∣∣∣∣
2
=
{
vol(E×A×F \A×E)2, if Ω = δ ◦NE/F ;
0, otherwise.
Finally it suffices to note that if Ω = δ ◦NE/F with δ ∈ Xun(F ) then the only
other quadratic character χ such that Ω = χ ◦ NE/F is χ = δη, and since δ is
assumed to be unramified, δη ∈ Xun(F ) if and only if η is unramified.
We now set about computing
Icusp(f) =
∑
π′∈Acusp(G)
Iπ′(f).
It is clear from the results of Section 3.2 and Fact 3.1 that if π′ ∈ Acusp(G) then
R(f) is zero on the space of π′ unless π′ ∈ F ′(N, 2k). Hence,
Lemma 3.7. For f ∈ C∞c (G(AF )) as in Section 3.1,
Icusp(f) =
∑
π′∈F ′(N,2k)
Iπ′(f).
It remains to compute Iπ′(f) for π′ ∈ F ′(N, 2k).
Lemma 3.8. For f ∈ C∞c (G(AF )) as in Section 3.1 and π′ ∈ F ′(N, 2k),
Iπ′(f) = λ(N, k)
L(1/2, πE ⊗ Ω)
L(1, π, Ad)
L(2, 1F )LS(Ω)(1, η)2
√
|∆F |
2
√
c(Ω)|∆E |
∏
v|N
(1−q−1v )
∏
v∈Σ∞
Γ(2kv)
πΓ(kv +mv)Γ(kv −mv)
,
where π = JL(π′).
Proof. Let π′ ∈ F ′(N, 2k) and let Vπ′ denote the space of π′. Let Wπ′ denote
the space of w ∈ Vπ′ such that,
1. π′(k)w = w for all k ∈ ∏v<∞R×v , and
20

and
C(k,Ω, fp)
2L(1, η)2
|∆F |
3
2L(2, 1F )
∏
v|N
1
qv − 1
∏
v∈Σ∞
4π
2kv − 1
,
where C(k,Ω, fp) is defined in Lemma 3.6.
4 Geometric side of the trace formula
Recall that
I(f) =
∫
A×F E×\A
×
E
∫
A×FE×\A
×
E
Kf(t1, t2)Ω(t−11 t2) dt1 dt2.
A quick calculation shows that the following matrices are a set of representatives
for the double cosets of E×\D×/E×:
{(
1 0
0 1
)
,
(
0 ǫ
1 0
)}
∪
{(
1 ǫx
x 1
)
: x ∈ E×/E1
}
.
The first two cosets are referred to as irregular cosets and the remaining cosets
are called regular cosets.
By the geometric expansion of the relative trace formula [JC01, Section 1
(8)],
I(f) =
∑
ξ∈ǫNE×
I(ξ, f) + vol(A×FE×\A×E)
[
I (0, f) + δ(Ω2)I (∞, f)
]
where for an idele class character χ, δ(χ) = 1 if χ = 1 and δ(χ) = 0 otherwise.
For ξ = ǫxx,
I (ξ, f) :=
∫
A×F \A
×
E
∫
A×F \A
×
E
f
(
t1
(
1 ǫx
x 1
)
t2
)
Ω(t1t2)dt1dt2,
I (0, f) :=
∫
A×F \A
×
E
f (t)Ω(t)dt,
and
I (∞, f) :=
∫
A×F \A
×
E
f
(
t
(
0 ǫ
1 0
))
Ω(t)dt.
We remark that these integrals factor as a product of local ones, which we
denote with similar notation. We note that the geometric expansion in [JC01]
has δ(Ω2) next to I(0, f) rather than I(∞, f). This is because Kf(t1, t2) is
integrated against Ω(t1t2)−1 rather than against Ω(t−11 t2).
In this section we compute the period integrals I(g, f) for f chosen as in
Section 3.1. First we make the necessary local calculations in Sections 4.1 and
4.2 and then we bring together the local calculations to write down the geometric
side of the global relative trace formula in Section 4.3. We note that the choice
22

of local function fv is intrinsic to G(Fv), as is the parameterization of the
double cosets. Hence for simplicity we can fix the following identifications for
the remainder of this section.
For v /∈ Σ∞,
Dv =
{(
α ǫvβ
β α
)}
, Ev =
{(
α 0
0 α
)}
, where ǫv = 1 if v ∤ N,
ǫv = ̟v if v | N,
and for v ∈ Σ∞,
Dv =
{(
α −β
β α
)}
, Ev =
{(
α 0
0 α
)}
.
For v that splits in E, let Ev = Fv ⊕ Fv. For v /∈ Σ∞, let τv be such that
OEv = OFv [τv] and τv is a uniformizer in Ev if Ev/Fv is ramified. If v is inert
in E, let ̟Ev = ̟v. If v ramifies, let ̟Ev = τv. If v splits let ̟Ev ∈ OEv be
such that v(N(̟Ev )) = 1. For a valuation v of F that is not split in E, let vE
be the corresponding valuation on E. Let
Rv =
{(
α ǫvβ
β α
)
: α ∈ 1
(τv − τv)̟n(Ωv)v
(
OFv +̟n(Ωv)v OEv
)
, α+ β ∈ OFv +̟n(Ωv)v OEv
}
.
This is a maximal order in Dv such that Rv ∩ Ev = OFv +̟n(Ωv)v OEv .
For the rest of this section we will drop the v from our notation and let
n = n(Ωv) when it is clear that we are working locally.
4.1 Irregular cosets
First we compute the orbital integrals associated to the irregular cosets.
Lemma 4.1. Let v divide N. Then
I(0, fv) = vol(F×v \F×v (1 +̟n(Ωv)v OEv ))×.
Proof. We have
I(0, fv) =
∫
F×v \E×v
fv
((
a 0
0 a
))
Ωv(a)d×a
= vol(F×v \F×v (1 +̟n(Ωv)v OEv))×.
Lemma 4.2. Let v divide N. Then
I (∞, fv) = 0.
23

Proof.
I (∞, fv) =
∫
F×v \E×v
fv
((
0 ̟va
a 0
))
Ωv(a)d×a.
Clearly (
0 ̟va
a 0
)
/∈ ZvR×v
because ZvR×v only contains matrices, g, where v(det g) is even.
Before we compute the irregular orbital integrals for finite v away from N,
we need the following technical lemma.
Lemma 4.3. Let v be a finite prime not dividing N. Let α = aα + bατv and
β = aβ + bβτv. Then F×v R×v αR×v = F×v R×v βR×v if and only if
v(N(α))−2min{v(aα), v(̟−n(Ωv)v bα)} = v(N(β))−2min{v(aβ), v(̟−n(Ωv)v bβ)}.
Proof. For the proof we can take D = M(2, F ) and embed E as
E →֒M(2, F ) : a+ bτ 7→
(
a+ bTrE/F (τ) b̟−n
−b̟nN(τ) a
)
.
We set R = M(2,OF ), then we have R ∩ E = OF +̟nOE . We recall that
GL(2, F ) =
⊔
m≥0
F×R×
(
̟m
1
)
R×,
and
g =
(
a b
c d
)
∈ F×R×
(
̟m
1
)
R×,
if and only if
m = v(det g)− 2min{v(a), v(b), v(c), v(d)}.
Thus we see that
α = a+ bτ ∈ F×R×
(
̟m
1
)
R×,
if and only if
m = v(N(α)) − 2min{v(a), v(̟−nb)}.
For fv ∈ H(G(Fv), ZvR×v ) we have
I(0, fv) =
∫
F×v \E×v
fv(a)Ωv(a) d×a
= vol(F×v \F×v (OFv +̟n(Ωv)v OEv)×)I˜(fv),
24

where
I˜(fv) :=
∑
α∈F×v (OFv+̟
n(Ωv)
v OEv )×\E×v
fv(α)Ωv(α).
For finite v away from N,
(
0 ǫ
1 0
)
∈ R×v , and hence
I(∞, fv) =
∫
F×v \E×v
fv
(
a
(
0 ǫv
1 0
))
Ωv(a)d×a
=
∫
F×v \E×v
fv (a)Ωv(a)d×a
= I(0, fv).
In the following three lemmas, we compute I˜(fv) for v inert, ramified and split
in E.
Lemma 4.4. Let v be a finite prime not dividing N, which is inert in E. For
fv ∈ H(G(Fv), ZvR×v ),
I˜(fv) =
{
fv(1) when n(Ωv) = 0
fv(1)− fv(1 + τv̟n(Ωv)−1v ) when n(Ωv) > 0.
Proof. Recall that n = n(Ωv). We note that a set of representatives for F×(OF+
̟nOE)×\E× is given by
{1 + bτ : b ∈ OF /pnF } ∪ {a+ τ : a ∈ pF /pnF} .
Applying Lemma 4.3 we see that
I˜(f) =
n∑
k=0
f(1 + τ̟k)
∑
b∈(pkF /pnF )×
Ω(1 + bτ) + f(τ)
∑
a∈pF /pnF
Ω(a+ τ).
The lemma follows by the observation that
∑
b∈(pkF /pnF )×
Ω(1 + bτ) =



0, if 0 ≤ k ≤ n− 1;
−1, if k = n− 1;
1, if k = n,
and that ∑
a∈pF /pnF
Ω(a+ τ) = 0.
Lemma 4.5. Let v be a finite prime which is ramified in E. For fv ∈ H(G(Fv), ZvR×v ),
I˜(fv) =
{
fv(1) + Ωv(τv)fv(τv) when n(Ωv) = 0
fv(1)− fv(1 + τv̟n(Ωv)−1v ) when n(Ωv) > 0.
25

Proof. We note that a set of representatives for F×(OF +̟nOE)×\E× is given
by
{1 + bτ : b ∈ OF /pnF} ∪ {a̟ + τ : a ∈ OF /pnF} .
Applying Lemma 4.3 we see that
I˜(f) =
n∑
k=0
f(1 + τ̟k)
∑
b∈(pkF /pnF )×
Ω(1 + bτ) + f(̟ + τ)
∑
a∈OF /pnF
Ω(a̟ + τ).
The lemma follows by the observation that
∑
b∈(pkF /pnF )×
Ω(1 + bτ) =



0, if 0 ≤ k ≤ n− 1;
−1, if k = n− 1;
1, if k = n.
And that ∑
a∈OF /pnF
Ω(a̟ + τ) =
{
Ω(τ), if n = 0;
0, if n > 0.
Lemma 4.6. Let v be a finite prime which is split in E. Let fv ∈ H(G(Fv), R×v ).
When n(Ωv) = 0 we have
I˜(fv) =
∑
α∈E×v /F×v O×Ev
Ωv(α)fv(α),
and when n(Ωv) > 0 we have
I˜(fv) = fv(1)− fv(1 + τv̟n(Ωv)−1v ).
Proof. In this case we take R = M(2,OF ) and embed E as
(a, b) 7→
(
a ̟−n(a− b)
0 b
)
.
By Lemma 4.3 we have
(a, b) ∈ F×R×
(
̟m 0
0 1
)
R×
if and only if m = v(ab)− 2min{v(a), v(̟−n(a− b))}. We recall
I˜(f) =
∑
α∈F×(OF+̟n(Ω)OE)×\E×
f(α)Ω(α)
=
∑
α∈UnF \F×
Ω(α, 1)f(α, 1).
26

Thus we see that if n = 0 then
I˜(f) = f(1) +
∞∑
m=1
(Ω(̟m, 1) + Ω(̟−m, 1))f(̟m, 1).
On the other hand if n > 0, then only α ∈ UF contribute to the sum, thus
I˜(f) =
∑
a∈UnF \UF
Ω(a, 1)f(a, 1)
= f(1, 1)− f(1 +̟n−1, 1).
We now compute the irregular orbital integrals at the archimedean places.
Lemma 4.7. Let v ∈ Σ∞. Then I (0, fv) = vol(F×v \E×v ).
Proof. We have
I (0, fv) =
∫
F×v \E×v
fv
((
a 0
0 a
))
Ωv(a)d×a
= vol(F×v \E×v ).
Lemma 4.8. Let v ∈ Σ∞. When α 6= 0 we have
fv
(
α −β
β¯ α¯
)
=
1
(αα¯ + ββ¯)kv−1
kv−|mv|−1∑
i=0
(−1)i
(kv +mv − 1
i
)(kv −mv − 1
i
)
(ββ¯)i(αα)kv−1−i
(α
α¯
)−mv
,
when mv = 0 we have
fv
(
0 −β
β¯ 0
)
= (−1)kv−1
and when mv 6= 0 we have
fv
(
0 −β
β¯ 0
)
= 0.
Proof. Let π be the discrete series representation of PGL(2,R) of weight 2k.
Recall that Ω is the character of C× given by
Ω : z 7→
(z
z¯
)m
27

where m is an integer with |m| < k. We have
D×(R) =
{(
α −β
β¯ α¯
)
∈ GL(2,C)
}
.
Viewing D×(R) ⊂ GL(2,C) gives an irreducible 2-dimensional representation
V of D×(R). We take π′ to be the representation of G(R) which corresponds
to π via the Jacquet-Langlands correspondence. Thus
π′ = Sym2k−2 V ⊗ det−(k−1).
We realize π′ on the space of homogeneous polynomials in x and y of degree
2k − 2, with g acting by
π′(g) : x 7→ αx+ β¯y, y 7→ −βx+ α¯y.
We set, for 0 ≤ i ≤ 2k − 2,
vi = xiy2k−2−i,
so that
π′
(
α 0
0 α¯
)
vi =
(α
α¯
)i−(k−1)
vi.
Hence we have
π′
(
α 0
0 α¯
)
vm+k−1 = Ω(α)vm+k−1.
We have, for g ∈ G(R),
f(g) = 〈π
′(g)vm+k−1, vm+k−1〉
〈vm+k−1, vm+k−1〉
.
We compute that
f
(
α −β
β¯ α¯
)
=
1
(αα¯ + ββ¯)k−1
k−|m|−1∑
i=0
(−1)i
(k +m− 1
i
)(k −m− 1
i
)
(ββ¯)iα¯m+k−1−iαk−m−1−i
and the lemma now follows.
From the definition of I(∞, fv) we have,
Corollary 4.9. For v ∈ Σ∞,
I(∞, fv) =
{
vol(F×v \E×v )(−1)kv−1, if mv = 0;
0, otherwise.
28

4.2 Regular cosets
Recall that
I(ξ, fv) =
∫
(F×v \E×v )2
fv
((
ab ab¯xǫv
a¯bx¯ ab
))
Ωv(ab)d×bd×a.
By a change of variables this equals
∫
F×v \E×v
∫
E1v
fv
((
a 0
0 a¯
)
γxz
)
Ωv(a)d×zd×a
where γxz =
(
1 xzǫv
xz 1
)
. For x ∈ E×v , we let
I(x,E1v ) :=
{
z ∈ E1v : ∃a ∈ E×v such that aγxz ∈ R×v
}
.
In this section we compute I(ξ, fv) under certain simplifying assumptions, and
provide a bound on |I(ξ, fv)| in all cases.
4.2.1 Exact calculations
We begin by calculating I(ξ, fv) when v divides N.
Lemma 4.10. Let v divide N. Then
I(ξ, fv) =
{
0 if v(ξ) ≤ 0
vol(F×v \E×v )2 if v(ξ) ≥ 1.
Proof. It is easy to see that aγxz ∈ F×R× exactly when v(x) ≥ 0. Hence v(ξ)
must be greater than or equal to one.
We now establish a vanishing result for I(ξ, fv) for finite v away from N.
Lemma 4.11. Let v be a finite prime not dividing N. Suppose that fv is the
characteristic function of the double coset F×v R×v γR×v for γ ∈ Rv. If v(1− ξ) >
v(dE/F c(Ω) det(γ)), then I(ξ, fv) = 0.
Proof. Clearly aγxz ∈ F×v R×v γR×v if and only if there exists a λ ∈ F×v such that
λaγxz ∈ R×v γR×v . If this matrix lies in R×v γR×v , then v(λ2aa(1−ξ)) = v(det(γ))
and λa ∈ 1(τ−τ)̟nv OE . Thus the orbital integral is zero unless v(1 − ξ) ≤
v(dE/F c(Ω) det(γ)).
In the next two lemmas we compute I(ξ,1v) when Ωv is unramified.
Lemma 4.12. Let v be a finite prime not dividing N, not split in E and such
that n(Ωv) = 0. If v ramifies in E, assume that the characteristic of the residue
field of Fv is odd. Then I(ξ,1v) equals
vol(F×v \F×v UEv ) vol(F×v \E×v )Ωv(̟
vE (1−ξ)
2
Ev )×



0 if v(1− ξ) > v(dE/F )
1 if v(1− ξ) ≤ 0
1
2 if v(1− ξ) = v(dE/F ) > 0.
29

Proof. Let x = (x1, x2) and k = v(1 − ξ). We look at our orbital integral,
I(ξ,1) =
∫
F×\E×
∫
E1
1
((
a azx
azx a
))
Ω(a)d×zd×a.
A matrix in the integrand of this integral lies in F×R× if and only if there is a
λ ∈ F× such that
1. v(λ2aa) = −v(1− ξ)
2. λa ∈ OE
3. λa(1 + zx) ∈ OE .
It is easy to see from these conditions that if v(1 − ξ) ≤ 0,
I(ξ,1) =
−k∑
l=0
∫
F×\F×(̟lUF×̟−k−lUF )
Ω(a)d×a
−k−l+v(x2)∑
m=−v(x1)−l
∫
(̟m,̟−m)UF
d×b.
If k = 0,
I(ξ,1) = (v(ξ) + 1) vol (UF )2 .
If k < 0, then v(x1x2) = k and
I(ξ,1) =
−k∑
l=0
Ω(̟k+2l, 1) vol (UF )2 .
Finally, we compute the regular orbital integral for v archimedean.
Lemma 4.14. For v ∈ Σ∞ and ξ ∈ Fv with ξ < 0 we have
I(ξ, fv) =
vol(F×v \E×v )2
(1 − ξ)kv−1
kv−|mv|−1∑
i=0
(kv +mv − 1
i
)(kv −mv − 1
i
)
(−ξ)i.
Proof. By definition of fv we have
fv
((
a 0
0 a¯
)
γ
(
b 0
0 b¯
))
= Ω−1v (ab)fv(γ).
Applying Lemma 4.8 gives the result.
31

4.2.2 Bounds on I(ξ,1v)
We now bound the regular orbital integral in the remaining cases. For a place
v of F , we fix x ∈ Ev such that ξ = ǫvxx¯.
Lemma 4.15. Let v be a finite prime. Then
|I(ξ,1v)| ≤ vol
(
F×v \(1 +̟n(Ωv)v OEv)×F×v
)
vol
(
I(x,E1v )
)
.
Proof. By definition,
I(ξ,1) =
∫
I(x,E1)
∫
F×\E×
1(aγxz)Ω(a)d×ad×z.
For each fixed z ∈ I(x,E1) we know that there exists an axz ∈ E× such that
axzγxz ∈ R×. By a change of variables with axz,
|I(ξ,1)| ≤
∫
I(x,E1)
∫
F×\E×
1(a)d×ad×z.
The lemma now follows by the fact that R ∩ E = OF +̟nOE .
Corollary 4.16. Let v be a finite prime not dividing N which is not split in E.
Then
|I(ξ,1v)|
≤ vol
(
F×v \(1 +̟n(Ωv)v OEv)×F×v
)
vol
({
z ∈ E1v : vE(1 + xz) ≥
vE(1− ξ)
2
})
.
Proof. If aγxz ∈ F×R× then there must be a λ ∈ F× such that v(λ2aa) =
−v(1− ξ) and λa(1 + zx) ∈ OF +̟nOE . Thus
I(x,E1) ⊆
{
z ∈ E1 : vE(1 + xz) ≥
vE(1− ξ)
2
}
.
Lemma 4.17. Let v be a finite prime which is split in E.
If v(1 − ξ) < 0, then
|I(ξ,1v)| ≤ (|v(ξ)|+ 1) vol
(
F×v \(1 +̟n(Ωv)v OEv )×F×v
)
vol (UFv ) .
If v(c(Ω)) ≥ v(1 − ξ) ≥ 0, then
|I(ξ,1v)| ≤
{
vol
(
F×v \(1 +̟n(Ωv)v OEv )×F×v
)
vol
(
E1 ∩ x−1(1 +̟ v(1−ξ)2 OEv)
)
, if v(1− ξ) is even;
0, if v(1− ξ) is odd.
32

Proof. Let a = (a1, a2), x = (x1, x2), z = (b, b−1) and k = v(1 − ξ). We take
τ = (1, 0). We look at our orbital integral,
I(ξ,1) =
∫
F×\E×
∫
E1
1
((
a azx
azx a
))
Ω(a)d×ad×z.
It is easy to see that aγxz ∈ R× if and only if the following conditions are
satisfied:
1. a2 ∈ 1̟nOF
2. a1 ∈ (−a2 +OF ) ∩̟−k−v(a2)UF
3. a2(1 + x2b−1) ∈ OF
4. a1(1 + x1b) ∈ a2(1 + x2b−1) +̟nOF .
For the intersection in condition (2) to be nonempty, we must have either v(a2) =
v(a1) = −k2 or −k ≥ v(a2) ≥ 0 and v(a1) = −k − v(a2).
Thus if k ≥ 0, then we must have v(a2) = v(a1) = −k2 and hence,
I(x,E1) ⊆ E1 ∩ 1x(1 +̟
v(1−ξ)
2 OEv).
If k < 0, then −k ≥ v(a2) ≥ 0 and v(a1) = −k− v(a2). In this case we must
have v(a2x2b−1) ≥ 0 and v(a1x1b) ≥ 0. Thus v(a2x2) ≥ v(b) ≥ −v(a1x1). Thus
vol
(
I(x,E1)
)
≤
−k∑
l=0
l+v(x2)∑
m=k+l−v(x1)
vol(UF ) = (|v(ξ)| + 1) vol(UF ).
For the last equality we are using the fact that if k < 0, then k = v(x1x2).
We complete the proof by applying Lemma 4.15.
We now bound the volume term that appears in the results above.
Lemma 4.18. Assume that v is unramified in E. Then,
vol(E1v∩x−1(1+̟kvOEv)) ≤



vol(E1v ∩ UEv)(1 + |v(xx¯)|), if k = 0 and v split;
vol(E1v ), if k = 0 and v not split;
vol(E1v )q−kv (1 + q−1v )−1, if k > 0 and v inert;
vol(E1v ∩ UEv)q−kv (1− q−1v )−1, if k > 0 and v split.
If v is ramified in E there exists a constant C(Ev, Fv) such that,
vol(E1v ∩ x−1(1 +̟kvOEv )) ≤ C(Ev, Fv)q−kv
for all k ≥ 0.
33

Proof. First we assume that k = 0. We note that when E/F is not split we
have
vol(E1 ∩ x−1OE) =
{
vol(E1), if x ∈ OE ;
0, otherwise.
Next we assume E/F is split. We write E = F ⊕ F and x = (x1, x2). Then we
have
x−1OE = {(α1, α2) : v(α1) ≥ −v(x1), v(α2) ≥ −v(x2)} .
Let (y1, y2) ∈ E1. Then we have v(y1y2) = 0 and hence y ∈ 1xOE implies that
−v(x1) ≤ v(y1) ≤ v(x2).
Hence the intersection is empty if v(xx¯) > 0, and if v(xx¯) ≤ 0 then
vol(E1 ∩ x−1OE) = (1 + |v(xx¯)|) vol(E1 ∩ UE).
We now assume k > 0. If we assume that E1 ∩ x−1(1 +̟kOE) 6= ∅ then we
clearly have
vol(E1 ∩ x−1(1 +̟kOE)) = vol(E1 ∩ (1 +̟kOE)).
So we need to compute the order of
(E1 ∩ UE)/E1 ∩ (1 +̟kOE) ∼−→(E1 ∩ UE)(1 +̟kOE)/(1 +̟kOE).
Now we have
(E1 ∩ UE)(1 +̟kOE) =
{
x ∈ UE : N(x) ∈ N(1 +̟kOE)
}
.
Hence we are looking to compute the size of the kernel of the map
N : UE/(1 +̟kOE) → UF /N(1 +̟kOE).
When E/F is unramified quadratic this map is surjective and we have
#UE/(1 +̟kOE) = q2k(1 − q−2)
and
#UF /N(1 +̟kOE) = #UF /(1 +̟kOF ) = qk(1− q−1)
and hence the kernel has order qk(1 + q−1). Next we assume that E = F ⊕ F ,
then the norm map is surjective and its kernel has order qk(1− q−1).
Finally we assume that E/F is ramified. We write the discriminant as
dE/F = pt+1. From [Ser62, Corollaire 1, pg 93] we deduce that when t is even
and k > t2 ,
N(1 +̟kOE) = 1 +̟k+
t
2OF ,
and when t is odd k ≥ t2 ,
N(1 +̟kOE) = 1 +̟k+
t+1
2 OF .
Since #UE/(1 +̟kOE) = q2k(1 − q−1) we see that when k > t2 , the order of
the kernel of N : UE/(1+̟kOE) → UF /N(1+̟kOE) is at least qk−
t+1
2 . Thus
there exists some constant C′(E) such that for all k ≥ 0 the order of this kernel
is at least C′(E)qk.
34

The following lemma is a direct consequence of Corollary 4.16 and Lemmas
4.17 and 4.18, and the fact that
vol(F×v \F×v (1 +̟n(Ωv)v OEv )×) = vol(UFv\UEv)q−n(Ωv)v L(1, ηv).
Lemma 4.19. Assume n(Ωv) > 0, and let k = vF (1−ξ)2 . There exists a constant
C(Ev, Fv) that is equal to one for all v unramified in E and such that
|I(ξ,1v)| ≤q−n(Ωv)v L(1, ηv) vol(UFv\UEv) vol(E1v ∩ UEv)C(Ev, Fv)
×



q−kv L(1, ηv) when k > 0
1 when k ≤ 0 and v is not split
1 + |vF (ξ)| when k ≤ 0 and v is split.
4.3 Global calculations
We recall that
I(f) =
∑
ξ∈ǫNE×
I(ξ, f) + vol(A×FE×\A×E)
[
I (0, f) + δ(Ω2)I (∞, f)
]
.
We now apply the local calculations of the previous sections to this global dis-
tribution. We denote by
Ireg(f) =
∑
ξ∈ǫNE×
I(ξ, f),
and
Iirreg(f) = vol(A×FE×\A×E)
[
I (0, f) + δ(Ω2)I (∞, f)
]
.
Thus
I(f) = Ireg(f) + Iirreg(f). (3)
We begin by computing Iirreg(f).
Proposition 4.20. For f as defined in Section 3.1,
Iirreg(f) =
2[F :Q]+1L(1, η)LS(Ω)(1, η)
√
|∆F |√
c(Ω)
√
|∆E |
(
1 + δ(Ω2)δ(N)
∏
v∈Σ∞
(−1)kv−1
)
I˜ (fp) ,
where δ(N) = 0 if N 6= OF and δ(OF ) = 1.
Proof. By Lemmas 4.1, 4.2, 4.4, 4.5, 4.6 and 4.7, and Corollary 4.9,
vol(A×FE×\A×E)
[
I (0, f) + δ(Ω2)I (∞, f)
]
is equal to
2L(1, η)I˜ (fp)
∏
v<∞
vol(F×v \F×v (1+̟n(Ωv)v OEv )×)
∏
v∈Σ∞
vol(F×v \E×v )
(
1 + δ(Ω2)δ(N)(−1)kv−1
)
.
35

Recall that for v ∈ Σ∞,
vol(F×v \E×v ) = vol(R×\C×) = 2,
and for v finite we have
vol(F×v \F×v (1+̟n(Ωv)v OEv )×) =
{
vol(UFv\UEv), if v 6∈ S(Ω);
vol(UFv\UEv)q−n(Ωv)v L(1, ηv), if v ∈ S(Ω).
The proposition now follows.
We now consider the regular terms in the trace formula. For each v ∈ Σ∞
we let ιv denote the corresponding embedding F →֒ R. We note that ξ ∈ ǫNE×
if and only if
1. ιv(ξ) < 0 for all v ∈ Σ∞,
2. v(ξ) is odd for all v | N, and
3. ηv(ξ) = 1 for all finite v ∤ N.
We define
Xp =
{
γ ∈ Rp : ̟−1p γ 6∈ R
}
.
For fp ∈ H(G(fp), R×p ) we define
n(fp) = max {vp(det γ) : γ ∈ Xp, fp(γ) 6= 0}
and set I(fp) = pn(fp).
We denote by S(Ω,N, fp) the set of ξ ∈ ǫNE× such that
1. v(ξ) ≥ 1 for all v | N, and
2. (1 − ξ)−1 ∈ (c(Ω)dE/F I(fp))−1.
Lemma 4.21. We have
Ireg(f) =
∑
ξ∈S(Ω,N,fp)
I(ξ, f).
Furthermore, the set S(Ω,N, fp) is finite, and empty when |N| ≥ dE/F (c(Ω)|I(fp)|)hF .
Proof. The fact that Ireg(f) is supported on S(Ω,N, fp) follows from Lem-
mas 4.10 and 4.11. Suppose now that ξ ∈ S(Ω,N, fp). We fix 0 6= x ∈
c(Ω)dE/F I(fp), then
(1− ξ)−1x = m ∈ OF
and hence
ξ = m− xm .
36

We now take v ∈ Σ∞ and consider ιv : F →֒ R, then since ιv(ξ) < 0 it follows
that
|ιv(m− x)| < |ιv(x)|.
The finiteness of S(Ω,N, fp) now follows from the finiteness of
{
y ∈ OF : |ιv(y)| < |ιv(x)| for all v ∈ Σ∞ and x ∈ c(Ω)dE/F I(fp)
}
.
We note that since v(ξ) ≥ 1 for all v | N we also require m − x ∈ N. Hence
S(Ω,N, fp) is empty whenever
{
y ∈ OF : |ιv(y)| < |ιv(x)| for all v ∈ Σ∞ and x ∈ c(Ω)dE/F I(fp)
}
∩N = {0},
which is clearly the case when |N| ≥ dE/F |c(Ω)I(fp)|hF .
Corollary 4.22. For N sufficiently large, e.g. for N with absolute norm at
least dE/F (c(Ω)|I(fp)|)hF , we have
I(f) = 2
[F :Q]+1L(1, η)LS(Ω)(1, η)
√
|∆F |√
c(Ω)
√
|∆E |
(
1 + δ(Ω2)δ(N)
∏
v∈Σ∞
(−1)kv−1
)
I˜ (fp) .
We note that under certain simplifying assumptions we can explicitly com-
pute the regular terms in the geometric side of the trace formula.
We define, for integers k and m with |m| < k, polynomials
Pk,m(x) =
1
(1 − x)k−1
k−|m|−1∑
i=0
(k +m− 1
i
)(k −m− 1
i
)
(−x)i.
For an ideal a ⊂ OF we define
RE(a) =
{
b ⊂ OE : NE/F (b) = a
}
.
For non-zero ideals a, b ∈ OF we define
σ(a, b) = # {c ⊂ OF : c|a + b} .
When Ω is unramified we regard its finite part as a character on the group
of fractional ideals of F .
Proposition 4.23. Assume that Ω is unramified, fp = 1p and E/F is unram-
ified at the even places of F . Let d ∈ OF be a generator of dE/F . Then we have
I(f) equal to the sum of
2[F :Q]+1L(1, η)
√
|∆F |√
c(Ω)
√
|∆E |
(
1 + δ(Ω2)δ(N)
∏
v∈Σ∞
(−1)kv−1
)
and
4[F :Q]
|∆F |
|∆E |
∑
n
|RE(nN−1)|σ(dE/F , (n+d))
∑
a∈RE((n+d))
Ω(D−1E/F a)
∏
v∈Σ∞
Pkv ,mv
(
ιv
( n
n+ d
))
,
with the outer sum taken over the finite set of n ∈ N such that
37

1. ηv
(
1 + dn
)
= 1 for all v | dE/F , and
2. ιv(n) lies between −ιv(d) and 0 for all v ∈ Σ∞.
Proof. Let ξ ∈ ǫNE× and let v be a finite place of F . When v splits in E we
have, with the notation of Lemma 4.13,
• v(1 − ξ) > 0 =⇒ I(ξ, fv) = 0,
• v(1 − ξ) = 0 =⇒ I(ξ, fv) = (1 + v(ξ)) vol(UFv )2,
• v(1 − ξ) < 0 =⇒ I(ξ, fv) =
∑|v(ξ)|
i=0 Ω(̟iv, ̟
|v(ξ)|−i
v ) vol(UFv )2.
When v ∤ N is inert and unramified in E we have Ωv trivial and, by Lemma
4.12,
• v(1 − ξ) > 0 =⇒ I(ξ, fv) = 0,
• v(1 − ξ) odd and ≤ 0 =⇒ I(ξ, fv) = 0,
• v(1 − ξ) even and ≤ 0 =⇒ I(ξ, fv) = vol(UFv\UEv)2.
When v | N we have Ωv trivial and, by Lemma 4.10,
• v(ξ) ≤ 0 =⇒ I(ξ, fv) = 0,
• v(ξ) ≥ 1 =⇒ I(ξ, fv) = vol(UFv\UEv)2.
When v is odd and ramified in E we have Ω2v trivial and, by Lemma 4.12,
• v(1 − ξ) > 1 =⇒ I(ξ, fv) = 0,
• v(1 − ξ) = 1 =⇒ I(ξ, fv) = vol(UFv\UEv)2Ω(̟Ev)−1,
• v(1 − ξ) ≤ 0 =⇒ I(ξ, fv) = 2Ω(̟Ev)
vE(1−ξ)
2 vol(UFv\UEv)2.
And at v ∈ Σ∞ we have, by Lemma 4.14,
I(ξ, fv) = vol(F×v \E×v )2Pmv ,kv (ιv(ξ)).
As in the proof of Lemma 4.21 we note that if I(ξ, f) 6= 0 then we have
ξ = nn+ d
with n ∈ N. Furthermore such an element lies in ǫNE× if and only if
• RE(nN−1) and RE((n+ d)) are non-empty,
• ηv
(
1 + dn
)
= 1 for all v | dE/F , and
• ιv(n) lies between ιv(0) and −ιv(d) for all v ∈ Σ∞.
38

5.1 An application of the Plancherel formula
We assume throughout this subsection that F is a non-archimedean local field
of characteristic zero. We let q denote the order of the residue field of F . We
let G′ = PGL(2, F ) and K = PGL(2,OF ). We fix a Haar measure on G′ which
gives K volume one. For n ≥ 0 we denote by fn the characteristic function of
K
(
̟n 0
0 1
)
K.
For s ∈ iR let πs denote the unramified principal series representation of G′
unitarily induced from (
a 0
0 b
)
7→
∣∣∣ab
∣∣∣
s
.
For f ∈ H(G′,K) and x ∈ [−2,+2] we define
fˆ(x) = Tr πs(f),
where x = qs + q−s. We have (see [RR05, Lemma 9]) fˆ0 ≡ 1 and for n > 0,
fˆn(x) = q
n
2
(
qns + q−ns + (1− q−1)(q(n−2)s + q(n−4)s + . . .+ q−(n−2)s)
)
. (4)
On the interval [−2,+2] we take the Sato-Tate measure
µ∞ =
√
4− x2
2π dx
and the spherical Plancherel measure on PGL(2, F ),
µq =
q + 1
(q 12 + q− 12 )2 − x2
µ∞.
By the Plancherel formula we have,
Lemma 5.1. For all f ∈ H(G′,K) we have
f
(
1 0
0 1
)
=
∫ 2
−2
fˆ(x) µq,
and
f
(
̟m 0
0 1
)
=
1
(1 + q−1)qm
∫ 2
−2
fˆ(x)fˆm(x) µq,
for m > 0.
In particular we note the following corollary.
40

Corollary 5.2. We have
f
(
1 0
0 1
)
+ δf
(
̟ 0
0 1
)
=
1
q + 1
∫ 2
−2
fˆ(x)(1 + q 12 δx+ q) µq,
and
f
(
1 0
0 1
)
− f
(
̟2 0
0 1
)
=
∫ 2
−2
fˆ(x) µ∞.
We define for δ ∈ C a distribution
Λδ : f 7→
∑
n∈Z
δnf
(
̟n 0
0 1
)
on H(G′,K). Then we have the following.
Lemma 5.3. For f ∈ H(G′,K) and δ with |δ| = 1,
Λδ(f) =
∫ 2
−2
fˆ(x) (1− q
−1)
(1 − δq− 12x+ δ2q−1)(1 − δ−1q− 12x+ δ−2q−1)
µ∞.
Proof. This follows by a short calculation using Lemma 5.1, Formula 4 and that,
f
(
̟m
1
)
= f
(
̟−m
1
)
.
Alternatively, one can argue as in [RR05, Section 6] in the case δ = 1.
5.2 The distribution I˜
For α, β ∈ C we let ρ(α, β) denote the unramified representation of GL(2, Fp)
with Satake parameters {α, β}. For x ∈ [−2,+2] we define αx ∈ C to be such
that αx + α−1x = x; of course αx is not well defined, however, all constructions
below will depend only on x, and not on the choice of αx. We then define
µp,E,Ω = L(1/2, ρ(αx, α−1x )Ep ⊗ Ωp)
√
4− x2
2π dx,
where ρ(αx, α−1x )Ep denotes the base change of ρ(αx, α−1x ) to GL(2, Ep). We
note that,
L(1/2, ρ(αx, α−1x )Ep ⊗ Ωp) = 1, when Ωp is ramified and,
L(1/2, ρ(αx, α−1x )Ep ⊗ Ωp) =


(1− xΩ(̟Ep)q
− 12
p +Ω(̟Ep)2q−1p )−1(1− xΩ(̟Ep )q
− 12
p + 1Ω(̟Ep )2 q
−1
p )−1, for p split,
((1 + q−1p )2 − x2q−1p )−1, for p inert,
(1− xΩ(̟Ep)q
− 12
p + q−1p )−1, for p ramified,
41

when Ωp is unramified, where ̟Ep is defined in Section 4. We note that
∫ 2
−2
µp,E,Ω =
{
L(1, ηp), if Ω is unramified at p;
1, otherwise.
We note for future reference the relationship between µp,E,Ω and the spherical
Plancherel measure µp on PGL(2, Fp) is given by
µp,E,Ω =
L(1/2, ρ(αx, α−1x )Ep ⊗ Ωp)
L(1, ρ(αx, α−1x ), Ad)
L(2, 1Fp)µp.
We recall I˜(fp) is defined in Section 4.1.
Lemma 5.4. For all fp ∈ H(G(Fp), ZpR×p ) we have
I˜(fp) =
∫ 2
−2 fˆp(x) µp,E,Ω
L(1, ηp)
,
if Ω is unramified at p and we have
I˜(fp) =
∫ 2
−2
fˆp(x) µp,E,Ω,
if Ω is ramified at p.
Proof. We fix an isomorphismDp ∼−→M(2, Fp) and an embedding Ep →֒M(2, Fp)
such that M(2,OFp)∩E = OF +̟
n(Ωp)
p OEp as in the proof of Lemma 4.3. We
now prove the lemma on a case by case basis.
First we assume that n(Ωp) > 0. Then we have, by Lemmas 4.4, 4.5, 4.6
and Corollary 5.2,
I˜(fp) = fp
(
1 0
0 1
)
− fp
(
̟2p 0
0 1
)
=
∫ 2
−2
fˆp(x)µ∞.
On the other hand in this case we clearly have µp,E,Ω = µ∞.
Next we assume that p is unramified and inert in E and Ω is unramified at
p. Then from Lemmas 4.4 and 5.1 we have
I˜(fp) =
∫ 2
−2
fˆp(x)µqp =
∫ 2
−2
fˆp(x)µp,E,Ω.
Next we assume that p splits in E and Ω is unramified above p. We write
Ωp = (χ, χ−1). Then from Lemmas 4.6 and 5.3 we have
I˜(fp) =
∑
m∈Z
χ(̟mp )fp
(
̟mp 0
0 1
)
=
∫ 2
−2
fˆp(x)µp,E,Ω.
Finally we assume that p is ramified in E and Ω is unramified at p. Then
from Lemma 4.5 and Corollary 5.2 we have
I˜(fp) = f
(
1 0
0 1
)
+Ω(τ)f
(
̟p 0
0 1
)
=
∫ 2
−2
fˆp(x)µp,E,Ω.
42

6 Main results
We now combine the calculations of Sections 3 and 4 to obtain the main results
of this paper.
6.1 Average L-values
By Propositions 3.9, 4.20 and Lemma 4.21 we see that we have an exact formula
for ∑
π∈F(N,2k)
L(1/2, πE ⊗ Ω)
L(1, π, Ad) fˆp(πp),
in terms of orbital integrals. We now write down the formula precisely under
certain further assumptions for which we have computed all the necessary orbital
integrals.
Combining Proposition 3.9, Corollary 4.22 and Lemma 5.4 we get the fol-
lowing.
Theorem 6.1. Let E be a CM extension of a totally real number field F . Let
N ⊂ OF be an ideal such that each prime dividing N is unramified and inert
in E and that the number of primes dividing N has the same parity as [F : Q].
Let Ω : A×FE×\A×E → C× be a character which is unramified outside of N
and such that for v ∈ Σ∞ the weight mv of Ωv is strictly less than kv. Let
fp ∈ H(G(Fp), ZpR×p ). Then for |N| ≥ dE/F (c(Ω)|I(fp)|)hF ,
2[F :Q]
|N|
(
2k− 2
k+ m− 1
) ∑
π∈F(N,2k)
L(1/2, πE ⊗ Ω)
L(1, π, Ad) fˆp(πp),
is equal to
4|∆F |
3
2LS(Ω)∪{p}(1, η)
(
1 + δ(Ω2)δ(N)
∏
v∈Σ∞
(−1)kv−1
)∫ 2
−2
fˆp(x)µp,E,Ω
− 4C(k,Ω, fp)LS(Ω)(1, η)2
√
c(Ω)|∆E |
L(2, 1F )
√
|∆F |
∏
v|N
1
qv − 1
∏
v∈Σ∞
2π
2kv − 1
,
where I(fp) is defined before Lemma 4.21, C(k,Ω, fp) is defined in Lemma 3.6
and the measure µp,E,Ω is defined in Section 5.2.
We note in particular that when fp is the identity in H(G(Fp), ZpR×p ) the
second term in the theorem above is equal to
4|∆F |
3
2LS(Ω)(1, η)
(
1 + δ(Ω2)δ(N)
∏
v∈Σ∞
(−1)kv−1
)
.
We recall that by the Ramanujan conjecture ap(π) ∈ [−2, 2] for all π ∈
F(N, 2k); see [Bla06] for the most general version. The distribution of the
43

Corollary 6.4. Let F = Q and J ⊂ [−2,+2]. Then
lim
N→∞
1
|F(N, 2k)|
∑
π∈F(N,2k)
ap(π)∈J
Lp(1/2, πE ⊗ Ω)
Lp(1, π, Ad)
is equal to
24
2k − 1
(
2k − 2
k +m− 1
)−1
LS(Ω)∪{p}(1, η)L(2, 1Qp)µp(J)
where the limit is taken over squarefree N such that ϕ(N) ∼ N and each prime
dividing N is inert and unramified in E and does not divide c(Ω).
In principal, one could use the formulas developed in this paper to study the
average computed by summing over the space of cusp forms and normalizing by
the dimension of this space. Averaging in this way may remove the condition
that ϕ(N) ∼ N .
Finally, when Ω is unramified we have the following exact formula for all
levels by combining Propositions 3.9 and 4.23.
Theorem 6.5. Let E be a CM extension of a totally real number field F . Let
N ⊂ OF be an ideal such that each prime dividing N is unramified and inert in
E and such that the number of primes dividing N has the same parity as [F : Q].
Let Ω : A×FE×\A×E → C× be a character which is everywhere unramified and
such that for v ∈ Σ∞ the weight mv of Ωv is strictly less than kv. Assume
furthermore that E/F is unramified at the even places of F . Let d ∈ OF be a
generator of dE/F . Then, with C(k,Ω,1p) defined as in Lemma 3.6,x
1√
c(Ω)dE/F |N||∆F |2
(
2k− 2
k+ m− 1
) ∑
π∈F(N,2k)
L(1/2, πE ⊗ Ω)
L(1, π, Ad)
+ C(k,Ω,1p)
4L(1, η)2
|∆F |
3
2L(2, 1F )
∏
v|N
1
qv − 1
∏
v∈Σ∞
π
2kv − 1
,
is equal to the sum of
4L(1, η)
√
|∆F |
2[F :Q]
√
c(Ω)
√
|∆E |
(
1 + δ(Ω2)δ(N)
∏
v∈Σ∞
(−1)kv−1
)
and
2
|∆F |
|∆E |
∑
n

|RE(nN−1)|σ(dE/F , (n+ d))
∑
a∈RE((n+d))
Ω(D−1E/F a)×
∏
v∈Σ∞
Pkv ,mv
(
ιv
( n
n+ d
))
 ,
with the outer sum taken over the finite set of n ∈ N such that
1. ηv
(
1 + dn
)
= 1 for all v | dE/F , and
2. ιv(n) lies between −ιv(d) and 0 for all v ∈ Σ∞.
45

6.2 Subconvexity
We now apply our calculations of the relative trace formula to the problem of
subconvexity.
Let π1 and π2 be cuspidal automorphic representations of GL(2,AF ). The
convexity bound for Lfin(1/2, π1 × π2) is that for ǫ > 0,
Lfin(1/2, π1 × π2) ≪ǫ C(π1 × π2)
1
4+ǫ,
where C(π1 × π2) = Cfin(π1 × π2)C∞(π1 × π2) is the analytic conductor of
π1 × π2; see [IS00a, Section 2.A]. Cfin(π1 × π2) is the conductor of π1 × π2
and C∞(π1 × π2) depends only on the infinity types of π1 and π2; we refer to
[IS00a, Section 2.A] for the precise definition. We note that when π1 and π2
have disjoint ramification Cfin(π1 × π2) = (Cfin(π1)Cfin(π2))2.
The problem of beating the convex bound, with π2 fixed, has been the study
of many authors. When F = Q and π1 and π2 have trivial central character the
convexity bound was beaten by Kowalski, Michel, and VanderKam [KMV02],
with the central character condition being relaxed by Michel and Harcos [Mic04],
[HM06]. In the case that Ω is trivial, so that the L-function factors as L(s, πE) =
L(s, π)L(s, π ⊗ η), the convexity bound was beaten by Duke, Friedlander and
Iwaniec [DFI94] in the level aspect over Q with π fixed and η varying. For
number fields other than Q the first subconvex result was obtained by Cogdell,
Piatetski-Shapiro and Sarnak [Cog03] in the case of a fixed Hilbert modular form
twisted by a ray class character. Further extensions of these subconvexity results
to cusp forms on arbitrary number fields have been obtained by Venkatesh
[Ven05].
We now continue with the usual assumptions on π ∈ F(N, 2k) and Ω as in
Section 2. We denote by σΩ the induction of Ω to an automorphic representation
of GL(2,AF ). Hence,
L(s, πE ⊗ Ω) = L(s, π × σΩ).
We have Cfin(π) = |N| and, by the formula for the conductor of an induced
representation (see for example [Sch02, Section 1.2]), Cfin(σΩ) = dE/F c(Ω).
Thus, Cfin(π × σΩ) = (|N|dE/F c(Ω))2, and the convexity bound in the level
aspect is given by
Lfin(1/2, π × σΩ) ≪k ,ǫ (|N|dE/F c(Ω))
1
2+ǫ.
We now proceed to apply our work to the problem of beating convexity for
these L-functions. By combining our calculations of the relative trace formula,
together with the bounds on the orbital integral integrals from Section 4.2.2,
we will get an estimate for Lfin(1/2, π× σΩ) (Theorem 6.8 below) which beats
the convexity bound as π and Ω vary in a hybrid range.
The relative trace formula provides an expression for the first moment of
L(1/2, π × σΩ) averaged over π ∈ F(N, 2k). The size of this family is approxi-
mately |N| while the conductor of the L-function is (|N|dE/F c(Ω))2, this allows
46

us to obtain estimates which beat convexity when |N| is of size around
√
c(Ω).
We note that over Q, Michel [Mic04] and Harcos-Michel [HM06] in their work
on subconvexity for L(s, π1 × π2) average over the family of modular forms of
level [Cfin(π1), Cfin(π2)] and bound the second moment of the L-function. By
shortening the family we are able to get away with an estimate for only the first
moment.
Recall that Ω is a unitary character on the idele class group of E that is
trivial when restricted to A×F , unramified above N and has weight |mv| strictly
less than kv for every infinite place v of F . We let S(Ω) = {p1, p2, ..., pm} denote
the places of F above which Ω is ramified, c(Ω) the norm of the conductor of
Ω in F and c(Ω) the absolute norm of c(Ω). Let m = |S(Ω)|, c(Ω) = ∏mi=1 p2nii
and NF/Qpi = qi.
In the remainder of this section we fix E and allow N and Ω to vary. We
recall that D ramifies precisely at the infinite places of F and the places dividing
N. Therefore as N varies, D varies and hence the image of E in D depends on
N. However E as a field extension of F does not depend on N. For the bounds
we prove in this section, the ≪E notation refers to constants that only depend
on E as a field, such at
√
|∆E |, and thus have no hidden dependence on N.
We begin with a necessary technical lemma which we will require in the
bounding of the geometric expansion of I(f). For a ∈ F and integers ri, ti ≥ 0
for 1 ≤ i ≤ m let
S(ri),(ti)(a) = {y ∈ N : vpi(y) = ri, vpi(y−a) = ti, |ιv(y)| < |ιv(a)|, i = 1, ...,m, v ∈ Σ∞}.
Lemma 6.6. S(ri),(ti)(a) is empty unless for each i = 1, ...,m,
1. ri < vpi(a) and ti = ri or
2. ri > vpi(a) and ti = vpi(a) or
3. ri = vpi(a) and ti ≥ vpi(a).
In addition,
|S(ri),(ti)(a)| ≤
c(F )|NF/Q(a)|
|NF/Q(Npmax{r1,t1}1 ...p
max{rm,tm}
m )|
where c(F ) is a constant that only depends on F .
Proof. It is clear that S(ri),(ti)(a) is empty unless conditions 1, 2 or 3 hold for
each i. We now proceed to prove the bound following the ideas of the proof in
[Lan94, §V.1, Theorem 0].
Since S(ri),(ti)(a) is finite there exists an integer n such that
n < |S(ri),(ti)(a)| ≤ n+ 1.
Pick a v0 ∈ Σ∞. Identify Fv0 with the real line. By the condition |ιv0 (y)| <
|ιv0(a)|, S(ri),(ti)(a) is contained in the interval centered at the origin of length
2|ιv0(a)|. By the Pigeon Hole Principle if we divide this interval into n equal
47

Proof. Let A1, ...,AhF be a fixed set of representatives of the ideal classes of F .
Then for any ideal I of OF there exists an i such that AiI is principal. Let
a ∈ F be such that
aOF = Aic(Ω)dE/F
for some i. Then
∣∣∣∣
NF/Q(a)
c(Ω)dE/F
∣∣∣∣ ≤ c(F ) = max{|Ai| : 1 ≤ i ≤ hF }. (5)
For ξ ∈ S(Ω,N, 1p) we have,
ξ = ξy :=
y
y − a ,
for some y ∈ N such that |ιv(y)| < |ιv(a)| for all v ∈ Σ∞. Hence we can partition
these y into the sets S(ri),(ti)(a). Thus
Ireg(f) =
∑
r1≥0
∑
r2≥0
...
∑
rm≥0
∑
t1≥0
∑
t2≥0
...
∑
tm≥0
∑
y∈S(ri),(ti)(a)
I(ξy , fS(Ω))
m∏
i=1
I(ξy , fpi).
(6)
First we consider the integrals I(ξy , fS(Ω)). For an ideal a ⊂ OF we define
RS(Ω)E (a) = {b ⊂ OE : NEv/Fv (bOEv) = aOFv for v /∈ S(Ω) and pv ∤ b for v ∈ S(Ω)}.
By the proof of Proposition 4.23 we see that
|I(ξy, fS(Ω))| ≤ C(E)|RS(Ω)E (yN−1)||R
S(Ω)
E ((y − a))||I(ξ, fΣ∞)|, (7)
where C(E) is a constant depending only on E. Furthermore, from Lemma
4.14, it is clear that
|I(ξy, fΣ∞)| ≤ C(k ), (8)
where C(k ) is a constant depending only on k .
Let σ0(x) denote the number of divisors of x. For any ideal a ∈ OF ,
|RS(Ω)E (a)| ≤ σ0(NF/Q(a))[F :Q] ≪ǫ |NF/Q(a)|ǫ. (9)
For any y ∈ S(ri),(ti)(a), |ιv(y)| < |ιv(a)| and |ιv(y−a)| < |ιv(a)| for all v ∈ Σ∞.
Thus |NF/Q(y)| =
∏
v∈Σ∞ |ιv(y)| ≤ |NF/Q(a)| and similarly |NF/Q(y − a)| ≤
|NF/Q(a)|.
By (7), (8) and (9),
|I(ξy , fS(Ω))| ≪F,E,k ,ǫ |NF/Q(a)|ǫ. (10)
49

Combining (6) and (10) we have,
|Ireg(f)| ≪F,E,k ,ǫ |NF/Qa|ǫ
∑
r1≥0
∑
r2≥0
...
∑
rm≥0
∑
t1≥0
∑
t2≥0
...
∑
tm≥0
∑
y∈S(ri),(ti)(a)
m∏
i=1
|I(ξy, fpi)|.
(11)
To bound |I(ξy , fpi)| we first note that
vpi(1− ξ) = vpi(a)− vpi(y − a).
Thus by Lemma 4.19, for y ∈ S(ri),(ti)(a),
|I(ξy , fpi)| ≤



C(Ev , Fv)q−nii L(1, ηv)2q
ri−vpi (a)
2
i 0 ≤ ri < vpi(a)
C(Ev , Fv)q−nii L(1, ηv)(1 + ti − ri) ri = vpi(a)
C(Ev , Fv)q−nii L(1, ηv)(1 + ri − vpi(a)) ri > vpi(a)
(12)
where C(Ev, Fv) is the constant in the statement of Lemma 4.19. In particular,
C(Ev, Fv) = 1 if v is unramified in E.
Because the bounds in Lemma 6.6 and (12) depend only on ri and ti for
each i,
|Ireg(f)| ≪F,E,k,ǫ |NF/Q(a)|ǫ
|NF/Q(a)|
|NF/QN|
m∏
i=1
∑
ri≥0
∑
ti≥0
q−max{ri,ti}i |I(ξyri,ti , fpi)|,
(13)
where we choose yri,ti to be any element in Fpi such that vpi(yri,ti) = ri and
vpi(yri,ti − a) = ti.
Again by Lemma 6.6 and (12), for i fixed
∑
ri≥0
∑
ti≥0
q−max{ri,ti}i |I(ξyri,ti , fpi)|
≤
∑
vpi (a)>ri≥0
C(Ev , Fv)q−nii L(1, ηv)2q
ri−vpi (a)
2
i q−rii
+
∑
ti≥vpi (a)
C(Ev, Fv)q−nii L(1, ηv)(1 + ti − vpi(a))q−tii +
∑
ri>vpi (a)
C(Ev, Fv)q−nii L(1, ηv)(1 + ri − vpi(a))q−rii
≤
∑
ri≥0
C(Ev, Fv)q−nii L(1, ηv)2q
ri−vpi (a)
2
i q−rii + 2
∑
ℓ≥vpi(a)
C(Ev, Fv)q−nii L(1, ηv)(1 + ℓ− vpi(a))q−ℓi
≤ C(Ev, Fv)L(1, ηv)q−ni−
vpi (a)
2
i

 L(1, ηv)
1− q−1/2i
+
2q
−vpi (a)
2
i
(1− q−1i )2


≤ 25q−ni−
vpi (a)
2
i C(Ev , Fv)
≤ 25q−2nii C(Ev , Fv).
50

Finally we can bound the product of these terms over 1 ≤ i ≤ m. We note
that
2|S(Ω)| ≤ σ0(c(Ω))[F :Q] ≪ǫ c(Ω)ǫ.
Also, because C(Ev , Fv) = 1 unless v ramifies in E,
m∏
i=1
C(Ev, Fv) ≪E,F 1.
By these facts,
m∏
i=1
∑
ri
∑
ti
q−max{ri,ti}i |I(ξy , fpi)| ≪ǫ,F,E
c(Ω)ǫ
c(Ω) . (14)
Combining (13) and (14) and applying (5) we conclude
|Ireg(f)| ≪F,E,k ,ǫ
c(Ω)ǫ
|N| .
Finally we combine the spectral expansion for I(f), Proposition 3.9, together
with the calculation of Iirreg(f), Proposition 4.20, with the bound on Ireg(f)
established above. In the theorem below the term |N|1+ǫc(Ω)ǫ comes from the
irregular term and |N|ǫc(Ω) 12+ǫ comes from the bounds on the regular orbital
integrals.
Theorem 6.8. Fix a totally real number field F and a CM extension E of F .
Let N be a squarefree ideal in OF such that the number of primes dividing N
has the same parity as [F : Q] and such that each prime of F dividing N is inert
and unramified in E. Let Ω be a character of A×FE×\A×E which is unramified
above N and has weights at the archimedean places strictly less than k. Then
for any ǫ > 0,
Lfin(1/2, π × σΩ) ≪F,E,k,ǫ |N|1+ǫc(Ω)ǫ + |N|ǫc(Ω)
1
2+ǫ,
for all π ∈ F(N, 2k).
Proof. We note that with the weight k fixed there are only finitely many possi-
bilities for the archimedean type of Ω, hence it suffices to prove the same bound
for the completed L-function.
We take fp = 1p then from the spectral expansion for I(f) we have, by
Proposition 3.9,
I(f) ≥ LS(Ω)(1, η)
2
2|∆F |2
√
dE/F c(Ω)
4[F :Q]
|N|
(
2k − 2
k + m − 1
) ∑
π∈F(N,2k)
L(1/2, πE ⊗ Ω)
L(1, π, Ad) .
(15)
51

On the geometric side, we recall that we have written
I(f) = Iirreg(f) + Ireg(f) (16)
in Section 4.3. By Proposition 4.20 we have,
|Iirreg(f)| ≪F,E,ǫ
c(Ω)ǫ√
c(Ω)
, (17)
and by Lemma 6.7 we have, for all ǫ > 0,
|Ireg(f)| ≪F,E,k ,ǫ
c(Ω)ǫ
|N| . (18)
Hence, combining (15), (16), (17) and (18), and noting that
1
LS(Ω)(1, η)
≤ 22|S(Ω)| ≤ σ0(c(Ω))2[F :Q] ≪ǫ c(Ω)ǫ,
we get,
∑
π∈F(N,2k)
L(1/2, πE ⊗ Ω)
L(1, π, Ad) ≪F,E,k,ǫ |N|c(Ω)
ǫ + c(Ω) 12+ǫ. (19)
Now using positivity of L(1/2, πE ⊗ Ω), which is clear from the period for-
mula, we have
L(1/2, πE ⊗ Ω)
L(1, π, Ad) ≤
∑
π∈F(N,2k)
L(1/2, πE ⊗ Ω)
L(1, π, Ad) (20)
for any π ∈ F(N, 2k ). We now use that for all ǫ > 0,
L(1, π, Ad) ≪k ,ǫ |N|ǫ; (21)
cf [IK04, Theorem 5.41]. Hence by (19), (20) and (21) we have, for all ǫ > 0,
L(1/2, πE ⊗ Ω) ≪k ,ǫ |N|ǫ
L(1/2, πE ⊗ Ω)
L(1, π, Ad) ≪F,E,k ,ǫ |N|
1+ǫc(Ω)ǫ + |N|ǫc(Ω) 12+ǫ.
Finally we explicate how this Theorem gives a subconvex bound as N and
Ω vary in certain ranges.
Corollary 6.9. For 0 ≤ t < 16 and ǫ > 0,
Lfin(1/2, π × σΩ) ≪F,E,k,ǫ (c(Ω)|N|)
1
2−t,
for π ∈ F(N, 2k); with N and Ω satisfying the conditions of Theorem 6.8 and
c(Ω)
2t+ǫ
1−(2t+ǫ) ≤ |N| ≤ c(Ω)
1−(2t+ǫ)
1+2t+ǫ .
52

Proof. To beat convexity we need,
|N|1+ǫc(Ω)ǫ ≤ |N| 12−tc(Ω) 12−t
and
|N|ǫc(Ω) 12+ǫ ≤ |N| 12−tc(Ω) 12−t.
Thus we need
c(Ω) 2ǫ+2t1−2t−2ǫ ≤ |N| ≤ c(Ω) 1−2t−2ǫ1+2t+2ǫ .
We note that the range for |N| is non-empty provided 16 > t ≥ 0.
6.3 Classical reformulation
To finish we work out Theorem 6.1 in the case F = Q classically. We fix an
imaginary quadratic field E = Q(
√
−d) of discriminant −d and let χ−d =
(−d
·
)
denote the associated quadratic Dirichlet character.
Let N be a squarefree integer which is the product of an odd number of
primes p satisfying χ−d(p) = −1. For a positive integer k we denote by F(N, 2k)
the finite set of normalized newforms of level N , weight 2k, trivial nebentypus
and which are eigenforms for all the Hecke operators. On F(N, 2k) we take the
Petersson inner product defined by
(f, f) =
∫
Γ0(N)\H
|f(x+ iy)|2 y2k dx dyy2 .
We note that,
L(1, πf , Ad) =
22k
N (f, f)
where πf denotes the automorphic representation of GL(2,AQ) generated by
f . We denote by L(s, f) the completed L-function of f , which has a functional
equation relating the value at s to 2k − s.
We now fix a character Ω : E×\A×E → C whose restriction to A×Q is trivial.
At infinity we have
Ω∞ : z 7→
(z
z
)m
,
with m ∈ Z. We recall that when Ω does not factor through the norm map
N : A×E → A×Q there is a modular form gΩ of level dc(Ω), weight 2|m|+ 1 and
nebentypus χ−d such that
L(s, gΩ) = L(s,Ω).
For f ∈ F(N, 2k) we let L(s, f × gΩ) denote the completed Rankin-Selberg L-
function which satisfies a functional equation relating the value at s to 2k +
2|m|+ 1− s.
We recall the well known facts that for the completed L-functions, L(2, 1Q) =
π
6 and
L(1, χ−d) =
h−d
u−d
√
d
,
53

where h−d denotes the class number of E and u−d = #O×E/{±1}.
Taking into account that the gamma factor for L(k, f) is 2(2π)−kΓ(k) and
the gamma factor for L(k + |m| + 12 , f × gΩ) is 4(2π)−2kΓ(k + m)Γ(k − m),
we now apply Theorem 6.1 to get the following following averages for the finite
parts of the L-functions.
Theorem 6.10. Let N be a squarefree integer as above with N > d. Then we
have
u−d
√
d
8π2
∑
f∈F(N,2)
Lfin(1, f)Lfin(1, f ⊗ χ−d)
(f, f) = h−d
(
1− 12h−du−dϕ(N)
)
,
where ϕ denotes the Euler totient function. When k > 1 we have
(2k − 2)!u−d
√
d
2π(4π)2k−1
∑
f∈F(N,2k)
Lfin(k, f)Lfin(k, f ⊗ χ−d)
(f, f) = h−d.
For a character Ω as above which does not factor through the norm we get for
N > dc(Ω),
(2k − 2)!u−d
√
dLS(Ω)(1, χ−d)
2π(4π)2k−1
∑
f∈F(N,2k)
Lfin(k + |m|+ 12 , f × gΩ)
(f, f) = h−d.
We note that when the level is prime the first part of this Theorem agrees
with Duke’s asymptotic result [Duk95, Proposition 2] and the first and second
parts agree with Michel and Ramakrishnan’s exact formula [MR]. One can see
[MR, p.5] for explicit examples verifying that this formula agrees with known
data.
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