The Second Moment of Rankin-Selberg L-Functions, Hybrid Subconvexity Bounds, and Related Topics

The Second Moment of Rankin-Selberg L-functions, Hybrid Subconvexity Bounds, and Related Topics

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Zhilin Ye, B.S.

Graduate Program in Department of Mathematics

The Ohio State University

2014

Dissertation Committee:

Roman Holowinsky, Advisor James Cogdell Wenzhi Luo Mark Berliner c Copyright by

Zhilin Ye

2014 Abstract

In this thesis, we study three problems related to subconvexity bounds of Rankin-

Selberg L-functions. Let M, N be two coprime square-free integers. Let f be either a holomorphic or a Maaß cusp form of level N. Using a large sieve inequality, we establish

P ( j) 2 a bound for an unampliﬁed 2nd moment average, such as g L (1/2 + it, f ⊗ g) t, j, M + M2/3−βN4/3 (MN) where β ≈ 1/500 and g ranges over an orthonormal basis of holomorphic cusp forms of level M and a ﬁxed weight for any j. As a consequence, we obtain a subconvexity bound for L( j) (1/2 + it, f ⊗ g) for any N < M satisfying the conditions above. Moreover, by symmetry, we establish a level aspect hybrid subconvexity bound for any coprime square-free M and N when both forms are holomorphic. We also establish a

k/2 1/2 −1/6+ sup-norm bound for holomorphic forms such that ky f (z)k∞ k N k f k2. Further- more, we establish an equidistribution property of mod c reciprocals which is natural to the subconvexity problem and second moments of Rankin-Selberg L-functions.

ii This is dedicated to the one I love.

iii Acknowledgments

I would like to express my special appreciation and thanks to my advisor Professor

Roman Holowinsky for his tremendous mentoring, encouraging, and priceless advice on my research. I would also like to thank my committee members, Professor James Cogdell,

Professor Wenzhi Luo, and Professor Mark Berliner for serving as my committee members even at hardship. I also want to thank my family, my friends and all of people who have been there to support me during my life as a Ph.D. candidate.

iv Vita

2006-2010 ...... B.S., Mathematics, Zhejiang University, Hangzhou, Zhejiang, P.R. China. 2010-2014 ...... Ph.D., Mathematics, The Ohio State University, Columbus, OH, USA.

Fields of Study

Major Field: Mathematics

Studies in Analytic Number Theory: Roman Holowinsky

v Table of Contents

Page

Abstract...... ii

Dedication...... iii

Acknowledgments...... iv

Vita...... v

1. Introduction...... 1

1.1 L-functions, their properties, and the subconvexity problem...... 1 1.2 Known subconvexity bounds and their applications...... 12 1.2.1 The Burgess’s bound and subconvexity bound for GL1 ...... 12 1.2.2 The subconvexity bound for GL2 case...... 18 1.2.3 The QUE conjecture and the subconvexity problem of higher rank 20

2. Preliminaries and Past Works...... 27

2.1 Automorphic forms and the normalization...... 27 2.2 Rankin-Selberg convolution and L-functions...... 31 2.3 Properties of Bessel functions...... 34 2.4 The Voronoi formula, trace formulae and large sieve inequalities..... 40 2.5 Jutila’s version of circle method...... 47 2.6 Past works on hybrid subconvexity bounds for Rankin-Selberg L-functions 48 2.7 Past works on Sup-norm bounds for modular forms...... 51 2.8 Past works on the distribution of inverses mod c ...... 54

3. Main Results...... 58

3.1 Hybrid subconvexity bounds for Rankin-Selberg L-functions...... 58 3.2 Sup-norm bounds for holomorphic modular forms...... 60 3.3 The distribution of inverses mod c ...... 61

vi 4. The Proof of Hybrid Subconvexity Bounds for Rankin-Selberg L-functions.. 63

4.1 The deduction of Theorem 3.1.1 and Corollary 3.1.2 from Proposition 3.1.1...... 63 4.2 A sketch proof of Propositon 3.1.1...... 66 4.3 The proof of Proposition 3.1.1...... 70 4.3.1 Step 1. Reducing to sums of Kloosterman sums via trace formula...... 71 4.3.2 Step 2. Removing large and small values of D...... 71 4.3.3 Step 3. Applying the Voronoi formula to convert Kloosterman Sums to Ramanujan sums...... 73 4.3.4 Step 4. Treating the zero shift...... 75 4.3.5 The sum of shifted sums...... 75 4.3.6 Step 5. Applying the circle method...... 76 4.3.7 Step 6. Applying Vornoi formula twice to regenerate Klooster- mann sums...... 78 4.3.8 Step 7. Applying the large sieve type inequality to the sum of Kloostermann sums...... 80 4.3.9 Conclusion and the ﬁnal bound...... 82 4.4 The estimation of weight functions...... 84 4.5 The proof of Theorem 3.1.3...... 91

5. The Proof of Sup-norm Bounds for Holomorphic Modular Forms...... 96

5.1 Preliminary...... 96 5.1.1 The Sup-norm via Fourier expansion...... 96 5.1.2 Pretrace formula for holomorphic cusp forms...... 96 5.1.3 Ampliﬁcation method...... 97 5.1.4 Counting lattice points...... 100 5.1.5 The estimation of parabolic matrices...... 104 5.2 The proof of Theorem 3.2.1...... 107

6. The Proof of the Distribution of Inverses mod c ...... 110

6.1 Wilton’s bound for Eisenstein series...... 110 6.2 The proof of Theorem 3.3.1...... 119

Bibliography...... 122

vii Chapter 1: Introduction

1.1 L-functions, their properties, and the subconvexity problem

The study of L-functions plays a key role in number theory. The most famous L-

function is the Riemman Zeta function, which is deﬁned as

X∞ 1 ζ(s):= ns n=1 when Re (s) > 1. It has a meromorphic continuation to any complex number s. Fur-

thermore, by the unique factorization of the integers, we have the Euler product when

Re (s) > 1 Y ζ(s) = 1 − p−s−1 , p primes which relates ζ(s) to prime numbers.

The locations of poles and zeros of the Riemann Zeta function carry highly nontrivial

information about the distribution of prime numbers. For example, we can prove that there

are inﬁnitely many prime numbers by showing that ζ(s) has a pole at s = 1. A deeper result

is the Prime Number Theorem

Theorem 1.1.1 (Prime Number Theorem). Let π(x) denote the number of primes p 6 x. As

x → ∞, we have x π(x) ∼ . log x

1 This theorem is equivalent to the fact that ζ(s) , 0 when Re (s) = 1. For a complete proof, see [IK04] Chapter 2.

These examples illustrate why the study of the locations of zeros of ζ(s) has attracted number theorists for over 100 years. In 1859, B. Riemann made a conjecture about the locations of these zeros. His conjecture, which is now called the Riemann Hypothesis, claims that all the solutions of the equation ζ(s) = 0 such that 0 < Re (s) < 1 lie on the vertical

Re 1 line (s) = 2 . This conjecture is still open.

In order to study prime numbers in arithmetic progressions, L. Dirichlet (1837) invented

multiplicative characters χ(mod q) and associated a Dirichlet L-functions to each of them.

The explicit deﬁnition is as follows.

Denote by Z the set of all integers. Denote by C× the set of all nonzero complex

numbers. Let q > 1 be a positive integer. Denote by (Z/qZ)× the modulo q multiplicative

× × group. Let χq : (Z/qZ) → C be a character. A Dirichlet L-function is deﬁned as

∞ X χq(n) L(s, χ ) = , q ns n=1 when Re (s) > 0. It also has a meromorphic continuation to any complex number s and can

be rewritten as the Euler product

− Y −s 1 L(s, χq) = 1 − χq(p)p . p primes Just like the the Riemann Zeta function, the zeros of Dirichlet L-functions carry information about prime numbers. For example, we can prove that there are inﬁnitely many primes p such that p ≡ 1(mod 3) by showing that L(s, χ3) , 0 when Re (s) = 1. For a complete proof, see [IK04] Chapter 2.

2 The Generalized Riemann Hypothesis suggests that all the zeros of L(s, χq) in the strip

Re Re 1 0 < (s) < 1 lie on the vertical line (s) = 2 for any Dirichilet character χq modulo q.

There are other arithmetic objects with L-functions associated to them. For example,

2 let SL2(R) be the group of 2-by-2 matrices with determinant 1. Let H be the upper-half ! a b plane of C. Any γ = ∈ SL (R) acts on H2 via Mobius¨ action c d 2

az + b γ.z = . cz + d

Let M > 0 be an integer and κ > 0 be an even integer. Let ( ! ) a b Γ (M):= ∈ SL (Z): c ≡ 0(mod M) 0 c d 2

be a congruence subgroup of SL2(Z).

The holomorphic cusp forms with weight κ, level M, and trivial nebentypus are holomorphic functions on the upper half-plane F : H2 → C satisfying

F(γ.z) = (cz + d)κF(z),

when ! a b γ = ∈ Γ (M), c d 0 and vanishing at every cusp, e.g. F(a.z) → 0 as Im (z) → ∞ for any a ∈ SL2(Z).

κ Let f (z) = y 2 F(z) where F is a holomorphic cusp form with weight κ, level M and trivial nebentypus. We call such f (z) a Hecke newform, if it is not generated by cusp

3 forms with lower levels (see [IK04] Chapter 14, [Cas73] for more details) and it is the

eigenfunction of Hecke operators Tm for all m > 0, where ! 1 X X az + b T f (z):= √ f . m d m ad=m 06b

Denote by λ f (m) the corresponding eigenvalues.

For a Hecke newform f , we can deﬁne the L-function that associates to it as below:

X λ f (n) L(s, f ):= . ns n

Similar to the Riemann Zeta function, we also have the Euler product (but now of degree

2 )

− − Y −s 1 −s 1 L(s, f ):= 1 − α f,1(p)p 1 − α f,2(p)p , p primes

for some complex numbers α f,i(·).

For two such forms f and g, we can deﬁne the Rankin-Selberg convolution as

− Y Y −s 1 L(s, f × g):= 1 − α f,i(p)αg, j(p)p . p primes i, j

This deﬁnition will be recalled in Chapter2.

The above deﬁnitions of L-functions can be generalized to any GLn automorphic representation (In this thesis, we focus on L-functions associated with representations on GL2.

However, we give a short introduction of L-functions related to GLn automorphic representations here ).

4 We ﬁrst introduce the GL1 case. Denote by Q the set of all the rational numbers. Denote

n by Zp the p-adic integers, which is the inverse limit of the rings Z/p Z:

n Zp = lim Z/p Z. ←−

Denote by Qp the fractional ﬁeld of Zp. Denote by AQ the adele ring of Q, which can be deﬁned as the restricted product

Y0 AQ = R × Qp p of all the p-adic completions Qp and the real numbers. In this case the restricted product means that for an adele (a∞, a2, a3, a5,... ), all but a ﬁnite number of ap’s are p-adic in-

× tegers. Let GL1(AQ) = AQ be the idele ring, which is the collection of all the invertible

× elements of AQ. There is a natural way to embed Q , the group of invertible elements in Q,

× × × into AQ via the map a ∈ Q to (a, a, a, a,... ) ∈ AQ.

Next, we relate our classical characters to automorphic representations on GL1. For

× × any character χq : (Z/qZ) → C , there exists an adelic lift to a unique adelic char-

× × × acter χ : AQ → C , which is invariant under multiplication by Q . Therefore, a clas-

× sical character χq can be considered as a C -valued multiplicative function deﬁned on

× × GL1 (Q) \GL1 AQ = Q \AQ. In terms of representation theory, these χs are irreducible automorphic representations of GL1 (Q) \GL1 AQ .

Then, we introduce automorphic representations on GLn, where GLn is the algebraic n-dimensional generalized linear group. An automorphic irreducible representation (π, V) can be realized as a set of certain functions deﬁned on GLn(Q)\GLn(AQ) equipped with right translation by GLn(AQ). We call such representations “global” for short, since it is

5 deﬁned over AQ. For the deﬁnition of automorphic representations, see [Cog04].

We then deﬁne the conductor of a representation. For each global representation π, we

can decompose it into a product of local representations as π = ⊗vπv, where v goes over

all the local places ∞, 2, 3, 5,... . Each πv is a local representation of GLn (Qv), equipped

with a vector space Vv. For a ﬁnite local place v = p, we can deﬁne a set of open compact

subgroup

(m) n m o Kp = g ∈ GLn Zp : eng ≡ (0, 0,..., 0, 1) (mod p ) ,

where en = (0, 0,..., 0, 1) is a 1 × n matrix ( Thus eng is simply the bottom row of the

(0) matrix g ). Kp = GLn Zp is the maximal open compact subgroups of GLn(Qp).

(m) Now, we denote by c(πp) the smallest number m such that there is a Kp -ﬁxed vector in

n (m)o the representation (πp, Vp), e.g. the set w ∈ Vp : π(k)w = w ∀k ∈ Kp is not trivial. And we deﬁne the local conductor Q(πp) as

c(πp) Q(πp) = p .

For an irreducible automorphic representation π, except at ﬁnitely many places, we have

Q(πv) = 1. Therefore, for such a global representation π, we can always deﬁne the global

conductor Q(π) as the product of all the local conductors, e.g.

Y Q(π) = Q(πv). v,∞

See [Cog04] for more details.

We are ready to give the general form of L-functions associated with GLn automorphic cusp forms. Due to the work of Ilya Piatetski-Shapiro, H. Jacquet, J. Shalika, R. Langlands

6 and many others, one can associate a well deﬁned L-function L(s, πcusp) to an automorphic

cuspidal representation πcusp such that

Y Y −s −1 L(s, πcusp):= Lv(s, πcusp,v) Pv(v )

v-Q(πcusp) v|Q(πcusp) n Y Y −s −1 Y −s −1 = (1 − απcusp,i(v)v ) Pv(v ) , v-Q(πcusp) i=1 v|Q(πcusp)

for some polynomials Pv, where απcusp,i(v) are local parameters. Moreover, the product of

−s −1 these Pv(v ) can be further decomposed as a product of local factors as

n Y Y −s −1 (1 − απcusp,i(v)v ) , v|Q(πcusp) i=1

for some complex numbers απcusp,i(v) (not necessarily nonzero ). So these L-functions still have Euler products.

Furthermore, for two automorphic representations π1 and π2, we can also deﬁne the

Rankin-Selberg L-function as

Y Y −s −1 L(s, π1 × π2):= Lv(s, π1,v × π2,v) Hv(v ) ,

v-Q(π1)Q(π2) v|Q(π1)Q(π2)

Qn −s −1 for some polynomials Hv, where Lv(s, π1,v × π2,v):= i, j=1(1 − απ1,i(v)απ2, j(v)v ) are the local factors. See [Cog04] for more details.

The Generalized Riemann Hypothesis still suggests that all the solutions of L(s, πcusp) =

0 in 0 < Re (s) < 1 lie on Re (s) = 1/2.

The L-functions we consider in this thesis will satisfy the following properties as listed in [IK04]:

7 (1) An L-function will be a Dirichlet series with Euler product of degree d > 1,

X −s Y −s −1 −s −1 L(s, f ) = λ f (n)n = (1 − α f,1(p)p ) ··· (1 − α f,d(p)p ) n>1 p

with λ f (1) = 1, λ f (n) ∈ C, α f,i(p) ∈ C. Moreover, the series and Euler product are

absolutely convergent when Re (s) > 1. The product

−s −1 −s −1 Lp(s, f ):= (1 − α f,1(p)p ) ··· (1 − α f,d(p)p )

is called the local product, and αi(p), 1 6 i 6 d are called the local parameters at the place

p. Furthermore, they satisfy

|α f,i(p)| < p for all p.

(2)L(s, f ) will have an associated gamma factor

d Y s + κ j γ(s, f ) = π−ds/2 Γ 2 j=1

where κ j ∈ C is called the local factor of L(s, f ) at inﬁnity. Moreover, Re (κ j) > −1 for

each j. This last condition shows that γ(s, f ) has no zero in C and no pole for Re (s) > 1.

(3) There exists an integer Q( f ) > 1, called the conductor of L(s, f ), such that α f,i(p) ,

0 for p - Q( f ) and 1 6 i 6 d. A prime p - Q( f ) is said to be unramiﬁed. The complete

L-function is deﬁned as

s Λ(s, f ) = Q( f ) 2 γ(s, f )L(s, f ).

Furthermore, it admits an analytic continuation to a meromorphic function for s ∈ C of

order 1 ( i.e. |Λ(s, f )| exp(|s|β) for any β > 1), with at most poles at s = 0 and s = 1.

Moreover, it must satisfy the functional equation

Λ(s, f ) = ( f )Λ(1 − s, f¯),

8 ¯ ¯ where f is an object associated with f (the dual of f ) for which λ f¯(n) = λ¯ f (n), γ(s, f ) =

γ(s, f ), Q( f¯) = Q( f ) and ( f ) is a complex number of absolute value 1, called the “root number” of L(s, f ).

Deﬁnition 1.1.1. In this thesis, we call L(s, f ) a “general” L-function if it satisﬁes all the three conditions above.

In order to introduce the subconvexity problem, we deﬁne the analytic conductor of

L(s, f ) as in [IK04] Chapter 5:

Yd Q(s, f ):= Q( f ) (|s + κi| + 3) . (1.1.1) i=1

The subconvexity problem is directly related to the Lindelof¨ Hypothesis for L-functions, which is a conjecture about the rate of growth of the L-function on the critical line Re (s) =

1/2.

Conjecture 1.1.1 (The Lindelof¨ Hypothesis). For any L(s, f ), we have

L(s, f ) Q(s, f ), for Re (s) = 1/2 with any > 0.

This conjecture is a consequence of the Generalized Riemann Hypothesis, see [IK04]

Corollary 5.20 for a proof.

For the the Riemann Zeta function, the Lindelof¨ Hypothesis is as follows.

9 Conjecture 1.1.2 (The Lindelof¨ Hypothesis for the Riemann Zeta Function).

1 ζ 2 + it t , for any > 0.

The Lindelof¨ Hypothesis for Dirichlet L-functions claims that

1 L 2 + it, χq (qt) , for any > 0.

For a holomorphic cuspidal newform f (see Chapter2 for the deﬁnition of newforms ) with level M and weight k, the Lindelof¨ Hypothesis says that

1 | | L 2 + it, f ((3 + t + k )M) .

The ﬁrst nontrivial upper bound of the central values of ζ(s) is due to E. L. Lindelof¨ 1 1/4+ himself, who proved that ζ 2 + it t in [Lin08] using the functional equation of ζ(s) and the Phragmen-Lindelof¨ convexity theorem. Using similar methods, we can also show

that

1 1 + 4 L 2 + it, χq (qt) ,

and 1 | | 1/4+ L 2 + it, f ((3 + t + k )M) .

10 In fact, for a larger group of general L-functions, we have the following bounds (for the explicit conditions on these L-functions, see [Mol02])

Theorem 1.1.2 (The Convexity Bound). For L(s, f ) satisfying conditions listed in [Mol02], we have

L(s, f ) Q(s, f )1/4+, for Re (s) = 1/2 with any > 0.

Remark 1.1.1. A more accurate convexity bound without the is given in [HB09], in which the author claimed that, as long as we have a suitable bound for Dirichlet series log L(s, f ),

1 1 Q 4 L(1/2, f ) ( 2 , f ) .

The subconvexity problem is to obtain non-trivial power savings in the conductor be-

yond the convexity bound.

Conjecture 1.1.3 (The Subconvexity Problem). For any general L(s, f ), there exists some

δ > 0 such that

L(s, f ) Q(s, f )1/4−δ,

for Re (s) = 1/2.

However, the subconvexity problem is still open in general. In this thesis, our main fo-

cus is the subconvexity problem for Rankin-Selberg L-functions of GL2 × GL2 and related

problems.

11 1.2 Known subconvexity bounds and their applications

In this section, we will introduce some existing subconvexity bounds and their applications.

1.2.1 The Burgess’s bound and subconvexity bound for GL1

For the Riemann Zeta Function, the value ζ(1/2+it) can be well approximated by ﬁnite sums XN n−1/2−it, n=1 with an integer N depending on t. We can estimate the above ﬁnite sum by considering the exponential sums of type XN e ( f (n)) , n=1 where f (x) = (2π)−1t log x. By relating the zeta function to such exponential sums, H

.Weyl used his method of shifting the argument and repeated squaring in estimating the exponential sums in [Wey21] to obtain the ﬁrst subconvexity bound

1 | | 1/6+ ζ 2 + it (1 + t ) .

After Weyl’s work, a lot of diﬀerent techniques have been developed to estimate the exponential sums in order to obtain better bounds for ζ(s), like the one in [BI86] for example. 1 32/205+ The best known subconvexity bound for ζ(s) so far is ζ 2 + it t by Huxley in [Hux02] and [Hux05].

12 For Dirichlet L-functions

X∞ χ(n) L(s, χ):= , ns n=1 the subconvexity bounds are obtained by studying character sums. In [Bur62b], [Bur62a], and [Bur63], the author studied a short sum of mod q characters χ and obtained a bound as follows.

Theorem 1.2.1 (Thm. 2 [Bur63]). Let q be any positive integer and let χq be a non- principal Dirichlet character belonging to the modulus q. Let be a ﬁxed positive number and let r be any ﬁxed positive integer. Then, if either q is cubefree or r = 2, for every pair of integers N, H(H > 0) we have

N+H X 2 χ (n) H1−1/rq((r+1)/4r )+, q n=N with the implied constant depending on and r.

Remark 1.2.1. The trivial bound for this sum is H, since |χq(n)| 6 1 for every n. Therefore, the above bound is nontrivial only if

r+1 1 + H q 4r > q 4 .

One application of this bound is to prove that the maximum number of consecutive

1 3 quadratic residues or non-residues (mod p) is O(p 4 (log p) 2 ), where p is a prime ([Bur62a]

Thm. 2 ).

13 Another application is to show that in any interval of length

1 + H > p 4 , for any ﬁxed > 0, there are

ϕ(p − 1) H 1 + O(p−δ) p − 1 primitive roots (mod p), where δ > 0 depends only on ([Bur62a] Thm. 3 ). Here ϕ(·) is the Euler’s totient function.

Finally, based on Theorem 1.2.1, Burgess obtained a subconvexity bound for Dirichlet

L-functions.

Theorem 1.2.2 (Thm. 3 [Bur63]). If χ is a non-principal character belonging to the modulus q, and if L(σ + it, χ) denotes the L-function corresponding to χ, where σ and t are

ﬁxed real numbers satisfying

0 < σ < 1, then we have for any ﬁxed > 0  q(4−5σ+)/8 for 0 < σ 6 1 , |L(σ + it, χ)|  2  (3−3σ+)/8 1 q for 2 6 σ < 1. and in particular

1 (3/16)+ L 2 + it, χ q .

To illustrate how to deduce this Theorem from Theorem 1.2.1, we introduce the Ap- proximate Functional Equation (a special case of this approximate functional equation will be recalled in Lemma 2.2.1).

14 Theorem 1.2.3 ([IK04] Theorem 5.3). Let L(s, f ) be an L-function. Let G(u) be any func-

tion which is holomorphic and bounded in the strip −4 < Re (u) < 4, even, and normalized

by G(0) = 1. Let X > 0. Then for s in the strip 0 6 Re (s) 6 1 we have     X λ f (n)  n  X λ f (n)  nX  L(s, f ) = V   + ( f, s) V   + R ns s  p  n1−s 1−s  p  n X Q( f ) n Q( f )

where Vs(y) is a smooth function deﬁned by Z 1 −u γ(s + u, f ) du Vs(y) = y G(u) 2πi (3) γ(s, f ) u

and

1 −s γ(1 − s, f ) (s, f ) = ( f )Q( f ) 2 . γ(s, f )

The last term R = 0 if Λ(s, f ) is entire, otherwise

Λ(s + u, f ) G(u) R = (res + res ) Xu. u=1−s u=−s Q( f )s/2γ(s, f ) u

Then, we sketch the proof of the subconvexity bounds for s = 1/2. By carefully choosing the weight functions G(u) as in Lemma 2.2.1, and noticing the fact that the complete

L-function of L(s, χ) is entire, we have R = 0 in the above theorem and (with X = 1 )

1/2+ X ! X ! qX 1 χ(n) n χ(n) n χ(n) L , χ = V 1 √ + (χ, 1/2) V 1 √ . 2 1 2 1 2 1/2 2 q 2 q n n n n n n=1

Applying Theorem 1.2.1 with r = 2 and summation by parts, we obtain the subconvexity bound 1 (3/16)+ L 2 , χ q .

The other cases are similar.

15 Another proof of the subconvexity bound for the GL1 case, came much later in [FI92],

where the authors established a subconvexity bound using, what is now called, an “ampli-

ﬁcation” method. The basic idea is as follows.

We still start with a sum of characters. Instead of bounding the sum directly, we esti-

mate the “average” of all the characters. The classical mean-value theorem for Dirichlet

polynomials ([FI92] ) asserts that 2 X X 1 −1 λnχ(n) = (1 + O(q N))kλk, ϕ(q) χ(mod q) 0

X 2 kλk = |λn| , 0

result gives an upper bound 2 1 X X χ(n) = (1 + O(q−1/2+))q. ϕ(q) n1/2 χ(mod q) 0

Now, we consider a weighted average as follows.

For any predetermined character ψ(mod q), let

X λn := ψ(k)ψ(l). klm=n 0

Corollary 1.2.1 (Coro. 1 [FI92]). Let ψ be a non principal character mod q. We then have, for M > q5/11+, X ψ(m) M1−σ, n6M where σ and the implied constant may depend on .

By choosing M = q1/2+ and applying an approximate functional equation argument similar to the one above, we have

Corollary 1.2.2 (Coro. 2 [FI92]). With ψ as above, we have

1 L(s, ψ) q5/22+ for Re (s) > , 2 with an implied constant depending on and s.

17 1.2.2 The subconvexity bound for GL2 case

The strength of the argument in [FI92] is that they may be generalized to other types of

L-functions. Using a similar “ampliﬁcation” idea, Duke-Friedlander-Iwaniec were able to prove subconvexity bounds for the GL2 case in [DFI93], [DFI94], and [DFI01].

We simply state the ﬁrst theorem in this series of papers:

Theorem 1.2.4 (Thm. 1 [DFI93]). Let χ be a primitive character of modulus q and let

Re 1 (s) = 2 . Let f be a ﬁxed holomorphic cusp form. Then we have

L(s, f × χ) |s|2q5/11τ(q)2 log q,

where τ is the divisor function and the implied constant depends only on f .

Remark 1.2.2. The convexity bound in this case is |s|1/2+q1/2+.

One important application of these subconvexity bounds is to reprove and reﬁne Duke’s equidistribution theorem [Duk88]. As discussed in [Har11], for a fundamental discriminant d < 0 (resp. d > 0) denote by Λd the set of Heegner points (resp. closed geodesics) of

2 discriminant d on the modular surface SL2(Z)\H . There is a bijection between Λd and the √ narrow ideal class group Hd of Q( d) (see [Har11] for details ). The total volume of Λd is

|d|1/2+o(1) by Siegel’s theorem ([Har11] Section 6 ). As in [Duk88], the problem of whether

Λd becomes equidistributed when d → ∞ relates to the estimation of Fourier coeﬃcients of half-integral weight Maaß forms. The connection with subconvexity comes from the work of Waldspurger [Wal81] on the Shimura correspondence, which shows that nontrivial bounds for these Fourier coeﬃcients are in fact equivalent to subconvexity bounds for

18 1 × d the central twisted values L 2 , f · as f ranges over the Hecke-Maaß cusp forms and

2 Eisenstein series on SL2(Z)\H .

Since the subconvexity bound for the central twisted values of Eisenstein series is the subconvexity problem for GL1 [Duk88], the equidistribution property of Heegner points

(resp. closed geodesics ) follows from a result in the joint work of V. Blomer, P. Michel, and G. Harcos [BHM07b].

Theorem 1.2.5 (Thm. 1.2 [Har11]). Let f be a primitive (holomorphic or Maaß) cusp form

Re 1 of level q and nontrivial nebentypus. Then for (s) = 2 one has

A 1 − 1 L(s, f ) (|s|(1 + |t f |)) q 4 1889 ,

where A > 0 is an absolute constant and t f is the local parameter at inﬁnity deﬁned in

Section 2.1.

All of the subconvexity bounds above are obtained using a trace formula, the ampliﬁ-

cation method, and a careful estimation of shifted sums. Another approach to this problem

is via a formula which relates the central value of triple product L-functions to integrals

of the corresponding producet of forms over a fundamental domain (an example of this

kind of formulae, which is called Watson’s formula, will be described in the next section ).

Towards this direction, P. Michel and A. Venkatesh gave a very general bound in [MV10].

Theorem 1.2.6 (Thm. 1.1 [MV10]). Let F be a ﬁxed number ﬁeld and AF is its ring of

ad`eles.There is an absolute constant δ > 0 such that: for π an automorphic representation of GL1(AF) or GL2(AF) (with unitary central character), one has

1/4−δ 1 Q 1 L 2 + it, π F 2 + it, π , 19 Q 1 where 2 + it, π is the analytic conductor (1.1.1).

Remark 1.2.3. The notation in [MV10] Theorem 1.1 is diﬀerent from our notation here, but they are equivalent by the discussion after [MV10] Theorem 1.1.

The bound above is a hybrid bound of all the aspects (t-aspect, spectral aspect, and level aspect). Therefore, the subconvexity bound of a single representation on GL2 and

GL1 over any number ﬁeld is obtained ( in the classical language, this result solves all the subconvexity problems of the form χ, f holomorphic or Maaß, and f × χ ). In their proof, they also used the idea of “ampliﬁcation”. However, other than studying a weighted sum, they considered the integral convolution of certain functions called “S -arithmetic function”.

Furthermore, they did not use a trace formula of any kind in the proof.

1.2.3 The QUE conjecture and the subconvexity problem of higher rank

Another application of subconvexity is the Quantum Unique Ergodicity (QUE) conjecture, formulated by P. Sarnak and Z. Rudnick [RS94].

2 Let H be the upper half plane of C. Γ0(M) ⊆ SL2(Z) is a congruence group as de-

2 ﬁned in Section 1.1. Γ0(M)\H is a noncompact Riemann surface. Consider the Laplacian

2 2 2 d d ∈ 2 y dx2 + dy2 deﬁned on this surface, where x, y are the real and imaginary part of z H .

2 2 A Maaß form is a L cusp form deﬁned on Γ0(M)\H which is an eigenfunction of the

Laplacian. Let λ f be the Laplacian eigenvalue of a Maaß form f . The full deﬁnition of

Hecke-Maaß forms is in Chapter2. Then, the QUE conjecture can be stated as follows.

20 Conjecture 1.2.1 (QUE Conjecture [Sar01]). Let f be a Hecke cusp form on a ﬁxed quo-

2 tient X0(M):= Γ0(M)\H such that k f k2 = 1. We consider two cases

(1) f is a holomorphic form of even integral weight k f .

(2) f is a Maaß form with eigenvalue λ f (in which case the weight k f = 0 ).

With f we associate the density µ f given by

dxdy µ := |F(z)|2 , f y2

where

k f F(z):= y 2 f (z).

Then, as k f → ∞ in the holomorphic case, or λ f → ∞ in the Maaß case, we have

1 dxdy µ → , f 2 Vol(X0(M)) y

in the weak sense, e.g. in the sense of integration against continuous functions of compact

support on X0(M).

The QUE conjecture can be considered as a consequence of a subconvexity problem of

triple product L-functions. For simplicity, let M = 1, so that Γ0(1) = SL2(Z). Let

1 dxdy dµ := 2 Vol(X0(1)) y

be the probability measure on X0(1). In order to study the convergence of dµ f , we need to

prove the limit of the integral Z gdµ f , 2 SL2(Z)\H for any compactly supported smooth function g(z) is

Z gdµ, 2 SL2(Z)\H

21 as k f or λ f approaches ∞. After the spectral decomposition ([IK04] Section 15 )

X X 1 Z g(z) = hg, u i u + hg, E ·, 1 + it i E z, 1 + it dt, j j 4π a 2 a 2 j>0 a R

1 where u0 is the constant 1, u j for j > 0 are Maaß cusp forms, and Ea z, 2 + it are Eisenstein series. By Weyl’s equidistribution criterion, we can study the case that g(z) is a Maaß cusp

form and the one that g(z) is an integral of Eisenstein series ([Sar01] ). When g is a Maaß

cusp form, we have the Watson’s formula ([Wat08] ) which says that Z 2 2 1 kgk2L , f × f × g gdµ ∼ 2 Q , f 2 f,g (1.2.1) 2 SL2(Z)\H L(1, Ad f ) L(1, Adg)

3 − 2 + where Q f,g is a ratio of gamma factors and t f . A generalized version of this formula can be found in [Ich08]. If we assume the subconvexity bound

1 3 −δ × × | | 2 L 2 , f f g (1 + t f )

−1 for some δ > 0, since L (1, Ad f ) (1 + |λ f |) (see [HL94] ), the right hand side goes to R zero when λ f tends to ∞. Moreover, since 2 gdµ = 0, we complete the proof for g SL2(Z)\H Maaß form case. The case of Eisenstein series is similar (see [Wat08] ).

1 × × However, such a subconvexity bound is not known for L 2 , f f g . Fortunately, for some special cases, we can decompose this L-function further. Since both f and g are

Hecke eigenforms, we have

L(s, f × f × g) = L(s, sym2 f × g)L(s, g).

Moreover, if f is a CM form, that is to say L(s, f ) = L(s, λ), for some Grossen character λ on a quadratic extension of Q, then L(s, sym2 f × g) factors further as L(s, λ2 × g)L(s, g × χ) for a suitable Dirichlet character χ. Now L(s, λ2) is the L-function of another modular cusp

22 form F. If f is holomorphic then so is F and its weight is twice the weight of f ; If f is a

1 2 1 2 Maaß form then so is F with eigenvalue 4 +(2r f ) when f has eigenvalue 4 +r f . Therefore, 1 × for CM forms, the subconvexity problem reduces to that of L( 2 , F g) in either the kF aspect or the λF aspect. See [Sar01] for further references.

This special case motivates the study of subconvexity problem for Rankin-Selberg L- functions. The spectral aspect subconvexity bounds can be found, for example, in [Sar01].

For the level aspect, the authors of [KMV02] proved the following bound for the j-th derivatives of Rankin-Selberg L-functions.

Theorem 1.2.7 (Thm. 1.1 [KMV02]). Let g be a primitive cuspidal holomorphic newform of integral weight, or a non-exceptional weight zero Maaß form, on Γ1(D) with D square- free. Then for all > 0, all integers j > 0, and for all holomorphic newform f with level q, weight k, and (q, D) = 1, | ( j) 1 × | | | B +1/2−1/80 L 2 + it, f g ,k, j,g (1 + t ) q ,

where the exponent B is absolute.

This theorem gives a subconveixty bound in the level aspect of f when g is ﬁxed. In

[MV10], the authors give a subconvexity bound in all aspects of the varying form when the

other form is ﬁxed.

Theorem 1.2.8 (Thm. 1.2 [MV10]). There is an absolute constant δ > 0 such that: for

π1, π2 automorphic representations on GL2(AF) we have:

1/4−δ 1 × Q 1 × L 2 , π1 π2 F,π2 2 , π1 π2 ;

more precisely, the constant implied depends polynomially on the discriminant of F (for F Q 1 varying over ﬁelds of given degree) and on 2 , π2 . 23 However, we have limited knowledge of hybrid subconvexity bounds when both repre-

sentations vary. The aim of this thesis is to establish results that contribute to the under-

standings of hybrid subconvexity bounds.

Returning to the QUE conjecture when f is non-CM, one would need a subconvexity 1 × × 1 2 × bound for the L-functions of triple products of the form L 2 , f f g or L 2 , sym f g ,

where f is varying and g is ﬁxed. Denote by φλi , a Hecke-Maaß cusp form with Laplacian eigenvalue λi. In [BR05], the authors proved:

Theorem 1.2.9 (Coro. 1.2 [BR05]). Let φ and φ0 be ﬁxed Hecke-Maaß cusp forms. For any > 0, there exists C > 0 such that the bound

1 × 0 × | |5/3+ L 2 , φ φ φλi 6 C λi

holds for any Hecke-Maaß form φλi .

The proof depends on a realization of the representation of a triple product and the am-

pliﬁcation method. However, it does not give a subconvexity bound which is suﬃcient for

proving QUE.

1 2 × The authors of [HMQ14] studied the subconvexity bound of L 2 , sym f g when both f and g vary:

Theorem 1.2.10 (Coro. 2.3 [HMQ14]). Let g be a newform of weight 2κ and level P. Let

f be a Hecke eigenform of even weight k > κ. Then we have

 13 25 13 4 k 29 P 29 (kP) , if P 64 < k 6 P 13 , 1 2 ×  L 2 , sym f g ,κ  13 3 4 3  P + k 7 P 7 (kP) , if P 13 < k 6 P 8 .

24 13 3 64 +δ 8 −δ 11 Their result gives a subconvexity bound when P < k < P for any 0 < δ < 128 .

2 By the symmetric lift, sym f may also be considered as a self-dual GL3 automor-

phic representation (see [Sou05] ). Therefore, one can study the subconvexity problem of 1 2 × L 2 , sym f g by establishing the subconvexity bound for Rankin-Selberg L-functions of

GL3 × GL2. Via a ﬁrst moment estimation, Li [Li11] was able to show that

Theorem 1.2.11 (Coro. 1.1 [Li11]). Let f be a ﬁxed self-dual Hecke-Maaß form for S L3(Z) and u j be an even Hecke-Maaß form for S L2(Z) corresponding to the Laplacian eigenvalue

1 2 4 + t j with t j > 0. Then for any > 0

1 11 + × | | 8 L 2 , f u j , f (1 + t j ) .

3 + 1 | | 2 × The convexity bound in this case is (1+ t j ) . The nonnegativity of L 2 , f u j plays a key role in their approach. Also, they obtained the t-aspect subconvexity bound.

Theorem 1.2.12 (Coro. 1.2 [Li11]). For f a self dual Hecke-Maaß form for S L3(Z),

1 11 + | | 16 L 2 + it, f , f (1 + t ) ,

3 + for any > 0 (the convexity bound is |t| 4 ).

With a modiﬁed circle method, the author of [Mun13a] is a able to extend the above

result to any Hecke-Maaß form which is not necessarily self dual.

Recall the Waston’s formula (1.2.1) Z 2 2 1 kgk2L , f × f × g gdµ ∼ 2 Q . f 2 f,g 2 SL2(Z)\H L(1, Ad f ) L(1, Adg)

25 Instead of obtaining power savings beyond the convexity bound, if we have both log sav-

ings for L-functions of triple products as well as a suitable lower bound for L(1, Ad f )2, it

is still possible to prove the QUE conjecture. We call the bounds with log savings beyond

the convexity bound the weak subconvexity bounds.

Towards this direction, Soundararajan [Sou10b] proved the following bounds:

Theorem 1.2.13 ([Sou10b]). Let f be a holomorphic Hecke eigenform of large weight k for

the full modular group S L2(Z). Let φ be a ﬁxed Hecke-Maass eigencuspform for S L2(Z).

Then

2 k L 1 , f × f × φ , 2 φ, (log k)1−

1+ for any > 0, where the convexity bound is Oφ,(k ).

Remark 1.2.4. If the Ramanujan conjecture holds for an irreducible cuspidal automorphic representation π of GLm over Q with unitary central character, then Soundararajan 1 × [Sou10b] also proved a weak subconvexity bound for L 2 , π0 π for any ﬁxed automorphic representation π0.

Based on this weak subconvexity bound, the combined work of K. Soundararajan and

R. Holowinsky [HS10] proved the QUE conjecture for the holomorphic modular form case.

Remark 1.2.5. The Maaß form case of QUE conjecture for S L2(Z) was proven via the com-

bined works of E. Lindenstrauss [Lin06] and K. Soundararajan [Sou10a], which avoids the

study of the subconvexity problem.

26 Chapter 2: Preliminaries and Past Works

2.1 Automorphic forms and the normalization

2 Let SL2(R) be the group of 2-by-2 matrices with determinant 1. Let H be the upper- ! a b half plane of C. Any γ = ∈ SL (R) acts on H2 via Mobius¨ action c d 2

az + b γ.z = . cz + d

Let M > 0 be an integer and κ > 0 be an even integer. Let ( ! ) a b Γ (M):= ∈ SL (Z): c ≡ 0(mod M) 0 c d 2

be a congruence subgroup of SL2(Z).

We recall the deﬁnition of holomorphic cusp forms and Maaß cusp forms.

The holomorphic cusp forms with weight κ, level M, and trivial nebentypus are holomorphic functions on the upper half-plane F : H2 → C satisfying

F(γ.z) = (cz + d)κF(z),

27 when ! a b γ = ∈ Γ (M), c d 0 and vanishing at every cusp, e.g. F(a.z) → 0 as Im (z) → ∞ for any a ∈ SL2(Z).

2 The Maaß cusp forms with Laplacian eigenvalue λ∞, weight 0 and level M are L func-

2 2 2 2 tions F : H → C satisfying ∆F = λF F for Laplacian ∆ = −y (∂x+∂y), F(z) = F(γ.z) for all

γ ∈ Γ0(M) and vanishing at every cusp, e.g. F(a.z) → 0 as Im (z) → ∞ for any a ∈ SL2(Z).

2 Denote by L0(M), the space of weight zero Maaß cusp forms with respect to the congruence subgroup Γ0(M). Following notations in [BH10b], we denote by Sκ(M) the linear

κ space of all the functions f (z) = y 2 F(z) where F is a holomorphic cusp form with weight

κ, level M and trivial nebentypus.

2 Both L0(M) and Sκ(M) are Hilbert spaces respect to the inner product

Z dxdy h f , f i f z f z . 1 2 := 1( ) 2( ) 2 2 Γ0(M)\H y

Hecke operators Tm are deﬁned by the formula ! 1 X X az + b T f (z):= √ f , (2.1.1) m d m ad=m 06b

2 2 for elements in L0(M) ∪ Sκ(M). If f ∈ L0(M) ∪ Sκ(M) is an eigenfunction of the Hecke

operator Tm, we denote by λ f (m) the corresponding eigenvalue.

28 2 We can choose orthogonal bases Bκ(M) and B(M), respectively, of Sκ(M) and L0(M) which consist of eigenfunctions of all the Hecke operators Tm with (m, M) = 1 (see [IK04]

Section 14 ). We call such forms Hecke eigenforms.

In order to treat both cases simultaneously, we write the Fourier expansion as

X ψ f (n) f (z) = √ W f (|n|y)e(nx) n n,0

where  1 2t f +1  − 2 2 −2πy Γ(2t f + 1) (4πy) e , f ∈ Sκ(M), W f (y) =  1 1 2  2 ∈ L y cosh( 2 πt f )Kit f (2πy), f 0(M).

The t f is the spectral (or archimedean) parameter of f which is deﬁned as   κ−1 , f ∈ S (M),  2 κ t f = q  − 1 ∈ L2  λ f 4 , f 0(M) satisfying ∆ f = λ f f.

2 We call cusp forms f ∈ L0(M) with real t f the non-exceptional Maaß cusp forms and

1 f with t f = ir for some 0 < r < 2 the exceptional Maaß cusp forms.

By the multiplicativity of Hecke operators (see [IK04] Section 14 ), one has that

X mn ψ (m)λ (n) = ψ (2.1.2) f f f d2 d|(m,n) for any m, n > 1 with (n, M) = 1 when f is in Bκ(M) or B(M). In particular, ψ f (1)λ f (n) =

ψ f (n) when (n, M) = 1.

∗ ∗ There are subsets Bκ(M) and B (M), respectively, of Bκ(M) and B(M) which consist of all the Hecke newforms, e.g. the forms which are not generated by cusp forms with

lower levels (see [IK04] Chapter 14, [Cas73] for more details). It is well known that Hecke

29 newforms are the eigenfunctions of all the Hecke operators Tm even for (m, M) , 1 (see

[BH10a] for further references ). For such forms, (2.1.2) holds for any m and n.

Moreover, when N is square-free, by local calculations (see [GH11] Section 11.12), we

have that

−1/2 λ f (L) = L for any L|N. (2.1.3)

In Chapter4, we will need the following Lemma:

2 Lemma 2.1.1. Let f ∈ Sk(N) ∪ L0(N). Then

X ψ f (n)e(nα) S ( f, X, α):= √ t f , k f k∞(NX) . n≤X n for any > 0.

Remark 2.1.1. We call this type of bound Wilton’s bound after J. R. Wilton who ﬁrst established a bound of a twisted sum of Fourier coeﬃcients and exponentials, see [Wil29].

Proof. When f ∈ Sk(N), as in [HM06] Section 2.6 (note that a diﬀerent normalization there is given in [HM06] Section 2.1 and 2.2), we have

1 Z 1 Z ∞ X ψ f (n)e(nα) Γ(k) 2 e(−Xt) − 1 = y−1+ f (t + α + iy)dydt. (2.1.4) 1 + k 2 0 1 − e(t) 0 16n6X n Γ 2 +

− 1 By the bound f (t + iy) k, |ψ f (1)|N y 2 ([HM06] Section 2.6 ), we have

Z ∞ Z 1 Z ∞ −1+ −1+ − 3 + y f (t+α+iy)dy k, y k f k∞dy+ y 2 |ψ f (1)|N dy k, k f k∞+|ψ f (1)|N . 0 0 1

Applying this bound in (2.1.4), and noticing that

Z 1 |ψ f (1)| k f (x + i)e(−x)dx k f k∞, 0 30 we get

X ψ f (n)e(nα) S (X):= 1 k f k∞(N) . 2 + 16n6X n By partial summation,

X ψ f (n)e(nα) X ψ f (n)e(nα) √ = 1 n 2 + 16n6X n 16n6X n X = S (X)X − S (l − 1)(l − (l − 1) ) k f k∞(NX) . 16l6X We complete the proof in this case.

2 When f ∈ L0(N),

X ψ f (n)e(nα) 1 2 + 16n6X n 1 2 + 1 Z 1 Z ∞ π cosh πt f e(−Xt) − 1 = 2 y−1+ ( f (t + α + iy) ± f (−α − t + iy)) dydt. 1 0 1 − e(t) 0 Φ 2 + , it f

1 where Φ( 2 + , it f ) is a function that is independent of N (see the proof of [HM06] Propo-

sition 2.4 ). The rest of the proof follows along the same line as in the case for Sk(N).

We also need a well-known Rankin’s result, see [DFI02] Section 19.

∗ ∗ Lemma 2.1.2. Let f ∈ Bk(N) or B (N), then

X 2 |λ f (n)| ,t f X(XN) , n≤X for any > 0.

2.2 Rankin-Selberg convolution and L-functions

Let N, M be two positive integers and κ, k be two ﬁxed positive even integers. Given

∗ S ∗ ∗ two Hecke newforms f ∈ Bk(N) B (N) and g ∈ Bκ(M), we consider the associated

31 Rankin-Selberg L-function

2 2 !−1 Y Y Y α f,i(p)αg, j(p) X L(s, f × g):= 1 − = ζ(NM)(2s) λ (n)λ (n)n−s ps f g p i=1 j=1 n>1 where {α f,i} and {αg, j} are the local parameters of L-functions associated to f and g respectively such that 2 !−1 Y Y α f,i(p) X L(s, f ) = 1 − = λ (n)n−s, ps f p i=1 n>1 2 !−1 Y Y αg, j(p) X L(s, g) = 1 − = λ (n)n−s, ps g p j=1 n>1 and !−1 Y 1 ζ(NM)(2s) = 1 − . p2s p-NM By [KMV02] Section 4, the local factor at inﬁnity is deﬁned as a product of gamma factors × −2s |k−κ| k+κ − ∈ B∗ L∞(s, f g):= (2π) Γ s + 2 Γ s + 2 1 for f k(N), × −2s κ+2it f −1 κ−2it f −1 ∈ B∗ L∞(s, f g):= (2π) Γ s + 2 Γ s + 2 for f (N).

The completed L-function is given as

s Λ(s):= Q 2 L∞(s, f × g)L(s, f × g), where Q := Q( f × g) is the conductor.

By the local Langlands correspondence, one can verify that (see [HM06])

(MN)2/(M, N)4 6 Q( f × g) 6 (MN)2/(M, N).

We also have the functional equation

Λ (s, f × g) = ε ( f × g) Λ 1 − s, f × g ,

32 where ε ( f × g) is the root number and f¯,g ¯ are the dual forms (see Section 1.1).

Based on the functional equation, one can obtain the following equation as in [IK04]

Section 5.2.

Lemma 2.2.1 (The Approximate Functional Equation). Let f, g be two Hecke newforms.

Let G(u) be any function which is holomorphic and bounded in the strip −4 < Re (u) < 4, even, and normalized by G(0) = 1. Then

∞ ! ∞ ! X λ f (n)λg(n) n X λ f (n)λg(n) n L (s, f × g) = V √ + ε ( f × g, s) V √ ns s n1−s 1−s n=1 Q n=1 Q

where Z 1 L∞(s + u, f × g) (NM) −u du Vs(y) = G(u) ζ (2s + 2u)y 2πi (3) L∞(s, f × g) u and

1 −s L∞(1 − s, f × g) ε ( f × g, s) = ε ( f × g) Q 2 . L∞(s, f × g)

One can choose !−16A0 πu G(u) = cos 4A0

where A0 is a positive integer. By [Mic04] Lem. 3.1 and Rem. 3.3, we have when Re (s) >

> 0, the test function Vs(y) satisﬁes

 −A0 j ( j) −1 j +  y  y V (y) log(1 + y )Q q (s) 2 1 +  (2.2.1) s A0, ∞  p  q∞(s)

4 for any > 0 and q∞(s) = |s| + |t f + tg| + 1 .

33 2.3 Properties of Bessel functions

Re − 1 Let v be a complex number with (v) > 2 . Let Jv(x) be the J-Bessel function of the ﬁrst kind. It can be deﬁned through Taylor series as,

X∞ (−1)m x2m+v J (x) = . v m!Γ(m + v + 1) 2 m=0

Let Kv(x) be the K-Bessel function deﬁned as v 1 1 ∞ ! !m π x X 1 1 1 K (x) = 2 2 − x2 . v sin(vπ) m!Γ(m − v + 1) m!Γ(m + v + 1) 4 m=0

The Y-Bessel function can be deﬁned as

J (x) cos(vπ) − J (x) Y (x) = v −v . (2.3.1) v sin(vπ)

For any ﬁxed v, Taylor series for Jv(x) is absolutly convergent for any x ∈ R. According to the Taylor expansion, when 0 < x 6 10, say

i (i) Re (v) x Jv (x) x , (2.3.2) where the implied constant depends on v and i.

For Kv(x) and Yv(x), by [HM06] Section 7 we have

−Re (v)− Kv(x), Yv(x) x , when x → 0+, and

v 0 v v 0 v (x Kv(x)) = x Kv−1(x), (x Yv(x)) = x Yv−1(x).

34 Thus, we have

j ( j) j ( j) −Re (v)− x Kv (x), x Yv (x) j,v x , (2.3.3) as x → 0+.

Moreover, we have the integral representation (see [Wat95] p. 206 )

1 −x Z ∞ v− 1 π 2 e −u v− 1 u 2 K (x) = e u 2 1 + du. v 2x 1 2x Γ(v + 2 ) 0

The relation between the J-Bessel and the K-Bessel function is

−vπi/2 −πi/2 vπi/2 πi/2 πiJv(x) = e Kv(xe ) − e Kv(xe ). (2.3.4)

So, we can write the J-Bessel function as

ix −ix Jv(x) = e Wv(x) + e Wv(x), (2.3.5)

where

i( π v− π ) r Z ∞ v− 1 e 2 4 2 iy 2 W (x) = e−y y 1 + dy. v 1 πx 2x Γ(v + 2 ) 0

Re − 1 Moreover, since (v) > 2 , the integral is absolutely convergent. The j-th derivative satisﬁes

1 x jW( j)(x) . (2.3.6) v v (1 + x)1/2

for x > 1.

For the K-Bessel function, we also have

−x ∗ Kv(x) = e Wv (x), (2.3.7)

35 where √ Z ∞ v− 1 ∗ π −u v− 1 u 2 W (x) = √ e u 2 1 + du. v 1 0 2x 2xΓ(v + 2 ) Furthermore, the j-th derivative satisﬁes

1 x jW∗( j)(x) . (2.3.8) v v (1 + x)1/2

Re − 1 for x > 1, when (v) > 2 . Next, we establish two lemmas bounding integrals of Bessel functions with ﬁxed weights.

The following lemma is similar to the one in [HM13].

Lemma 2.3.1. Let k, κ > 2 be ﬁxed integers. Let a, b, x, y > 0. Deﬁne

Z ∞ p p I(x, y):= h(ξ)Jk−1 4πa xξ Jκ−1 4πb yξ dξ 0

( j) j where h is a smooth function compactly supported on [1/2, 5/2] such that h j Z for all √ √ j and some Z > 0. Then, when a x − b y > 2, we have

√ √ − j j I(x, y) j (1 + Z) a x − b y ,

for any j > 0. Moreover,

i j √ i j ∂ ∂ 1 i √ j x y I(x, y) i, j √ √ (1 + a x) (1 + b y) . ∂xi ∂y j (1 + a x)1/2(1 + b y)1/2

Proof. The change of variables, ξ = ω2, gives

Z ∞ √ 2 √ I(x, y):= 2 ωh(ω )Jk−1 4πa xω Jκ−1 4πb yω dω. 0 √ √ When a x, b y > 1, using (2.3.5), I(x, y) may be written as a sum of four similar terms, one of them being

Z ∞ √ √ √ 2 √ e 2ω a x − b y ωh(ω )Wk−1 4πa xω Wκ−1 4πb yω dω. 0 36 √ √ Repeated integration by parts gives the ﬁrst statement. When a x 6 1 ( resp. b y 6 1 ), using the Taylor expansion (2.3.2) of Jk−1 ( resp. Jκ−1 ), one can still get the desired bound.

For the second statement, diﬀerentiate the J-Bessel function and then use either (2.3.6) for

Wk−1 or (2.3.2) for Jk−1 case by case as above.

In order to analyze the Maaß form case, we need a similar bound as follows.

Lemma 2.3.2. Let κ > 2 be a fixed integer and let t be a fixed real number (with r = 0 ) or a fixed pure imaginary number such that t = ir with 0 < r < 1/4. Let a, b, x, y > 0. Define

Z ∞ p p p I0(x, y):= h(ξ) Y2it(4πa xξ) + Y−2it(4πa xξ) Jκ−1 4πb yξ dξ, 0 Z ∞ p p I1(x, y):= h(ξ)K2it(4πa xξ)Jκ−1 4πb yξ dξ, 0

( j) j where h is a smooth function compactly supported on [1/2, 5/2] such that h j Z for

some Z > 0. Then,

√ √ √ − j −r− j Iι(x, y) j (1 + Z) 1 + (a x) a x − b y ,

and √ i j −r− √ i j ∂ ∂ 1 + (a x) i √ j x y Iι(x, y) i, j √ √ (1 + a x) (1 + b y) , ∂xi ∂y j (1 + a x)1/2(1 + b y)1/2 for any i, j > 0, where ι = 0 or 1. √ Proof. When a x > 1, by (2.3.4) and (2.3.1)

−vπi/2 −πi/2 vπi/2 πi/2 −πYv(x) = e Kv(xe ) + e Kv(xe ).

Then, we can obtain a similar expression as (2.3.5) for Y-Bessel functions. We also have √ the expression (2.3.7) for K-Bessel functions. When a x 1, we have (2.3.3) just like

the J-Bessel function case. Therefore, the proof follows the same line as the one in the

previous lemma.

37 − − 1 Remark 2.3.1. We require that r be in the range 0 < r < 1/4, i.e. 2r > 2 , since we need to apply (2.3.8) in our proof.

Next, we establish bounds for integrals of two Bessel functions with one weight ﬁxed

and the other varying.

Lemma 2.3.3. Let H(x) be a function supported on [X/2, 5X/2]. Let κ ≥ 1 be an integer.

i i Z Let X ≥ 10κ such that H (x) X for any i. Let l be a positive integer. Let t be a positive number or a purely imaginary number such that t = ir, where 0 < r < 1/2. Then for any positive A,  Z ∞ (Z + 1) log |X|/X dx  H(x)Jl(x)Jκ(x) A, A (2.3.9) x  (Z+1)X 0  l2 (Z + 1)X if l > 2κ.  Z ∞ (Z + 1) log |X|/X if t > 0 or t = ir, i dx  (J2it(x) − J−2it(x)) Jκ(x)H(x) A, A 2 sinh(πt) x  (Z+1)X 0  t2 (Z + 1)X if t > 1. (2.3.10)

Proof. The proof follows the same line as the one in [DI83] Lemma 7.1 and [BHM07a].

For the ﬁrst inequality in (2.3.9), we consider integral representations of Bessel functions

([DI83] Section 7 )

Z 2π 1 i(x sin u−lu) Jl(x) = e du, (2.3.11) 2π 0

and (2.3.5)

ix −ix Jκ(x) = e Wκ(x) + e Wκ(x).

Through integration by parts and (2.3.6), we obtain

Z ∞ Z ∞ ± !0 ix sin u H(x) i ix(sin u±1) H(x)Wκ (x) e Jκ(x)dx e dx 0 x sin u ± 1 0 x (Z + 1)X−3/2| sin u ± 1|−1,

38 + − where Wκ := Wκ and Wκ := Wκ. Otherwise, by (2.3.5) and (2.3.6), we have the trivial bound

− 1 Jκ(x) (1 + x) 2 , which gives Z ∞ ix sin u H(x) −1/2 e Jκ(x)dx X . 0 x Thus,

Z ∞ Z 2π Z ∞ 1 −ilu ix sin u H(x) H(x)Jκ(x)Jl(x)dx = e e Jκ(x)dxdu 0 2π 0 0 x Z 2π (Z + 1) log |X| X−1/2 min{1, (Z + 1)X−1| sin u ± 1|−1}du . 0 X The proof of the ﬁrst case in (2.3.10) follows the same line. We have the integral representation ([DI83] Section 7 )

J (x) − J (x) 4i Z ∞ 2it −2it = cos(x cosh u) cos(2tu)du. (2.3.12) sinh πt π 0 When t is real,

Z ∞ π J2it(x) − J−2it(x) dx H(x)Jκ(x) sinh πt 0 2i x Z ∞ Z ∞ H(x) = 2 cos(2tu) cos(x cosh u)Jκ(x) dxdu 0 0 x Z ∞ (Z + 1) log |X| X−1/2 min{1, (Z + 1)X−1| cosh u ± 1|−1}du . 0 X When t = ir, since 0 < r < 1/2, the second to last inequality above becomes

Z ∞ Z ∞ H(x) 2 cos(2iru) cos(x cosh u)Jκ(x) dxdu 0 0 x Z ∞ (Z + 1) log |X| X−1/2e2ru min{1, (Z + 1)X−1| cosh u ± 1|−1}du . 0 X For the second case in (2.3.10), we need the diﬀerential equation of which the J-Bessel function Jα(x) is a solution of d2y dy x2 + x + (x2 − α2)y = 0. dx2 dx 39 Like (2.15) in [BHM07a], one can check that for any compactly supported function

f (x) in C∞ ((0, ∞)), we have

∞ ∞ !00 !0! Z Z x2 f (x) x f (x) f x J x dx − J x dx. ( ) α( ) = 2 2 + 2 2 α( ) 0 0 x − α x − α Let ϕ(x) be a compactly supported smooth function. Applying the formula above to

2 2 (x − l )ϕ(x)Jκ(x) and the diﬀerential equation of Jκ(x), we obtain Z ∞ 1 Z ∞ 1 Z ∞ ϕ x J x J x dx D (ϕ) J x 0 J x dx D (ϕ) J x J x dx, ( ) κ( ) l( ) = 2 2 11 κ( ) l( ) + 2 2 12 κ( ) l( ) 0 κ − l 0 κ − l 0 where 0 00 h 2 i 0 2 D11 (ϕ) := 2 xϕ − x ϕ , D12 (ϕ) := (xϕ) − x ϕ .

By a similar argument, we have

Z ∞ 1 Z ∞ 1 Z ∞ ϕ x J x 0 J x dx D (ϕ) J x 0 J x dx D (ϕ) J x J x dx, ( ) κ( ) l( ) = 2 2 21 κ( ) l( ) + 2 2 22 κ( ) l( ) 0 κ − l 0 κ − l 0 where

0 2 00 2 2 0 D21 (ϕ) := −(xϕ + x ϕ ), D22 (ϕ) := 2(xϕ + (x − κ )ϕ ).

Finally, let ϕ = H, which has type (1 : Z) (see Section 4.4 for the deﬁnition) and the

support on [X/2, 5X/2] such that X > 10κ. Therefore, Di j (ϕ) has type (X(Z + 1) : Z + 1)

by the lemmas in Section 4.4. By repeated use of the above two formulae A times (with

ϕ → Di j(ϕ)), we then apply the integral representations (2.3.11) and (2.3.12) as above to

obtain the bound for the second case in (2.3.9). The proof of (2.3.10) follows the same

line.

2.4 The Voronoi formula, trace formulae and large sieve inequalities

We recall the Voronoi summation formula ([KMV02] Appendix A.3 and A.4). Let q be

an integer. Let N be a square-free integer. Denote by N2 := N/(N, q). For a cusp form f

40 ± and y > 0, deﬁne J f as follows.

+ − J f (y):= 2πJ2t f (4πy), J f (y):= 0,

∗ if f ∈ Bk(N), where t f = (k − 1)/2 is the spectral parameter, and

+ π − J f (y):= Y2it f (4πy) + Y−2it f (4πy) , J f (y):= 4 cosh(πt f )K2it f (4πy), cosh(πt f )

∗ if f ∈ B (N) is a Maaß newform with spectral parameter t f .

Remark 2.4.1. Our notations are as the same as the ones in [BH10b] Section 3. Bessel

± functions J f in [BH10b] equal the ones deﬁned in [KMV02]. Our t f is as the same as the t in [BH10b], which is equivalent to the r in [KMV02].

± Remark 2.4.2. We list the properties of J f (y) for a ﬁxed cusp form f as follows (see [BH10b] Section 4, or [HM06] Section 7).

∗ When f ∈ Bk(N),  yk−1 as y → 0+, J +(y)  f y−1/2 as y → ∞.

When f ∈ B∗(N), let

r f := Im (t f ) + .

Then   −2r f → + + y as y 0 , J (y) f y−1/2 as y → ∞.

And   −2r f + − y as y → 0 , J (y) f e−y as y → ∞.

41 Lemma 2.4.1. (The Voronoi Summation Formula) Let (a, q) = 1. Let h be a smooth

∗ ∗ function, compactly supported on (0, ∞). Let f ∈ B (N) ∪ Bk(N) be a form with square-

± free level N . Set N2 := N/(N, q). Then there exist two complex numbers η both with norm

1, depending on f , and a Hecke newform f ∗ of the same level N such that

!   Z ∞ 2 2 ! X λ f (n) a X ± X λ f ∗ (n)  aN2  ξ q N2 ± √ e n h(n) = 2 η √ e ∓n  h J (ξ)dξ q  q  n f n n ± n n 0 where x¯ denotes the multiplicative inverse of x mod q.

± ∗ Remark 2.4.3. The explicit expression of η is obtained in [KMV02]. When f ∈ Bk(N),

± k η = i η(N2), where η(N2) is the pseudo-eigenvalue of f under the Atkin-Lehner operator

∗ + − WN2 . When f ∈ B (N), η = η(N2) and η = f η(N2), where f is the eigenvalue of f under

± the reﬂection operator. Therefore, in any case, η only depends on N2 and f .

For any m, n, c ∈ N, denote by S (m, n; c) the Kloosterman sum

X na + ma¯ X∗ na + ma¯ S (m, n; c):= e = e . c c 0

Lemma 2.4.2. (The Petersson Trace Formula) Let M, κ > 1 be integers. For any n, m > 1, we have √ ! X X 1 4π nm ω−1ψ (n)ψ (m) = δ(n, m) + 2πi−κ S (n, m; c)J , g g g c κ−1 c g∈Bκ(M) c>0 c≡0(M)

where the weights ωg are given by

κ − 1 ω := hg, gi , g 4π

δ(n, m) = 1 if n = m, and δ(n, m) = 0 otherwise.

42 This is proved in [Iwa97] Theorem 3.6 for Fourier coeﬃcients normalized as in [Iwa97]

p. 52.

Let r, s > 0 be two coprime integers. Let q = rs and Γ = Γ0(q). Let a = ∞ and b = 1/s

be two cusps of Γ. Let c be a cusp of Γ. Denote by ψ jk(c, n), and ρ jc(n) the n-th Fourier

2 coeﬃcients of a L -normalized, e.g. h f j, f ji = 1, holomorphic cusp form f j with weight k and index j, and Maaß cusp form f j with spectral t f and index j, respectively, at cusp c.

Denote by ϕ ja (n, s) the n-th Fourier coeﬃcient of a normalized Eisenstein series with index P j at cusp a (for this part, see [DI83] and [BH10b] Section 2). In the following lemma, j means summing over a orthonormal basis of holomorphic cusp forms, Maaß cusp forms, and Eisenstein series respectively.

Under the normalization above, we state the Kuznietsov Trace formula [DI83] Theorem

1 with cusps a and b.

Lemma 2.4.3. (The Kuznietsov Trace Formula) Let m, n be two positive integers and ϕ a

C3-class function with compact support on (0, ∞). One has √ ! X 1 s¯ 4π mn X X √ e n S (mr¯, n; sC)ϕ √ = 4 ψ (∞, m)ψ ( 1 , n)ϕ ˜(k) r jk jk s C>0 s rC s rC k≡0(2) j (C,r)=1 X + 4π ρ (m)ρ 1 (n)ϕ ˆ(t j) j∞ j s j>1 ∞ ! ! X Z 1 1 + ϕ m, + it ϕ 1 n, + it ϕˆ(t)dt, j∞ 2 j 2 j −∞ s

43 and √ ! X 1 s¯ 4π mn X √ e n S (mr¯, −n; sC)ϕ √ = 4π ρ (m)ρ 1 (n)ϕ ˇ(t j) j∞ j r s C>0 s rC s rC j>1 (C,r)=1 ∞ ! ! X Z 1 1 + ϕ m, + it ϕ 1 n, + it ϕˇ(t)dt j∞ 2 j 2 j −∞ s

1 where the coeﬃcients on the eighth hand side indexed by ∞ and s correspond to the cusps

1 a = ∞ and b = s respectively, and the Bessel transforms are deﬁned by

Z ∞ l dy ϕ˜(l) = i Jl(y)ϕ(y) , 0 y i Z ∞ dx ϕˆ(t) = (J2it(x) − J−2it(x)) ϕ(x) , 2 sinh(πt) 0 x 4 Z ∞ dx ϕ t πt K x ϕ x . ˇ( ) = 2 cosh 2it( ) ( ) π 0 x

Remark 2.4.4. In [DI83] Thm. 1, for a cusp a, coeﬃcients are normalized such that

!1/2 !1/2 cosh(πt j) cosh(πt j) (q|t |)− |ρ (1)| (q|t |) , j q ja j q

(qk)− (qk) |ψ jk(a, 1)| . (qΓ(k))1/2 (qΓ(k))1/2 However, in the above lemma, by [HL94], coeﬃcients are normalized such that

− (q|t j|) (q|t j|) |ρ (1)| , q1/2 ja q1/2

(qk)− (qk) |ψ (a, 1)| . q1/2 jk q1/2

For trace formula under our normalization, see [BH10b].

Remark 2.4.5. The sum over j for the Eisenstein series part is a ﬁnite sum over a “suit-

able” parametrization of Eisenstein series. The classical parametrization, which is also

the one in [DI83], is the set of the cusps of Γ0(q). In this thesis, we use another natural

44 basis from the adelic reformulation of the theory of cusp forms, see [GJ79], [BHM07b],

and [BH10b] for details. By choosing this basis, we have follows ([BHM07b] Section 2).

(1) The sum ∞ ! ! 1 X Z m−it 1 1 ϕ m, + it ϕ 1 n, + it ϕˆ(t)dt π n j∞ 2 j 2 j −∞ s is a ﬁnite sum in both cases.

(2) There are complex numbers λ j(n, t) τ(n) and a character χ j such that ! ! 1 X nm 1 λ (n, t)ϕ m, + it = χ (d)ϕ , + it , j j∞ 2 j j∞ d2 2 d|(m,n)

when (n, q) = 1.

Next, we need an estimation of weight functions that appeared in the trace formula. We

quote a part of [Pit13] Lemma 2.1 as follows.

Lemma 2.4.4. ([Pit13] Lemma 2.1) Let X > 0,Z > 1, k ∈ N. If ϕ is supported on [X, 2X] v (v) Z with derivatives of orders v = 0, 1, 2,... 2k bounded by ϕ X , then the following bounds hold.

(a) For any real t and any l 6 k,

!l ! Z2 + t2 1 + | log(X/Z)| ϕˆ(t), ϕˇ(t), ϕ˜(t) . X2 1 + X/Z

(b) For all real t with |t| > max{2X, 1} and for all j 6 2k,

! j ! Z 1 X ϕˆ(t), ϕˇ(t), ϕ˜(t) + (1 + log |t|) . |t| |t|1/2 |t|

(c) For exceptional eigenvalues λ = 1/4 + (it)2, we take t ∈ (0, 1/2),   Z if X > Z  X ϕˆ(it), ϕˇ(it) 2t  Z Z {1 + log X } log X if X < Z.

45 For those coeﬃcients that appeared in the Kuznietsov Trace formula, we have the fol-

lowing large sieve type inequality.

Lemma 2.4.5. ([DI83] Theorem 2) Let T, N > 1. Set Γ = Γ0(q). Let a = u/w be a cusp of Γ where w|q and (u, w) = 1. Let ρ ja, ψ jk, ϕ ja be the ones in Lemma 2.4.3. Then, for any sequence of complex numbers {an} ,

2 X X a ρ (n) n ja |t j|6T n∼N 2 X X X a ψ (a, n) n jk k=0(2),k6T j n∼N 2 X Z T X ! 1 anϕ ja n, + it dt 2 j −T n∼N

2 1+ 2 −1 are all bounded by O T + µ(a)N ||an||2 , where µ(a) = (w, q/w)q .

Furthermore, we have

Lemma 2.4.6. ([KMV02] Prop. 5.1) Let X, Y, C > 0 be real numbers. Let N > 0 be an

( j) − j integer. Let η be a smooth function supported on [C/2, 5C/2] such that kη k∞ j C for

all j ≥ 0. For any sequences of complex numbers xn, ym we have √ ! X X X η(c) 4π mn x y S (n, m; c)J n m c k−1 c n≤X m≤Y c>0 c≡0(N) √ −  k 3/2 1/2 1/2  XY  X Y ,k C   1 + 1 + kxnk2kymk2  C  N N with any > 0. Moreover the exponent k − 3/2 may be replaced by 1/2.

46 2.5 Jutila’s version of circle method

Denote by

X∗ X := . a mod q 0

For a collection of integers Q ⊆ [1, Q], and a positive real number δ such that Q−2

δ Q−1, we deﬁne the function

1 X X∗ I˜Q,δ(x):= I[ a −δ, a +δ](x), 2δΛ q q q∈Q a mod q P where Λ = q∈Q ϕ(q), ϕ(·) is the Euler totient function, and I[a,b](x) is the characteristic

function of the interval [a, b]. I˜Q,δ(x) is an approximation of I[0,1] in the following sense:

Lemma 2.5.1. Z ∞ 2+ 2 Q I x − I˜ x dx . [0,1]( ) Q,δ( ) 2 −∞ δΛ

Proof. This proof is given in [Mun13b] Lemma 4. Let

Z 1   Z δ 1 X  X∗  an := I˜Q,δ(x)e(−nx)dx =  eq(−an) e(−nx)dx 2δΛ   0 q∈Q a mod q −δ ˜ P∗ be the n-th Fourier coeﬃcient of IQ,δ. The sum a mod q eq(−an) is the Ramanujan sum cq(n), P which can be bounded by d|(n,q) d. We have that a0 = 1, and for general an we have the

bound 1 X X 1 X X 1 X X Q|n| |a | d d d 1 . n Λ Λ Λ Λ q∈Q d|(n,q) q6Q d|(n,q) d|n q6Q d6Q q≡0(mod d) R δ e −nx dx O |n|−1 On the other hand, via integration by parts, we can also bound −δ ( ) by ( ). Then we have Q|n| |a | . n δΛ|n| 47 Therefore, by Parseval’s formula, we have

2 Z 1 Z 1 2 X I − I˜ dx = a e(nx) dx [0,1] Q,δ n 0 0 n,0 !2 !2 X X Q|n| X Q|n| Q2+ = |a |2 + . n Λ δΛ|n| δΛ2 n,0 |n|6δ−1 |n|>δ−1

2.6 Past works on hybrid subconvexity bounds for Rankin-Selberg L- functions

Let f and g both be GL2 Hecke cusp forms with levels M and N respectively. We consider the Rankin-Selberg L-function L(s, f × g). For most cases, this is an L-function of a GL4 automorphic form (see [Ram00] for further references ). In the classical language, a newform f (resp. g) with level M (resp. N) means that the automorphic representation generated by f (resp. g) has conductor M (resp. N). See [Cas73] for further references.

Thus, the Lindelof¨ Hypothesis suggests that

1 × L 2 , f g (MN) ,

and the convexity bound only gives

1 1 + × 2 L 2 , f g (MN) ,

at least when M, N are square-free and coprime.

1 × Our main goal in this thesis is to establish a hybrid subconvexity bound for L 2 , f g , at least when M, N are both square-free and coprime.

48 Some authors have successfully established level aspect subconvexity results via an

ampliﬁcation method when one form is ﬁxed. For example, if f is a Hecke cusp form of a

ﬁxed level M and g is a Hecke cusp form of a varying level N, then various bounds of the

form

1 1 −δ × 2 L 2 , f g f N

for some absolute positive constant δ have been shown by Kowalski-Michel-VanderKam

[KMV02], Michel [Mic04], and Harcos-Michel [HM06].

The basic idea is using a second moment estimation, the ampliﬁcation method, and a

trace formula. We study the sum of norm squares of L-functions over a set of Hecke cusp

forms F (In general, F will be a basis of holomorphic cusp forms of a ﬁxed weight or Maaß

cusp forms with spectral parameters in certain range ):

X 2 S 1 × := L 2 , f g . f ∈F By the Approximate Functional Equation (Lemma 2.2.1 ), we have 2 X X λ f (n)λg(n) S M √ . n f ∈F nN1+ Then, opening the squares and applying the trace formula (see Chapter2 ), we are left to estimate the following twisted sum of Kloosterman sums and λg(·) (the explicit formula will be given in Chapter4).

2 X |λ f (n)| X X 1 λ f (n)λ f (m) |F| + S (n, m; d) √ (2.6.1) n d nm nN1+ n,mN1+ dN1+ d≡0(M) In the above formula, we call the ﬁrst term the Diagonal term, and the second one the

Oﬀ-diagonal term. By a well-known Rankin’s bound (Lemma 2.1.2 ), we have X |λ (n)|2 f N2. n nN1+ 49 For the Oﬀ-diagonal term, we can still bound it by NA|F|−B for some positive numbers A

and B (see [KMV02] ). Therefore, for a ﬁxed M, without using the ampliﬁcation method,

we can only obtain the Lindelof¨ Hypothesis on average and convexity bounds for any indi-

vidual L-function as F1/2+ N. Instead, we consider a weighted sum

2 ∗ X X xnλ f (n)λg(n) S := √ . n f ∈F nN1+

Just like the phenomena described in the GL1 case (Section 1.2.1 ), by carefully choosing

the weight xn, we can expect further savings and obtain subconvexity bounds.

Furthermore, based on a ﬁrst moment method, the subconvexity bounds for two inde-

pendently varying forms have been established in the works of Michel and Ramakrishnan

[MR12], Feigon-Whitehouse [FW09], Nelson [Nel13] and Holowinsky-Templier [HT14]

in situations where positivity of the central L-values is known. The positivity in their works 1 × is necessary since they estimated L 2 , f g by considering the sum of ﬁrst moment of L- functions. Indeed, one can only claim that

X 1 × 1 × L 2 , f g 6 L 2 , f g , f ∈F when each term in the sum on the right is nonnegative.

1 × For a second moment argument of the hybrid subconvexity problem for L 2 , f g , we only have conditional results. For example, Holowinsky and Munshi [HM13] proved a hybrid subconvexity bound when the levels of f and g satisfy certain conditions via a second moment estimation.

50 The difficulty of this argument is roughly that, when we consider the Off-diagonal term in (2.6.1) with upper bound O(NA|F|−B), it is hard to minimize A and to maximize B simultaneously when both N and |F| approach infinity. In this thesis, we will resolve this difficulty and give a hybrid subvonvexity bound via a second moment argument when the two levels are coprime and both square-free.

2.7 Past works on Sup-norm bounds for modular forms

In our proof of the hybrid subconvexity bound, the level aspect sup-norm bound plays an essential role.

2 Let f be a Maaß form on SL2(Z)\H such that k f k2 = 1. The general result on Rie-

1/4 mann surfaces claims that the sup-norm k f k∞ λ , where λ f is the Laplacian eigenvalue associated to f (see, for example, [SS89]). In [IS95], the authors gave the ﬁrst nontrivial

5/24+ upper bound k f k∞ λ f using a pretrace formula and the ampliﬁcation method. Their strategy can be sketched as follows.

By the pretrace formula (see [IS95]), we roughly have that

X 2 X | f (z)| ∼ uT (ρ(γ.z, z)),

|λ f |∼T γ∈SL2(Z) where uT (·) is a rapid decaying function depending on T, and ρ(z, w) is the hyperbolic distance between two points z, w in H2 deﬁned as

|z − w¯ | + |z − w| ρ(z, w):= ln . |z − w¯ | − |z − w|

51 Observing the fact that ρ(γ.z, z) → ∞ when γ → ∞ in SL2(R), we can bound the right

hand side from above. By carefully estimating the sum on the right, for bounded z, one obtains X | f (z)|2 T 1/2+,

|λ f |∼T which corresponds to the best possible sup-norm bounds O(λ f ) on average and the convex-

1/4+ ity bound O(λ f ) for an individual f .

Now, we can expect further savings using an ampliﬁcation method. Applying the Hecke operator Tl to the pretrace formula, we can obtain that

X 2 X 1 λ f (l)| f (z)| ∼ √ uT (γ.z, z).

|λ f |∼T γ∈M2(Z) l det γ=l Then, consider the weighted sum

X X 2 X X 1 xlλ f (l)| f (z)| ∼ xl √ uT (γ.z, z).

l∼L |λ f |∼T l∼L γ∈M2(Z) l det γ=l

By carefully choosing the weight xl and the integer L, one achieves the Iwaniec-Sarnak

bound

5/24+ k f (z)k∞ λ f .

1/6− It is also known that the lower bound exists such that k f k∞ λ by estimating the

ﬁrst few Whittaker functions in the Fourier expansion of f , see [Tem14] for example.

For a holomorphic modular form with weight k, the best possible bound is established

1/4− k/2 1/4+ in [Xia07], where the author proved that k h f, f i ky f (z)k∞ k h f, f i for any

52 positive .

For the level aspect sup-norm, the case is more complicated. For a Maaß or holomorphic modular form f with weight k f and normalization Z yk f −2| f (z)|2dxdy = 1, 2 Γ0\H

the “trivial” bound is k f k∞ N . The ﬁrst breakthrough in the nontrivial level aspect sup-

−25/914+ norm was achieved in [BH10b] recently as k f k∞ N for N square-free. Improved

bounds were then obtained in [Tem10][HT12], and the best known bound so far is as

follows.

Proposition 2.7.1. (Theorem 1.1 [HT13]) Let f ∈ B∗(N) be a Maass Hecke newform of

square-free level N. Then for any > 0 we have

1 − 6 + 1/2 k f k∞ t f , N h f, f i ,

where the implied constant depends continuously on t f .

1− 1+ Remark 2.7.1. Under the normalization ψ f (1) = 1, we have N h f, f i N by

[HL94], so that

1 3 + k f k∞ t f , N .

The square-free condition is important in all the proofs of level aspect bounds. For the

−1/2+ level aspect sup-norm, it was conjectured that k f k∞ N (see [JK04] Rem. 3.3).

However, when N is a perfect square, N. Templier constructed a modular form with sup-

norm at least k1/4−t1/6− N−1/4− in [Tem14], where k is the weight and t is the eigenvalue.

This construction gave a counterexample to the conjecture above for N not square-free. It

53 is still not known that whether the conjecture is true in square-free case.

1/2 −1/6+ 1 In this thesis, we will give a proof of k f k∞ k N h f, f i 2 , when f is a holomorphic modular form with weight k and square-free level N (This bound is ﬁrst announced in

[HT13], however, there is no written proof ). The proof appears to be more diﬃcult than

the one in [HT13] for Maaß form case. Additional counting techniques are needed in this

case due to diﬀerent weight functions in the pretrace formula.

2.8 Past works on the distribution of inverses mod c

For any integers c and a such that (a, c) = 1, denote bya ¯ the mod c reciprocal such that aa¯ = 1(mod c) and 0 < a¯ < c.

Let F and G be two sets of basis of GL2 holomorphic Hecke cusp forms with ﬁxed

weights and coprime levels M and N respectively. By considering the second moment of

L-functions, the Approximate Functional Equation, and the trace formula, one roughly has that (see Chapter4 for details )

1 X X 2 X X 1 L 1 , f × g 1 + S (n, m; c)S (n, m; d)Φ(n, m, c, d), |F||G| 2 cd f ∈F g∈G n,m(MN)1+ M|c N|d for some fast decaying weight function Φ, where S (n, m; ·) is the Kloosterman sum that will be deﬁned in Chapter2.

54 Opening the two Kloosterman sums by their deﬁnition, we turn to study the twisted

exponential sums ! ! ! ! X X X X a b a¯ b¯ e − n e − m . c d c d n,m(MN)1+ M|c a(mod c) b(mod d) N|d (a,c)=1 (b,d)=1

Summing over m, n ﬁrst, and using both the summation formula for geometric series as

well as |e (x) | 6 1 for any real number x, one obtains an upper-bound of the sum above as

 −1 X X X ( −1)  ¯  1+ a b  1+ a¯ b  min (MN) , − min (MN) , − , c d  c d  M|c a(mod c) b(mod d)   N|d (a,c)=1 (b,d)=1

where kαk is the distance between the number α and the set of integers Z.

Therefore, in order to bound the above triple sum, one can study the distribution of

− a b −1 a¯ b¯ 1 a a¯ c − d and c − d . In this sense, the distribution of ( c , c ) is closely related to the study of second moment of Rankin-Selberg L-functions.

There is a famous congruence question about the distribution of pairs (a, a) on [1, c] ×

[1, c] (see [Shp12] ).

Question 1. Given any > 0. Is it true that, for any modulus c > 1 and any integer b with (b, q) = 1, the congruence equation q1q2 ≡ b(mod c) is solvable for some

1/2+ 1 6 q1, q2 c ?

Davenport [Dav40] used Kloosterman sum estimates to show that the above question is true for all > 1/3. Using Weil’s bound on Kloosterman sums, Davenport’s argument implies the truth of the above question for all > 1/4. More explicitly, his argument implies that:

55 Proposition 2.8.1 (Lemma 5 [Mer96]). Suppose that I (resp. J) is an interval in [0, 1]

n a a o with length A (resp. B), then the cardinality of the set a : c , c ∈ I × J is given by ABc + O c1/2 log c2 .

Remark 2.8.1. This proposition estimates the number of solutions for Question 1 when b = 1, q1 = a, and q2 = a.¯ The best result so far of the above question is established in

[Gar06] for all > 1/4. However, because of the error term O c1/2 log c2 , the above proposition can only show

−1/2 a a¯ that when I, J are long enough (when AB > c ), the number of pairs c , c ∈ I × J is proportional to its area.

n a a o We would like to have a sharper estimation such as # a : c , c ∈ I × J ∼ ABc, when A, B are small. Unfortunately, it is not an easy question for any AB < c−1/2. It involves the

deep arithmetic property of (Z/cZ)×. However, if we consider the question on average over

b or over all the intervals I and J, the question can be answered for almost all cases, see

[Cha11] and a survey on this question [Shp12].

In this thesis, we consider the same question over a larger group of modulus c, so that

the distribution question can be answered on average. Since we are going to consider more

then one modulus, we restate the problem in a Diophantine approximation fashion. In the

a¯ following statement,a ¯ in c always means the inverse mod c.

Remark 2.8.2. For simplicity, we only consider the case of b = 1 in this thesis. Our method

can be applied to any b.

56 Now, the expected estimation can be formulated as ( ! ) a a # (a, c): X < c < 2X, , ∈ I × J ∼ (1 + o(1))ABX2, (2.8.1) c c

where A, B are lengths of I, J respectively. However, this estimation cannot be always true

for any I, J with A, B 1/X because of the following counterexample.

f g A counterexample to (2.8.1). Assume that I, J are centered at h , h respectively, both with length 2X−1, where f + g = h. Then consider those pairs of rational numbers f hK+1 ghK+1 ≡ 2 h2K , h2K . We can check that (ghK + 1)( f hK + 1) 1mod h K. Therefore, when- X f hK+1 ghK+1 ∈ × X ever K 6 h2 , h2K , h2K I J. So, in this particular case, we have at least h2 many pairs (a, c) = ( f hK + 1, h2K) such that X < c < 2X. However, (2.8.1) claims that there are

only (1 + o(1))ABX2 = O(1) many such pairs.

This counterexample indicates that when the centers of I, J are close to rational num-

bers with small denominators, the pairs of inverses mod c are not equidistributed on aver-

age. By the results we will introduce in the following chapter, this is the only case where

the equidistribution property does not hold.

57 Chapter 3: Main Results

3.1 Hybrid subconvexity bounds for Rankin-Selberg L-functions

Proposition 3.1.1. Let f be a holomorphic (resp. non-exceptional Maaß) Hecke newform with weight k > 2 (resp. spectral parameter t f and weight 0) and level N. Let h be a

1 5 k ( j)k j smooth function supported on [ 2 , 2 ] such that its j-th derivative h ∞ j Zh for all j > 0. Assume that N is square-free and (M, N) = 1. Then we have 2 X − X λ f (n)ψg(n) n ω 1 √ h g n X g∈Bκ(M) n∈Z   r   X X (1 + Z )26 N4/3  X  1 + + + h 1 +  ((1 + Z )XMN), , f,κ  MN M1+β M1/3+β  MN  h

11 κ−1 h i where β = 5184 and ωg := 4π g, g .

4 Remark 3.1.1. For k = 2 and f holomorphic, we can only obtain the bound with N 3 re-

3 placed by N 2 in the 4-th term on the right.

Theorem 3.1.1. Let M, N be positive coprime integers and both square-free. Then, for

any holomorphic (resp. non-exceptional Maaß) Hecke newform f with weight k > 2 (resp. spectral parameter t f and weight 0) and level N, we have

  1   X 2   N 3   L( j) 1 + it, f × g M + 1 + (1 + |t|)52  M1−βN (1 + |t|)2+ (MN), 2 , f,κ, j   1   ∗ M 3 g∈Bκ(M) 11 where β = 5184 . 58 ( j) 1 × Remark 3.1.2. L 2 + it, f g is the j-th derivative of L as a function of s. Neither the exponent (1 + |t|)52 nor the power saving β is optimal.

Theorem 3.1.2. Under the condition of Theorem 3.1.1, and if N < M, there are eﬀective constants α, B > 0 such that

( j) 1 B 1 −α × | | 2 L 2 + it, f g , j, f,κ (1 + t ) (MN) ,

where the implied constant does not depend on the non-archimedean conductor N of f .

Remark 3.1.3. By combining our results and the proofs in [KMV02], α can be at least

1/1602. We can expect a sharper bound by using the ampliﬁcation method in Section 4.3.

Another approach to the hybrid subconvexity problem can be found in [MNV], where the

authors are able to establish subconvexity for more general cases.

The key ingredient to establish the theorems above is the following Large Sieve type

inequality.

Theorem 3.1.3. Let Z, V, H, Q, D > 1 be real numbers and let u(v, h, q, d) be a smooth

function supported on [V, 2V] × [H, 2H] × [Q, 2Q] × [D, 2D] with the (i, j, k, l)-th derivative

(i, j,k,l) i+ j+k+l −i − j −k −l ||u ||∞ 6 Z V H Q D

for all i, j, k, l > 0. Let r, s be positive integers s.t. (r, s) = 1. Let w be a positive integer such that (w, rs) = 1. Let a(v), b(h, d) be two ﬁnite sequences of complex numbers. Then X X 1 S := S (vr, ±hw; sq)a(v)b(h, d)u(v, h, q, d) ± q q∈Z d,h,v∈Z (q,r)=1 − θ  Ξ 2   r   r  √ 1 +  V H  Z      8 θ+ s r   Z + Ξ +  Z + Ξ +  ka(v)k2kB(h)k2(1 + Z )w (VHZ) ,  Z + Ξ   rs  rs

59 √ VHw√ P | | where Ξ = s rQ ,B(h) = D6d62D b(h, d) and θ is the Ramanujan bound for cusp forms on Γ(rs)\H2.

Remark 3.1.4. By Kim-Sarnak [Kim03], θ could be as small as 7/64 for any r, s. If we

assume the Ramanujan Conjecture for the discrete group Γ(rs), then the statement holds

true with θ = 0. Moreover, the condition (w, rs) = 1 is necessary in our proof. This

inequality is inspired by results in [DI83], [Blo04] and [Pit13].

Remark 3.1.5. This inequality is a generalization of [DI83] Theorem 13 (which is the case

w = 1 and D = 2/3). However, after directly applying the result in [DI83], one can only q q Hw H θ obtain Z + Ξ + sr in the 3rd parenthesis on the right, rather than Z + Ξ + sr w . Since we are going to apply this large Sieve type inequality when w is very large, our improvement is crucial.

Remark 3.1.6. There is no saving in the sum over d. However, we need the d-sum here for our application, since the weight function u contains an additional variable d whose support depends on the other variables v and h.

3.2 Sup-norm bounds for holomorphic modular forms

Theorem 3.2.1. (Sup-norm for holomorphic case) Let F be a holomorphic cusp form with square-free level N, weight k > 2. Let f (z) ∈ Bk(N) such that

k/2 X − 1 k k−1 y F(z) = f (z) = Γ(k) 2 (4πy) 2 ψ f (n)n 2 e(nz). n>0

Then for any > 0, we have

1 − 1 + 1/2 k f (z)k∞ k 2 N 6 h f, f i . (3.2.1)

60 1 1/2 Remark 3.2.1. The trivial sup-norm bound is k 2 N h f, f i in this case. This result is ﬁrst claimed in [HT12]. However the author of this thesis is not aware of any written proof.

Remark 3.2.2. Under the normalization ψ f (1) = 1, we have

N1− h f, f i N1+

by [HL94]. So that

1 1 + k f (z)k∞ k 2 N 3 .

3.3 The distribution of inverses mod c

a¯ In the following theorem, denote by c the rational number with denominator c, and integer numeratora ¯ such that aa¯ ≡ 1(mod c). Let ϕ(·) be the Euler totient function.

Theorem 3.3.1. Let X 1 and 0 < α, β < 1 be real numbers. For any X1, X2 X,

p1 let q1, q2 be the smallest positive integers such that there are p1, p2 satisfying |α − | q1 1 , |β − p2 | 1 . Then for any smooth functions h(·, ·) and g(·) with compact supports on X1 q2 X2 R2 and R respectively, such that h(i, j)(·, ·) Zi+ j and g(i)(·) Zi, we have ∞ P c X X a a¯ c c ϕ(c)g h X − α , X − β g = h(x, y)dxdy X 1 c 2 c X X X c>0 0

Remark 3.3.1. One can see from the example in Section 2.8, the optiomal bound is ∞ P c ! c ϕ(c)g X h(x, y)dxdy X + O −∞ X1X2 q1q2 " Remark 3.3.2. This theorem gives an answer to a smoothed version of the counting problem such that, for which A and B, we have

a a¯ # (a, c): X < c < 2X, ∈ [α − A, α + A], ∈ [β − B, β + B] ≈ 2−1ABX2. c c

61 −1+δ Our result suggests that as long as A, B > X /q1q2 for any ﬁxed δ, the above estimation

holds true. The above question is the question 15 in [Shp12].

Finally, α is said to be Diophantine, if there is a positive constant C(α, ) which only

depends on α and , such that a C(, α) α − > q q2+

a −1+δ for any rational number q . Then, from the above Theorem 3.3.1 with A = B = X , we have the following corollary.

Corollary 3.3.1. If both α and β are Diophantine, we have ! !! P c X a a¯ c c ϕ(c)g h X1−δ − α , X1−δ − β g = X2δ h(x, y)dxdy X + O Z6Xδ , q q X X2 c>0 0

for any δ > 0.

Remark 3.3.3. The set of Diophantine numbers has full measure in the real numbers

[RS98].

62 Chapter 4: The Proof of Hybrid Subconvexity Bounds for Rankin-Selberg L-functions

4.1 The deduction of Theorem 3.1.1 and Corollary 3.1.2 from Propo- sition 3.1.1

In this section, we prove Theorem 3.1.1 and Corollary 3.1.2 by assuming Proposition

3.1.1.

∗ ∗ Proof of Theorem 3.1.1. Fix positive even integers k and κ. Let f ∈ Bk(N) ∪ B (N) and

∗ g ∈ Bκ(M). Let M and N be coprime integers with N square-free. In this case, Q := Q( f × g) = (NM)2 is the conductor of L (s, f × g) by Section 2.2.

1 |Re | 2 Assume that s = 2 + µ in Lemma 2.2.1, where (µ) < log Q . From the Approximate Functional Equation (Lemma 2.2.1 ), we have

1 × L 2 + µ, f g = X∞ ! X∞ ! λ f (n)λg(n) −µ n 1 λ f (n)λg(n) µ n √ n V 1 √ + f × g, + µ √ n V 1 √ . 2 +µ 2 2 −µ n=1 n Q n=1 n Q

1 5 Assume that h0(x) is a positive smooth function, compactly supported on [ 2 , 2 ] with bounded derivatives. And when X runs over values 2v with v = −1, 0, 1, 2,... , for any

x > 1, X x h = 1. 0 X X 63 Remark 4.1.1. Such function h0 exists. For example, one can choose  t(4x − 3) x ∈ ( 3 , 1)  4 1 x ∈ [1, 3 ] h (x) =  2 0  − − ∈ 3 1 t(2x 3) x ( 2 , 2)  0 otherwise.

where ! πx−1 t(x) = exp tan . 2

After applying this smooth partition of unity, one obtains that

X 1 × × 1 ± L 2 + µ, f g t f ,κ,A, 1 + ( f g, 2 + µ) L f ×g(X) , ± X=2v where ! ± X λ f (n)ψg(n) n ±µ n L f ×g(X):= √ h0 n V 1 ±µ √ , X 2 n n Q

since ψg(n) = λg(n).

× 1 By the deﬁnition of ( f g, 2 + µ) and Stirling’s formula (see [Spi71])

∼ − 1 − 1 −1 ln Γ(z) z 2 ln z z + 2 ln(2π) + O(z ), we have that

Q4 1 ΓR − µ + µ f ×g,i(∞) 1 −µ i=1 2 ( f × g, + µ) = Q ,t f ,tg (1 + |µ|) . 2 Q4 1 ∞ i=1 ΓR 2 + µ + µ f ×g,i( ) Let ! ± x x ±µ x h := h0 x V 1 ±µ √ . X X 2 Q

64 From (2.2.1), we have

 −A0 ± x −1  X  h log(1 + Xx )(Q(|µ| + 1)) 1 + √  . X (|µ| + 1)2 Q √ 2 Then, for X (1 + |µ|) QM , we choose A0 = 100/ to obtain     X X  L 1 + µ, f × g ((1 + |µ|)Q)  L± (X) + M−50 . 2 t f ,κ,  f ×g   √   ± X(1+|µ|)2 QM  X=2v √ Also for such h±(x), and X (1 + |µ|)2 QM, by (2.2.1), one can verify that

j ±( j) −1 2 j x h (x) j log(1 + x )(Q(|µ| + 1)) (1 + |µ|) .

∗ Next, applying Cauchy-Schwartz ﬁrst and summing over all g ∈ Bκ(M), we have

X 2 −1 1 × ωg L 2 + µ, f g ∗ g∈Bκ(M)     X X X 2  ((1 + |µ|)Q)  ω−1 L (X) + M−50 . t f ,κ,  g f ×g   √   ± X(1+|µ|)2 Q(M) g∈Bκ(M)  X=2v Eventually, applying Proposition 3.1.1 with weight functions h±, we have

X 2 −1 1 × ωg L 2 + µ, f g ∗ g∈Bκ(M)   1     N 3   1 + 1 + (1 + |µ|)48  M−βN (1 + |µ|)2+ (MN) . (4.1.1)   1   M 3

∗ 1− Since g ∈ Bκ(M) is normalized such that ψg(1) = 1, we have that M ωg = κ−1 h i 1+ 4π g, g M (see [HL94]). Therefore, we can obtain the bound in Theorem 3.1.1 for j = 0.

Finally, since L (s, f × g) is holomorphic ([Mic04] Section 3 for further references ), by

Cauchy integral

Z 2 2 (L (s + w, f × g)) L( j) (s, f × g) j dw. = ! j+1 |w|=log−1 Q w

65 1 ∈ B∗ We then complete our proof by taking s = 2 + it, summing over g κ(M), and applying

1 (4.1.1) for µ = 2 + it + w.

Proof of Corollary 3.1.2. Assume that N < M. Then from [KMV02], there are eﬀective

numbers A, C > 1 and δ > 0 such that

1 C A 1 −δ+ × | | 2 L 2 + it, f g (1 + t ) N M , (4.1.2)

where A > 1.

Theorem 3.1.1 gives that

1 49 1 + − 1 − β × | | 2 2 2 L 2 + it, f g (1 + t ) (MN) N + M , (4.1.3)

where β = 11/5184.

1 δ x δ 4 − 4(A+δ) + δ Let, N = M . When x 6 A ,(4.1.2) is bounded by Q . When A < x 6 1, (4.1.3)

βA 1 + − δ − is bounded by Q 4 Q 4(A+δ) + Q 4(A+δ) .

1 Remark 4.1.2. By carefully going through the proof in [KMV02], we can choose δ = 80

1 − 1 + and A = 10. So the ﬁnal bound can be Q 4 3204 . One may use the ampliﬁcation method in

our argument to get a sharper bound.

4.2 A sketch proof of Propositon 3.1.1

In this section, we provide a sketch of the proof. It follows the same lines as in [HM13]

until step 7.

For simplicity, we assume that M = Q, N = P are two diﬀerent primes in this section.

∗ Let f ∈ Bk(P) be a holomorphic Hecke newform. Its Fourier coeﬃcients are normalized

66 1/2 such that ψ f (n) = λ f (n). Furthermore, we restrict to the case of X ∼ Q = PQ. Therefore, we will only show the following in this sketch

2 X X λ (n)ψ (n) 1/3 ! −1 f g 1 1 P ω √ ,k,κ P + + (PQ) , g n P Qβ Q1/3+β g∈Bκ(Q) n∼PQ

where n ∼ PQ means that n varies from PQ to 2PQ.

Step 1. Reducing to a sum of Kloosterman sums via the trace formula. Now, after

expanding the square and applying Petersson’s trace formula (Lemma 2.4.2), we have

2 X − X λ f (n)ψg(n) ω 1 √ = g n g∈Bκ(Q) n∼PQ 2 √ ! X |λ f (n)| X X 1 λ f (n)λ f (m) 4π mn + S (m, n; d) √ J . n d κ−1 d n∼PQ d≡0(Q) n∼PQ nm m∼PQ

Rankin’s bound (Lemma 2.1.2 ) and summation by parts tell that the ﬁrst term is

bounded by a constant of size at most (PQ).

Step 2. Removing large and small values of D. Now, through the idea in section

4.3, we can truncate our d-sum into the range such that d ∼ PQ with a loss at most of size √ 1 1 ∼ P P + Qβ . Next, mn/d 1 in this range, which is also the transition range for the Bessel function. Therefore, we may assume that Jκ−1 behaves like the constant function 1 in this

range.

Thus, we may simply consider

X X 1 λ f (n)λ f (m) R := S (m, n; d) √ . f d d≡0(Q) n∼PQ nm d∼PQ m∼PQ

67 Step 3. Applying the Voronoi formula to convert Kloosterman Sums to Ramanujan

sums. Now, applying the Voronoi summation formula (Lemma 2.4.1) on the n-sum, we

have

X X 1 X∗ am¯ + an λ f ∗ (n)λ f (m) R = e √ f d d d≡0(Q) n∼PQ a(d) nm d∼PQ m∼PQ

X X 1 X∗ am¯ an¯ λ ∗ (n)λ (m) ≈ e e − f √ f d d d d≡0(PQ) n∼PQ a(d) nm d∼PQ m∼PQ   X X 1 X∗ am¯  aPn λ f ∗ (n)λ f (m) + e e −  √ d d  d  nm d≡0(Q) n∼P2Q a(d) d.0(P) m∼PQ

X X λ f ∗ (n)λ f (m) X X λ f ∗ (n)λ f (m) ≈ √ + √ . (4.2.1) d≡0(PQ) n≡m(d) nm d≡0(Q) Pm≡n(d) nm d∼PQ n,m∼PQ d∼PQ n∼P2Q d.0(P) m∼PQ The ﬁrst sum contains only a bounded number of terms with respect to d, and therefore

the sum is bounded by (PQ) by Rankin’s result (Lemma 2.1.2 ) and Cauchy’s inequality.

Remark 4.2.1. The main diﬀerence between the trivial nebentypus case and nontrivial

nebentypus case is that, after applying Voronoi summation formula in the ﬁrst case, we get

Ramanujan sums rather than Gaussian sums. See [HM06] for the case of Gaussian sums.

Step 4. Treating the Zero Shift. For the second term in (4.2.1), we consider ﬁrst the

case that Pm = n. we have

X X λ f ∗ (n)λ f (m) √ (PQ). d≡0(Q) Pm=n nm d∼PQ n∼P2Q d.0(P) m∼PQ Here we used Rankin’s bound (Lemma 2.1.2 ), multiplicativity of Hecke-eigenvalues (2.1.2),

−1/2 and the fact λ f (P) = P (2.1.3).

68 Step 5. Applying the Circle Method. Let Pm − n = rd for some r. We are left with

the case that r , 0.

We apply the circle method to detect the relation Pm−n = rd for some nonzero integers r (Section 2.5 ) and have ! X X λ ∗ (n)λ (m) X X X 1 X∗ a(Pm − n − rd) λ ∗ (n)λ (m) f √ f ≈ e f √ f . cC c d≡0(Q) Pm≡n(d) nm d≡0(Q) 0<|r|P c∼C a(c) nm d∼PQ n∼P2Q d∼PQ n∼P2Q d.0(P) m∼PQ d.0(P) m∼PQ Since we are using Jutila’s version of circle method (Section 2.5 ), one can assume that

C is suﬃciently large and that (c, P) = 1. This choice of c is crucial in our proof.

Step 6. Applying Voronoi formula twice to regenerate Kloostermann sums. We

apply Voronoi summation formula (Lemma 2.4.1) for both the n and the m sums to get

X X X 1 λ f ∗ (n)λ f (m) S P2(m − Pn), rd; c √ . cC d≡0(Q) 0<|r|P c∼C nm d∼PQ n∼C2/PQ d.0(P) m∼C2/Q

Next, set v = m − Pn, so that v C2/Q and our sum becomes   X X X X 1  X λ f (n)λ f ∗ (m) S P2v, rd; c  √  . cC  nm  d≡0(Q) 0<|r|P v∼C2/Q c∼C m−Pn=v d∼PQ d.0(P)

Step 7. Applying the large sieve type inequality to the sum of Kloostermann sums.

At this stage of [HM13], the authors simply apply the Weil’s bound for Kloosterman sums.

In order to obtain Proposition 3.1.1, we apply the large sieve type inequality (Proposition

3.1.3) witht h = rd/Q, w = Q, r = P2, and s = 1. The sum above becomes   X X X 1  X λ f (n)λ f ∗ (m) b(h) S P2v, hQ; c  √  cC  nm  h∼P2 v∼C2/Q c∼C m−Pn=v

69 where X b(h) = 1 ≤ τ(h). rd0=h r,d0∼P d0.0(P) Notice that

 2 X X ∗  λ f (n)λ f (m) 2 +   3  √  P (4.2.2) v m−Pn=v nm

by the argument in Section 4.3.8. And

X X |b(h)|2 ≤ τ(h)2 P2+. h∼P2 h∼P2

Thus, recalling that C is suﬃciently large, R f is bounded by

√  r   r  4 2 2 −1 2 p √ 3 P  C Q   P  θ 2 + P 3 1 +  1 +  Q P 3 P2+ (PQC) (PQC) .  2   2  1 −θ C P P Q 2

Remark 4.2.2. In the last step above, through the large sieve inequality in [DI83], one can

only get the convexity bound. So our generalization of the large sieve inequality is crucial.

Also, (4.2.2), which is a direct consequence of the nontrivial sup-norm bound Theorem

3 −θ θ− 1 3.2.1, plays an essential role here. Since we need our ﬁnal bound to be less than P 2 Q 2 ,

1 −θ 1 a sup-norm bound better than P 2 is required ( the trivial one is P 2 under our normaliza-

tion ).

4.3 The proof of Proposition 3.1.1

Let f be a Hecke newform with level N. Let t f > 0 be a ﬁxed number which is the

spectral parameter of f when f is holomorphic or a non-exceptional Maaß form. Also

assume that N is square-free, M and N are coprime. Let h be a smooth function, compactly

1 5 ( j) j supported on [ 2 , 2 ] such that h Zh for a positive constant Zh and all j > 0. Let X > 1.

70 4.3.1 Step 1. Reducing to sums of Kloosterman sums via trace formula.

By Petersson’s trace formula (Lemma 2.4.2), we have 2 X − X λ f (n)ψg(n) n ω 1 √ h g n X g∈Bκ(M) n>1 2 √ ! X |λ f (n)| n X X 1 λ f (m)λ f (n) 4π mn n m = h2 + S (m, n, d) √ J h h . n X d κ−1 d X X n d≡0(M) n,m>1 nm Due to Rankin’s bound (Lemma 2.1.2 ), the ﬁrst term is bounded by O((XN) ). Now

we consider the second term.

P d v First, we use the partition of unity 1 = D h0 D , where D runs over values 2 with v = −1, 0, 1, 2,... . Furthermore, we can assume that h0(x) is a smooth function, compactly

1 5 supported on [ 2 , 2 ] with bounded derivatives. Finally, let

! √ ! X X 1 λ f (m)λ f (n) d m n 4π mn R (X, D):= S (m, n; d) √ h h h J , f d 0 D X X κ−1 d d≡0(M) n,m>1 nm such that 2 X X λ (n)ψ (n) X −1 f g n ω √ h = O((XN) ) + R f (X, D) (4.3.1) g n X g∈Bκ(M) n>1 D where D runs over values 2v with v = −1, 0, 1, 2,... .

4.3.2 Step 2. Removing large and small values of D.

In this section, we prove the following,

Lemma 4.3.1. 2 X − X λ f (n)ψg(n) n ω 1 √ h (4.3.2) g n X g∈Bκ(M) n>1 −β X ,t f ,κ 1 + M (XMN) + max R f (X, D) (XMN) . M X(M)−β

By (4.3.1), it suﬃces to estimate R f (X, D) when D is large and small.

71 Eliminating R f (X, D) when D is large

Assume that D > X(M)2β for some positive β.

λ (n) λ (m) √f n √f m Let xn = n h X and ym = m h X . We apply Lemma 2.4.6 to R f (X, D). Therefore d √ ! X X h0 4π mn X 1/2 X R (X, D) = D S (m, n; d)x y J D 1 + kxk kyk f d n m κ−1 d ,κ D M 2 2 d≡0(M) m,n>1 2 X X |λ f (n)| n X M−β 1 + (XM) h2 1 + M−β (XMN). ,κ M n X ,κ M n

Eliminating R f (X, D) when D is small

Now, assume that D < X(M)−β for the same β as in the previous case.

For ﬁxed m and n, consider the test function √ ! 4π mn W (x):= J (x)h , m,n κ−1 0 Dx and rewrite R f (X, D) as √ ! X X 1 λ f (m)λ f (n) m n 4π mn S (m, n; d) √ h h W . d X X m,n d d≡0(M) m,n>1 mn

πX 10πX Notice that Wm,n is supported on the interval [ D , D ]. Applying Kuznietsov Trace

Formula (Lemma 2.4.3 with r = 1, s = q = M, and Γ = Γ0(M), so that b = 1/q is equivalent to ∞ as cusps of Γ ) to R f (X, D), we obtain

X λ f (m)λ f (n) m n R (X, D) = √ h h × f X X m,n mn X∞ X Z ∞ ˆ ˆ 1 1 4π W(t j)ρ j∞(m)ρ j∞(n) + W(t)ϕ j∞ m, 2 + it ϕ j∞ n, 2 + it dt j=1 j −∞ X X ! + 4 W˜ (l) ψ jl(∞, m)ψ jl(∞, n) .

72 √ 4π mn −β By Lemma 2.3.3 with H(x) = h0 Dx , we know that when D < X(M) the weight functions satisfy   X −1+  if t > 0 or t = ir,  Wˆ (t), W˜ (t)  D  X A X  if t > 2κ.  Dt2 D Consider ﬁrst the contribution from the sum over the Maass forms. By the Cauchy-

Schwarz inequality, we have that

∞ X λ f (m)λ f (n) m n X √ h h Wˆ (t )ρ (m)ρ (n) (4.3.4) X X j j∞ j∞ m,n mn j=1 2 X X X λ f (n) n Wˆ (t j) √ h ρ j∞(n) n X T T≤|t j|<2T n    X X  X X X |λ (n)|2 n   ˆ 2 f 2  +  max {|W(t j)|} T + (XM) h   T≤|t j|<2T M n X  X 1/2+ X 1/2+  T≤|t j|<2T n T≤( D ) T>( D )

where T goes over powers of 2. For the last step above, we used Lemma 2.4.5.

Applying the bounds for Wˆ (t), we therefore obtain an upper bound for (4.3.4)

− ! − X 1+ X 1+2 X X 100 X + (XMN) + 1 + (XMN) ,κ D D M D M X 1 + M−β (XMN). ,κ M

Similarly, we have the same bound for ϕ j and ψ jl. Therefore

X R (X, D) 1 + M−β (XMN). f ,κ M

4.3.3 Step 3. Applying the Voronoi formula to convert Kloosterman Sums to Ramanujan sums.

Let D be such that

X(M)−β < D < X(M)2β. (4.3.5)

73 As was done in [HM13], we will apply the Voronoi formula (Lemma 2.4.1) to the m-sum.

N Assume that (d, N) = R and LR = N. Set N2 = (N,d) = L. Since N is square-free, (d, L) = 1. We have   ! X X X X X∗ ∗ ± 1 λ f (m)  aLm an λ f (n) R f (X, D) =2 η √ e ∓  e √ L d  d  d LR=N ± d≡0(RM) n,m a(d) m n (d,L)=1 √ ! ∞ !   d n Z Ld2t2 4π nLt × J ±   h0 h h f (t)Jκ−1  √  dt D X 0 mX m

X X X X 1 λ f (n)λ f ∗ (m) = S (0, m ∓ nL; d) √ I± (m, n, d) d L,X,D LR=N ± d≡0(RM) n,m nm (d,L)=1

X X X X cµ(b) X λ f (n)λ f ∗ (m) = √ I± (m, n, d) + R d L,X,D 0 ± LR=N d≡0(RM) bc=d n,m nm (d,L)=1 m∓nL≡0(c) m∓nL,0 where √ ! ∞ !   n d Z Ld2t2 4π nLt ± ± J ±   IL,X,D(m, n, d) = 2ηLh h0 h f (t)Jκ−1  √  dt, (4.3.6) X D 0 mX m , and the zero shift

X X X ϕ(d) X λ f (n)λ f ∗ (m) R = √ I± (m, n, d), 0 d L,X,D ± LR=N d≡0(RM) m∓nL=0 nm (d,L)=1 where ϕ(·) is the Euler totient function and µ(·) is the Mobius¨ function. Here we used the identity for Ramanujan sum ! X d S (0, k; d) = cµ . (4.3.7) c c|(d,k) ± Remark 4.3.1. As stated in Lemma 2.4.1, ηL only depends on L and f and has norm 1 in this case.

± From the estimation of IL,X,D(m, n, d) in Lemma 4.4.4, we have that the contribution from those m satisfying r √ mX nX − (1 + Zh) M LD2 D

74 is negligible. Thus, we can truncate m such that m MλLD where λ is deﬁned to be

D X λ = max , (1 + Z )2 . (4.3.8) X D h

2β 2 Then via (4.3.5) we have λ ≤ M (1 + Zh) .

Hence one can break apart the sum over m dyadically such that

R f (X, D) = (4.3.9)

∗ X X X cµ(b) X λ f (n)λ f (m) ± m −100 √ I (m, n, d)h0 + R0 + O(M ). d nm L,X,D S S =2i d≡0(RM) bc=d n,m S M λLD (d,L)=1 ± m∓nL≡0(c) LR=N m∓nL,0

4.3.4 Step 4. Treating the zero shift.

In this section, we will treat the m − nL = 0 case ( we always have that m + nL > 0 ).

∗ ∗ X X X λ f (n)λ f (n)λ f (L) + R0 = √ IL,X,D(nL, n, d) LR=N d≡0(RM) n Ln (d,L)=1 2 2 X X λ f (n) + λ f ∗ (n) X (XMN) (XMN)2, NM n NM nX

−1/2 where the last two steps follow from Lemma 4.4.4, 2.1.2 and the bound λ f (L) = L

(2.1.3).

4.3.5 The sum of shifted sums.

0 Let m ∓ nL = rc in (4.3.9). Since bc = d and M|d, let c0 := (c, M) such that c = c c0.

Let h = rc0 and d = d0RM. Denote by ! X c0 Md0R b (h, d0):= µ , R d0R c c0 c0|(d0R,h) 0 when (d0, L) = 1, and 0 otherwise.

75 Then, after rewriting (4.3.9),

R f (X, D) (4.3.10) 1+ ! X X c X X λ (n)λ ∗ (m) m X 0 b h, d0 f √ f I± m, n, d0RM h O . R( ) L,X,D( ) 0 + 1− M 0 nm S (MN) S =2i c0|M d m,n,h,0 S M λLD m∓nL=hc0 LR=N

Therefore, it is natural to study the sum of shifted sums. We have the following proposition.

Proposition 4.3.1. Let f1 and f2 both be Hecke newforms with the same level N and spec-

0 tral parameters t f1 , t f2 > 0. Let I(x, y, d ) be a smooth function supported on [1/2S 1, 5/2S 1]×

0 [1/2S 2, 5/2S 2] × [1/2D1, 5/2D1] with (x, y, d )-type (1 : Z, Z, Z) with Z > 1 (see Section

0 4.4). Let b(h, d ) be a series of complex numbers. Let c0 be an integer coprime with N. Let

100 Q = (Z|l1l2|S 1S 2D1c0N) . Then

X X λ f (n)λ f (m) b(h, d0) 1 √ 2 I(m, n, d0) (4.3.11) nm d0 m,n,h,0 l1m∓l2n=hc0 ( 1/2 !) θ− 1 5 p kB(h)kH H 2 6 11 ,t f ,tg max c N l1l2 √ S 1,2 + √ Z Q H<3(S l +S l )/c 0 1 1 2 2 0 H Nl1l2 q 1 1 where S 1,2 = (S 1l1 + S 2l2) + , and S 1l1 S 2l2

X 2 X 2 B(h):= |b(h, d)|, kB(h)kH := |B(h)| . d h∼H

We will prove this proposition in the following few steps.

4.3.6 Step 5. Applying the circle method.

By the support of I(x, y, d), we have |h| < 3(l1S 1 + l2S 2)/c0. Set

b(h, d0) = 0 (4.3.12)

76 when |h| > 3(l1S 1 + l2S 2)/c0. We shall apply Jutila’s version of circle method to detect the relation l1m ∓ l2n = hc0.

−1 100 Using the notations in Section 2.5, we choose δ = Q , Q = (Z|l1l2|S 1S 2D1c0N) , and

2− P = {q : Q < q < 2Q, (q, l1l2N) = 1}, so that Λ Q . Thus, by Jutila’s circle method

(see Section 2.5 ), the inner sum on the left of (4.3.11) gives

Z 1 X X 0 λ f1 (n)λ f2 (m) 0 b(h, d ) √ I(m, n, d ) e((l1m ∓ l2n − hc0)x)dx nm n,m h,0 0 ! 1 X X∗ X X (l m ∓ l n − hc )a λ f (n)λ f (m) = b(h, d0)e 1 2 0 1 √ 2 I(m, n, d0) Λ q nm q∈P a(q) n,m h,0 Z δ 1 0 × e(∆x)dx + E(l1, l2, d , c0) 2δ −δ where ∆ = ∆(l1m, l2n, hc0) = l1m ∓ l2n − hc0, and Z 1 0 X 0 λ f1 (n)λ f2 (m) 0 E(l1, l2, d , c0) = b(h, d ) √ I(m, n, d ) (1 − I˜P,δ(x))e((l1m ∓ l2n − hc0)x)dx n,m,h nm 0

Z 1 X λ f (n)λ f (m) ≤ b(h, d0) 1 √ 2 I(m, n, d0) |1 − I˜ (x)|2dx P,δ n,m,h nm 0

X λ (n)λ (m) Q2+ 0 f1 f2 0 0 −1 ≤ b(h, d ) √ I(m, n, d ) ≤ kb(h, d )k2Q . δΛ2 n,m,h nm

The last inequality above follows from Lemma 4.4.4 and 2.1.2. Here we used a similar argument to the one in [Blo04].

Let 1 Z δ wδ (∆) := e(∆x)dx. 2δ −δ

Set

R f1, f2 := (4.3.13)

∗ ! 1 X X X X 0 (l1m ∓ l2n − hc0)a λ f1 (n)λ f2 (m) 0 b(h, d )e √ I(m, n, d )wδ (∆) . Λ q nm q∈P a(q) n,m,d0 h,0

77 4.3.7 Step 6. Applying Vornoi formula twice to regenerate Klooster- mann sums.

We only treat the − sign case, the + sign case can be treated similarly. Recall that

2− (q, l1l2N) = 1 for any q ∈ P. So that N2 = N/(N, q) = N. Also, we have that Λ Q .

Applying Voronoi formula (Lemma 2.4.1) to both the m and the n sums in (4.3.13), we

get

R f1, f2 =

1 X X X X λ f ∗ (n)λ f ∗ (m) 0 1 2 ±1,±2 0 b(h, d )S ((∓1l2m ±2 l1n)l1l2N, hc0; q) √ H (m, n, h, d , q), Λ 0 nm ±1,±2 q∈P n,m,d h,0

where H±1,±2 (m, n, h, d0, q) is given by

H±1,±2 (m, n, h, d0, q):= (4.3.14) ∞ ! !! ξ2q2N µ2q2N ξ2q2N µ2q2N 4η±1 η±2 I , , d0 w ∆ , , hc J ±1 (ξ)J ±2 (µ)dξdµ. g f δ 0 f2 f1 0 m n m n " ±2 ±1 and η f , ηg depend on f1, f2, and N only.

−1 100 Recall that δ = Q , Q = (|l1l2|S 1S 2D1c0N) . By Lemma 4.4.5, we can truncate the

sum over m and n such that

NZQ2+ NZQ2+ m , n . S 1 S 2

Breaking apart the m and n sums dyadically, we can assume that the sizes of m, n are

2+ 2+ A, B respectively with A NZQ /S 1 and B NZQ /S 2.

78 Let

Re(A, B):=

X X X X λ f ∗ (n)λ f ∗ (m) 1 0 1 2 ±1,±2 0 b(h, d )S ((∓1l2m ±2 l1n)l1l2N, hc0; q) √ HeA,B (m, n, h, d , q), Λ 0 nm ±1,±2 q∈P n,m,d h

where m n He±1,±2 (m, n, h, d0, q) = H±1,±2 (m, n, h, d0, q)h h . A,B 0 A 0 B

Then, we have

R f , f max{Re(A, B)}(ZQ) . (4.3.15) 1 2 A,B

P ± ± ±1,±2 in Re(A, B) contains four terms. We only consider the case with 1, 2 both being

± +. The proofs of other cases will be similar. Let l2m − l1n = v. Set t (v) to be functions such that t±(v) = 1 when ±v > 2/3 and t±(v) = 0 when ±v 6 1/3. By Abel’s summation formula, we have

Re+,+(A, B)

X X X X λ ∗ (n)λ ∗ (m) 1 0 f1 f2 +,+ 0 = b(h, d )S (0, hc0; q) √ HeA,B(m, n, h, d , q) Λ 0 nm d q∈Q h l2m=l1n Z ∞ X X X X X λ f ∗ (n)λ f ∗ (m) 1 0 1 2 + 0 − b(h, d ) S (vl1l2N, hc0; q) √ u (v, h, q; d , x)dx Λ 1 nm d0 q∈Q h 2 v>0 m6x l2m−l1n=v

X X X Z ∞ X X λ ∗ (n)λ ∗ (m) 1 0 f1 f2 − 0 − b(h, d ) S (vl1l2N, hc0; q) √ u (v, h, q; d , x)dx Λ 1 nm d0 q∈Q h 2 v<0 m6x l2m−l1n=v

= I0 − I+ − I−, (4.3.16)

where ! − 0 d +,+ l1 x − v 0 − u (v, h, q; d , x):= HeA,B x, , h, d , q t (v), dx l2

79 and ! + 0 d +,+ v + l1 x 0 + u (v, h, q; d , x):= HeA,B , x, rc, d , q t (v). dx l2 Let

X λ f ∗ (n)λ f ∗ (m) A(v; x):= 1 √ 2 . m6x nm l1m−l2n=v Remark 4.3.2. In Theorem 3.1.3, we need w to be coprime with the level rs. In our case,

c0 is coprime with the level Nl1l2 that we will consider. So, c0 will play the role of w, and

will give us the saving that we need.

4.3.8 Step 7. Applying the large sieve type inequality to the sum of Kloostermann sums.

In this section, we will bound I0 and I±.

First, consider I0 as deﬁned in (4.3.16). By (4.3.7), Lemma 4.4.5, the Cauchy-Schwarz inequality and Rankin’s bound (Lemma 2.1.2), we have

X X X X X λ ∗ (n)λ ∗ (m) 1 0 q f1 f2 +,+ 0 I0 = b(h, d ) µ u √ HeA,B(m, n, h, d , q) Λ 0 u nm d q∈Q h u|(hc0,q) l1m=l2n 2 2 1 X X |λ f ∗ (n)| + |λ f ∗ (n)| τ hc b h, d0 1 2 kB h k Q−1/2 1− ( 0) ( ) ( ) 2 Q 0 n h,d nl1A+l2 B

I + 0 1 5 Secondly, consider +. By Lemma 4.4.6, u (v, h, q; d , x) is zero if x < [ 2 B, 2 B], and it has (v, h, q, d0)-type Z : Z, 1, Z, Z . B

Break apart the v and h-sums dyadically such that v ∼ V, h ∼ H. By Lemma 4.4.6, we

0 have V Al2 and x B. By the choice of b(h, d )(4.3.12), we have H < 3(S 1l1 +S 2l2)/c0.

80 √ Let Ξ = √ VHc0 . Then, one may apply Theorem 3.1.3 to show that Nl1l2Q

Z ∞ X X X X X λ f ∗ (n)λ f ∗ (m) 1 0 1 2 + 0 I+ = b(h, d ) S (vl1l2N, hc0; q) √ u (v, h, q; d , x)dx Λ 1 nm d0 q∈Q h 2 v>0 m6x l2m−l1n=v (4.3.17) √ ! n Nl l Z V1/2+ max 1 2 B 1 + (Ξ)−2θ Z + Ξ + √ VAl 2 (Z + Ξ) Q B Nl1l2 H(S 1l1+S 2l2)/c0 1/2+ ! H θ o 8 × Z + Ξ + √ c0 max{||A(v, x)||}kB(h)kH Z Q xB Nl1l2 P where B(h) = d |b(h, d)|, and

2 X 2 kB(h)kH = A (h). h∼H Furthermore, we have

X λ f ∗ (n1)λ f ∗ (m1)λ f ∗ (n2)λ f ∗ (m2) ||A(v, x)||2 = 1 √2 1 2 (4.3.18) n1m1n2m2 n1,n2>0 m1,m26x l2m1−l1n1=l2m2−l1n2>0 Z 1 X λ f ∗ (m1)e(m1l2α)λ f ∗ (m2)e(−m2l2α) = 2 √ 2 0 m1m2 m1,m26x

X λ f ∗ (n1)e(−n1l1α)λ f ∗ (n2)e(n2l1α) × 1 √ 1 dα, n1n2 n1,n26l2 x/l1 which, by Lemma 2.1.1, Theorem 3.2.1 and Remark 3.2.2 (or Propositiion 2.7.1 and Re- mark 2.7.1 ), is 2 Z 1 X λ f ∗ (n)e(−nl1α) ∗ 2 2 kS ( f1 , x, l2α)k∞ √ dα 0 n n6l2 x/l1 2 X |λ f ∗ (n)| N2/3(Nx) 2 N2/3(xNl l ). n 1 2 n6l2 x/l1 2+ By Lemma 4.4.6,(4.3.17), and (4.3.18), recalling that A ZQ N/S 1 and B

2+ ZQ N/S 2, we determine that I+ is bounded by  √  cθ N5/6Z9 l l 1/2+ ! 1/2+ !   0 1 2 −2θ V H  max 1 + (Ξ) Z + Ξ + √ Z + Ξ + √ kB(h)kH Q . V,H    (Z + Ξ) Q Nl1l2 Nl1l2 

81 2 When VHc0 6 Nl1l2Q , recalling that H (S 1l1 + S 2l2)/c0 and Q is suﬃciently large , we

have  √  cθ N5/6Z8 l l 2 !θ 1/2+ ! 1/2+ !  0 1 2 Nl1l2Q V H  I+ max Z + √ Z + √ (Q) V,H    Q VHc0 Nl1l2 Nl1l2   θ− 1 √  c 2 N5/6 l l kB(h)k 1/2+ !  0 1 2 H H  8 max √ 1 + √ Z Q . H(S l +S l )/c   1 1 2 2 0  H Nl1l2 

2 2+ −1 −1 When VHc0 > NLQ , recalling that V Al2 ZQ Nl1l2 (S 1l1) + (S 2l2) , we have

 θ− 1 √   2 5/6 1/2+ ! c N l1l2kB(h)kH H  I max  0 √ S + √  Z11Q, +  1,2  H  H Nl1l2  q 1 1 where S 1,2 = (S 1l1 + S 2l2) + . S 1l1 S 2l2

The estimation of I− is similar. Thus, we combine these bounds with (4.3.15) and

(4.3.16) to compete the proof of Proposition 4.3.1.

4.3.9 Conclusion and the ﬁnal bound.

0 By the results of Section 4.3.5, we can apply Propostion 4.3.1 to R f (X, D) with I(x, y, d ) =

± 0 m IL,X,D(m, n, d RM)h0 S and ! X c0 MRd0 b(h, d0):= b (h, d0) = µ , R d0R c c0 c0|(d0R,h) 0 when (d0, L) = 1, and b(h, d0):= 0 otherwise.

By Lemma 4.4.4, the fact that S MλLD, and X λD (4.3.8), I(x, y, d0) has

(x, y, d0)-type

  1    −   SX 2   min 1,  : Z, Z, Z , (4.3.19)   2     LD  

where Z = λ.

82 For ﬁxed c0 and R, 2 2 X X X X X 0 0 c kB(h)kH = |bR(h, d )| ≤ d0R h d0∼D/RM h (d0,L)=1 c0|(d0R,h) (d0,L)=1 2 X X X s H(HD). d0R h s|h d0R≡0(s)

In our case, we have that S 1 = S, S 2 = X, D1 = D/RM, l1 = 1, l2 = L. By Propostion

4.3.1 and (4.3.10), we have 1 +θ 5  1  2  − ! X X c N 6  SX 2  S + XL S + XL R X, D 0  ,  √ √ Z11 ZXMN f ( ) ,t f ,κ min 1 2  + ( ) M  LD  SX c0N S =2i c0|M   S M λLD LR=N

  r   X λ13N4/3  X   + 1 +  (λXMN).  MN M1/2−θ  MN 

Remark 4.3.3. We used the condition that f is non-exceptional here in (4.3.19) to bound the sum above when S is small.

7 11 Using the result of Kim-Sarnak in [Kim03], we take θ = 64 and choose β = 5184 . Then,

2β 2 from (4.3.2) and λ < M (1 + Zh) , we have 2 X − X λ f (n)ψg(n) n ω 1 √ h g n X g∈Bκ(M) n>1   r   X X (1 + Z )26 N4/3  X  1 + + + h 1 +  (XMN). ,t f ,κ  MN M1+β M1/3+β  MN 

This concludes the proof of Proposition 3.1.1.

Remark 4.3.4. The choice of β is not optimal here when X ∼ MN,N < M. In this case,

1 − − logM N 76/75 −1/64 one can choose 25β = 2 θ 3 . As a result, the bound would be 1 + N M .

Remark 4.3.5. In this proof, M is not necessarily square-free. It is also possible to show

that when (M, N) is very small, a subconvexity bound still holds. However, if M is not

83 square-free, the size of Q( f × g) is not known.

4.4 The estimation of weight functions

In this section, we will use the lemmas in Section 2.2 to study various weight func-

± ±1,±2 0 tions appearing in our analysis in Section 4.3 such as IL,X,D(x, y, d), H (m, n, h, d , q) and u±(v, h, q; d0, x).

In order to simplify our notation, we introduce the following deﬁnition for the “type”

of a function.

Deﬁnition 4.4.1. Let x = (x1, x2,..., xn) be a vector of real numbers with each xi , 0. Let

F(x) be a function. If there are nonnegative functions ZF(x), F1(x), F2(x),..., Fn(x) such

that

|xi1 ... xin ∂i1 . . . ∂in F x | Z x F x i1 ... F x in 1 n x1 xn ( ) i1...in F( ) 1( ) n( )

for every x, where the implied constant depends on i1,..., in only, then we say that F(x)

has x = (x1, x2,..., xn)-type

(ZF(x): F1(x), F2(x),..., Fn(x)).

Moreover, when F(x) does not depend on xl for some l, we let Fl(x) = 0.

j j j j For example, let F(x, y):= e(x). Since x ∂xF(x, y) = (2πi) x e(x) and F(x, y) is independent of y, we have that F(x, y) has (x, y)-type (1 : |x|, 0).

First, we establish some basic properties about types. These properties will be used in the study of our weight functions.

84 m Let F and G be R -to-R functions with types (ZF : F1,..., Fm) and (ZG : G1,..., Gm) with each Fi, Gi nonnegative. Let x = (x1,..., xm) with each xi , 0.

Lemma 4.4.1. ∂xk F(x) has x-type (ZF Fk/xk : F1,..., Fm).

Proof. Without loss of generality, let k = 1.

1 xi1 xi2 ... xim ∂i1 . . . ∂im ∂ F x xi1+1 xi1 ... xim ∂i2 . . . ∂im ∂i1+1F x 1 2 m x1 xm x1 ( ) = 1 1 m x2 xm x1 ( ) x1 Z(x)F (x) 1 i1 im F1(x) ... Fm(x) . x1

We now use Lemma 4.4.1 to establish the type of F(x)G(x).

Lemma 4.4.2. F(x)G(x) has x-type (ZFZG : F1 + G1,..., Fm + Gm).

Proof. We proceed by induction on (i1, i2,..., im). Assume that for any (i1, i2,..., im) <

(k1, k2,..., km) (e.g. i j 6 k j for any j and i j0 < k j0 for some j0) we have that

xi1 xi2 ... xim ∂i1 . . . ∂im FG x Z Z x F G x i1 ... F G x im 1 2 m x1 xm ( ) F G( )( 1 + 1)( ) ( m + m)( ) holds for any F, G satisfying the assumptions as before.

Without loss of generality, assume that k1 > 0, such that

xk1 xk2 ... xkm ∂k1 . . . ∂km FG x xk1 xk2 ... xkm ∂k1−1 . . . ∂km G x ∂ F x F x ∂ G x . 1 2 m x1 xm ( ) = 1 2 m 1 m ( ) x1 ( ) + ( ) 1 ( )

Then, by Lemma 4.4.1 and induction, we complete the proof.

85 n Lemma 4.4.3. Let F(y) be a R -to-R map, with type (ZF : F1, F2,..., Fn). Let G(x) =

m n (G1(x),..., Gn(x)) be a R to R map with each Gk(x) , 0 for any x. Moreover, assume

that Gk(x) has x-type (ZGk : Gk1,..., Gkm) with each Gk j(x) nonnegative.

m Then F(G(x)) is a R -to-R map, with x-type (ZF(G) : F(G)1,..., F(G)m), where ZF(G) =

ZF(G(x)) and X Fk(G(x))ZG (x) + Gk(x) Gk j(x) F(G) = k . j G (x) k k

Proof. We proceed by induction on (i1, i2,..., im). Assume that for any (i1, i2,..., im) <

(k1, k2,..., km) (e.g. i j 6 k j for any j and i j0 < k j0 for some j0), we have that

 i j Y X F (G(x))Z (x) + G (x) G (x) i1 i2 im i1 im  k Gk k k j  x x ... x ∂ . . . ∂ F(G(x)) ZF(G(x))   1 2 m x1 xm  G (x)  j k k

holds for any F, Gk satisfying the assumptions as above.

Without loss of generality, one can assume that k1 > 0. Then

xk1 xk2 ... xkm ∂k1 . . . ∂km F G x xk1 xk2 ... xkm ∂k1−1 . . . ∂km ∂ F G x . 1 2 m x1 xm ( ( )) = 1 2 m x1 xm x1 ( ( ))   X  k1 k2 km k1−1 km   =x x ... x ∂ . . . ∂  (∂ jF)(G(x))∂x G j(x) . 1 2 m x1 xm  1  j  i j Y X Fk(G(x))ZG (x) + Gk(x) Gk j(x) Z (G(x))  k  F  G (x)  j k k

by Lemmas 4.4.1, 4.4.2 and induction.

± ±1,±2 0 Now, we are ready to study the weight functions IL,X,D(x, y, d), H (m, n, h, d , q) and u±(v, h, q; d0, x).

86 Let t f > 0 (this is true when f is holomorphic or non-exceptional Maaß ). After chang-

ing variables in (4.3.6),

r ! ∞  r   r  Xx d y Z h (u)  Xxu  Xyu I± x, y, d η± h h √ J ±   J  π  du. L,X,D( ) = 2 L 2 0 f  2  κ−1 4 2  Ld D X 0 2 u  Ld   d 

By Lemmas 2.3.1, 2.3.2, 4.4.2, and 4.4.3, one has

± Lemma 4.4.4. IL,X,D(x, y, d) has (x, y, d)-type

  1 − r r   2    xX  xX yX  min 1, : + 1, + 1 + Zh, 1 + Zh ,   LD2  LD2 D2  and is supported on R+ × [1/2X, 5/2X] × [1/2D, 5/2D]. Moreover, it also satisﬁes !  r r −m xX  xX yX  I (x, y, d) (1 + Z )m 1 +  −  . L,X,D m h LD2  LD2 D2 

for any m > 0. All the above implied constants may depend on the spectral parameters t f

and κ.

By changing variables in (4.3.14), one has

∞ ! q2Nξ2 q2Nµ2 H±1,±2 (m, n, h, d0, q):= F , , hc , d0 J ±1 (ξ) J ±2 (µ) dξdµ,(4.4.1) 0 f1 f2 0 m n " where

F(x, y, hc , d0) = 4η±1 η±2 I x, y, d0 w (∆ (x, y, hc )) , 0 f1 f2 δ 0

R δ x, y, hc l x − l y − hc w ( ) 1 e ( u) du ∆( 0) = 1 2 0, and δ ∆ = 2δ −δ ∆ .

±1,±2 0 Lemma 4.4.5. When both t f1 , t f2 > 0, for any given integers m, n > 0,H (x, y, h, d , q)

has (x, y, h, d0, q)-type

(BH(m, n): Z, Z, 1, Z, Z) ,

87 for any m, n > 0, where

 1 −  1 −  m  n  ! 2   ! 2   xS 1   yS 2   Z   Z  BH(m, n):= min 1, min 1,     ,  2   2   p 2   p 2   q N   q N  xS 1/q N yS 2/q N

100 and Q = (Zl1l2S 1S 2D1c0N) as deﬁned in Proposition 4.3.1.

0 Proof. Set a = (x, y, h, d , q). Let I = (ix, iy, ih, id0 , iq) be a vector of nonnegative integers.

i i 0 i I ix iy ih 0 id0 iq I ix y ih d q Let a := x y h (d ) q , ∂a := ∂x ∂y ∂h ∂d0 ∂q .

In order to establish the (x, y, h, d0, q)-type of H±1,±2 (x, y, h, d0, q), we need to bound

I I ±1,±2 0 a ∂aH (x, y, h, d , q) (4.4.2) ∞ ! q2Nξ2 q2Nµ2 = aI∂I F , , hc , d0 J ±1 (ξ) J ±2 (µ) dξdµ. a 0 f1 f2 0 x y " We ﬁrst study the type of the integrand by rewriting it as a composition of two maps.

Let b = (ξ, µ, x, y, h, d0, q) be a vector in R7, and ! q2Nξ2 q2Nµ2 G(b):= (G (b),..., G (b)) = , , hc , d0 1 4 x y 0 be a R7-to-R4 map.

Thus, ! q2Nξ2 q2Nµ2 ∂I F (G(b)) = ∂I F , , hc , d0 . a a x y 0

4 Let x = (x1, x2, x3, x4) be a vector in R . By (4.4.1),

F(x , x , x , x ) = 4η±1 η±2 I (x , x , x ) w (∆ (x , x , x c )) 1 2 3 4 f2 f1 1 2 4 δ 1 2 3 0

is supported on (x1, x2, x3) ∈ [S 1/2, 5S 1/2] × [S 2/2, 5S 2/2] × [D1/2, 5D1/2].

−1 −100 By the fact that δ = Q = (Zl1l2S 1S 2D1c0N) and the support above, wδ (∆ (x1, x2, x3c0)) has x-type (1 : δl1 x1, δl2 x2, δx3c0, 0), which is “bounded” by (1 : 1, 1, 1, 0). I (x1, x2, x4) has x-type (1 : Z, Z, Z). Hence, by Lemma 4.4.2, 4.4.4, F(x) has x-type

(1 : Z, Z, 1, Z) .

88 It is easy to check that G1(b) has b-type (G1(b) : 1, 0, 1, 0, 0, 0, 1), G2(b) has b-type

(G2(b) : 0, 1, 0, 1, 0, 0, 1), G3(b) has b-type (G3(b) : 0, 0, 0, 0, 1, 0, 0), and G4(b) has b-type

(G4(b) : 0, 0, 0, 0, 0, 1, 0).

Then, by Lemmas 4.4.3, F(G(b)) has b-type

(1, Z, Z, Z, Z, 1, Z, Z)

Set

Fx = Fy = Fd0 = Fq = Z, Fh = 1.

Also set

I ix iy ih id0 iq F := Fx Fy Fh Fd0 Fq .

I Then by Lemma 4.4.1, ∂aF (G(b)) has b-type

I I F /a : Z, Z, Fx, Fy, Fh, Fd0 , Fq .

Now, let

I I Ka,I(ξ, µ):= a ∂aF (G(b)) .

Then Ka,I(ξ, µ) has (ξ, µ)-type FI : Z, Z .

Next, we study the integral in (4.4.2). We will use a similar method as the one in the proof of Lemma 2.3.1 to establish the bound for m = 1 and n = 0.

We proceed by considering the integral over those (ξ, µ) where KI(ξ, µ) , 0. Diﬀerent techniques will be applied when ξ > 1 and when 0 < ξ < 1.

q2Nξ2 q2Nµ2 Note that Ka,I(ξ, µ) = 0 if x , y < [1/2S 1, 5/2S 1] × [1/2S 2, 5/2S 2]. Then we

p 2 p 2 can restrict to the case that ξ ∼ xS 1/q N and µ ∼ yS 2/q N. We ﬁrst treat the case

89 p 2 that xS 1/q N 1. When f1 is holomorphic, by (2.3.5), the right hand side of (4.4.2) becomes

∞ K (ξ, µ) eiξW (ξ) + e−iξW (ξ) J ±2 (µ) dξdµ. a,I 2t f1 2t f1 f2 0

∞ " iξ ±2 We consider Ka,I(ξ, µ)e W t (ξ)J (µ) dξdµ. The other term can be treated similarly. 0 2 f1 f2 From (2.3.5), (2.3.6), Remark 2.4.2 and integration by parts, one gets the upper bound as !  1    ! 2 −  yS 2   Z  min 1,  ·   FI.  2   p 2   q N  xS 1/q N

p 2 We then consider the case that xS 1/q N 1. By Remark 2.4.2, we can bound the integral trivially as

 1   1  ! 2 − ! 2 −  xS 1   yS 2  min 1,  min 1,  FI  q2N   q2N 

For a Maaß cusp form f , we can still do integration by parts for J + , but using the 1 f1 trivial bound for J − (y), which gives exponential decay for large values of y. By repeating f1 this process, we obtain the desired bound.

The following lemma is a consequence of Lemmas 4.4.1 and 4.4.5.

+,+ 0 +,+ 0 m n ± Lemma 4.4.6. Let HeA,B(m, n, h, d , q):= H (m, n, h, d , q)h0 A h0 B . Set t (v) to be functions such that t±(v) = 1 when ±v > 2/3 and t±(v) = 0 when ±v 6 1/3. Let ! + 0 d v + l1 x 0 + u (v, h, q; d , x):= HeA,B , x, h, d , q t (v), dx l2 ! − 0 d l2 x − v 0 − u (v, h, q; d , x):= HeA,B x, , h, d , q t (v). dx l1

+ 0 Then u (v, h, q; d , x) = 0 when 20Al2 < l1B. It is supported on the region such that

∈ 5 ∈ 1 5 0 v [0, 2 Al2] and x [ 2 B, 2 B]. Moreover, it has (v, h, q, d )-type

Z : Z, 1, Z, Z . B 90 − 0 Similarly, u (v, h, q; d , x) = 0 when Al2 > 20Bl1. It is supported on the region such that

− ∈ 5 ∈ 1 5 0 v [0, 2 Bl1] and x [ 2 A, 2 A]. Moreover, it has (v, h, q, d )-type

Z : Z, 1, Z, Z . A

4.5 The proof of Theorem 3.1.3

In this section, we prove the large sieve inequality Theorem 3.1.3. Our arguments are

motivated by ideas demonstrated in [Blo04], [DI83] and [Pit13].

Proof of Theorem 3.1.3. Consider the case of

X X 1 S = S (vr, +hw; sq)a(v)b(h, d)u(v, h, q, d). + q q∈Z d,h,v∈Z (q,r)=1

The treatment of S− is similar.

As in [DI83], we consider the Fourier transform of u as √ ! 4π x1 x2w G(t1, t2, t3; x) = u x1, x2, √ , x3 e(−t1 x1 − t2 x2 − t3 x3)dx1dx2dx3, R3 s rx $ so that √ ! 4π x1 x2w u x1, x2, √ , x3 = G(t1, t2, t3; x)e(t1 x1 + t2 x2 + t3 x3)dt1dt2dt3. s rx R3 $ Furthermore, one obtains

∂p p G(t1, t2, t3; x) ∂x √ p +p +p +p ! ∂ 1 2 3 4π x1 x2w =(2πit )p1 (2πit )p2 (2πit )p3 u x , x , √ , x 1 2 3 p1 p2 p3 p 1 2 3 R3 ∂x1 ∂x2 ∂x3 ∂x s rx

× e(−t1 x1 − t2 x2 − t3 x3)dx$1dx2dx3 √ √ −p1 −p2 −p3 −p (t1V/Z) (t2H/Z) (t3D/Z) ( VHwZ/s rQ) VH

by integration by parts. Also note that G(t1, t2, t3; x) is compactly supported in terms of x.

91 Thus,

X X X 1 S (vr, hw, sq)a(v)b(h, d)u(v, h, q, d) (4.5.1) q q h,d v (q,r)=1  √  X X X 1  4π vhw = S (vr, hw, sq)a(v)b(h, d)G t1, t2, t3;  3 q q R q h,d v $ (q,r)=1 × e(t1v + t2h + t3d)dt1dt2dt3.

We now study the integrand  √  X X X 1  4π vhw S (vr, hw, sq)a(v)b(h, d)G t1, t2, t3;  e(t1v + t2h + t3d), q  q  q h,d v (q,r)=1

saving integration over t1, t2, t3 for later. √ √ Let Ξ = VHw/s rQ, and for ﬁxed p1, p2, p3, deﬁne  √   √  4π vhw 4π vhw   −1 p1 p2 p3   ϕ  √  := (DVH) (t1V/Z) (t2H/Z) (t3D/Z) G t1, t2, t3; √  ,  s rq   s rq 

so that ϕ(x) is supported on [Ω−1Ξ, ΩΞ] for some absolute positive number Ω. Moreover,

ϕ(i)(x) (Z/Ξ)i.

s¯ We absorb the e(t1v + t2h + t3d) factor into a(v), b(h, d) and pull out a factor e v r from a(v). Since these all have norm 1, this will not affect our final bound. Therefore, it suffices to bound √ X X s¯ X 1 4π vhw a(v)e v b(h, d) √ S (vr, hw; sq)ϕ( √ ) (4.5.2) r h,d v q>0 s rq s rq (q,r)=1

92 Denote by a = 1/s and b = ∞ two cusps of Γ0(rs). Applying Lemma 2.4.3, the sum above equals

X X X 4 a(v)b(h, d)ψ jk(a, v)ψ jk(b, hw)ϕ ˜(k) k=0(2) j v,h,d X X + 4π a(v)b(h, d)ρ ja(v)ρ jb(hw)ϕ ˆ(t j) j>1 v,h,d X Z ∞ X 1 1 + a(v)b(h, d)ϕ ja v, 2 + it ϕ jb hw, 2 + it ϕˆ(t)dt. j −∞ v,h,d

Applying Cauchy’s inequality, we get that

2 2 X X X X S2 ϕˆ(t ) a(v)ρ (v) ϕˆ(t ) b(h, d)ρ (hw) j ja j jb j>1 v j>1 h,d 2 2 X X X X X X + |ϕ˜(k)| a(v)ψ (a, v) |ϕ˜(k)| b(h, d)ψ (b, hw) jk jk k=0(2) j v k=0(2) j h,d 2 2 X Z ∞ X X Z ∞ X + |ϕˆ(t)| a(v)ϕ v, 1 + it dt |ϕˆ(t)| b(h, d)ϕ hw, 1 + it dt. ja 2 jb 2 j −∞ v j −∞ h,d

We will bound

2 2 X X X X ϕˆ(t ) a(v)ρ (v) , ϕˆ(t ) b(h, d)ρ (hw) . j ja j jb j>1 v j>1 h,d

Consider the sum over v ﬁrst. Set T = Z + 2Ξ + 1. We have

2 2 2 X X X X X X ϕˆ(t ) a(v)ρ (v) = ϕˆ(t ) a(v)ρ (v) + ϕˆ(t ) a(v)ρ (v) . j ja j ja j ja j>1 v |t j|6T v |t j|>T v

From Lemma 2.4.4 and 2.4.5, we have

2  −2θ  X X 1 + log Ξ + Ξ  1+ !  Z Z  2 V 2 ϕˆ(iκ j) a(v)ρ ja(v)   T + ka(v)k .  1 + Ξ  rs |t j|

93 For the second term, we split the sum dyadically to obtain

2 X X X !2i 4 ! 1+ ! 1 Z 1 Ξ(1 + log T) 2 V 2 ϕˆ(iκ j) a(v)ρ ja(v) + T + ka(v)k 2 T T 1/2 T rs |t j|>T v i>0 ! ! Z 4 1 Ξ(1 + log T) V1+ + T 2 + ka(v)k2. T T 1/2 T rs

Therefore, by noticing that Z > 1, we have the bound

2  −2θ  X X 1 + Ξ  1+ !  Z  2 2 V 2 ϕˆ(t j) a(v)ρ ja(v)   Z + Ξ + ka(v)k Z(ΞZ) . (4.5.3)  Z + X  rs j>1 v   Next, we estimate the sum over n. For this term, we use the multilicativity of Hecke- eigenvalues

X h w ρ j∞(hw) = ρ j∞ u λ j u u|(h,w) when (w, rs) = 1.

Thus, one can estimate this sum in a similar way as

2 2 X X X X X ϕˆ(t ) b(h, d)ρ (hw) = ϕˆ(t ) b(h, d) ρ h λ w j jb j jb u j u j>1 h,d j>1 h,d u|(h,w) 2 X X 2 X X X ϕˆ(t ) λ w b(h, d)ρ h j j u jb u j>1 u|w u|w h≡0( mod u) d  −2θ   Ξ  1+ ! 1 + Z  H   Z2 + Ξ2 + w2θkB(h)k2Z(ΞZw).  Z + Ξ  rs

For the last step, we used (4.5.3) and the Kim-Sarnak bound |λ(w)| 6 τ(w)wθ (see [Kim03]).

The holomorphic case is similar. In order to treat the case of Eisenstein series, we need the Hecke eigenvectors of the space generated by incomplete Eisenstein series. As in Remark 2.4.5, we refer to the notation in [GJ79], where we have a basis of Eisenstein series indexed by a ﬁnite set

{(t, χ1, χ2, b)|t ∈ R, χ1χ2 = 1, b ∈ B(χ1, χ2)}.

94 Furthermore, as explained in section 2.6-2.8 [BH10a] and in [BHM07b] , the Hecke

multiplicativity of coeﬃcients of Eisenstein series and the large sieve inequality also hold.

As a consequence, a similar bound holds true for this part as well. We can also bound the

continuous part via a direct computation (see [BHM07a] Section 3.2).

Thus, one can get the ﬁnal bound of (4.5.2) as

 −2θ  1 + Ξ  1/2+ ! 1/2+ !  Z  V H θ+ 2   Z + Ξ + Z + Ξ + w ka(v)k2kB(h)k2Z (ΞZ) . (4.5.4)  Z + Ξ  (rs)1/2 (rs)1/2

Now assume that

 −2θ   Ξ  1/2+ ! 1/2+ ! 1 + Z  V H B(Ξ, Z, V, H, Q) =   Z + Ξ + Z + Ξ + wθ+(ΞZ),  Z + Ξ  (rs)1/2 (rs)1/2

For ﬁxed t1, t2, t3, using the bound (4.5.4), (4.5.1) is bounded above by

−p1 −p2 −p3 2 DVH(t1V/Z) (t2H/Z) (t3D/Z) B(Ξ, Z, V, H, Q)Z

1/2 1/2 $ X X X × |a(v)|2 | b(h, d)e(t d)|2 dt dt dt 3 1 2 3 v h d 8 B(Ξ, Z, V, H, Q)(1 + Z )ka(v)kkB(h)k.

−1− −1− −1− In the last step, we chose p1 = p2 = p3 = 0 when t1 < ZV , t2 < ZH , t3 < ZD and p1 = p2 = p3 = 2 otherwise. Therefore, we complete the proof.

95 Chapter 5: The Proof of Sup-norm Bounds for Holomorphic Modular Forms

The proof follows along the same lines as these seen in [HT12] and [HT13]. Let N be a

2 positive square-free integer. Let F be a holomorphic newform with weight k on Γ0(N)\H .

k Let f be the “normalized” cusp form f (z):= y 2 F(z).

5.1 Preliminary

5.1.1 The Sup-norm via Fourier expansion

We establish a bound for f when y is large.

Proposition 5.1.1.  k1/4+y−1/2 + y1/2k−1/4, if y k, f (z) h f, f i−1/2 N1/2  k1/4+y−1/2 + 2k/2k(2πy)k/2+e−2πyΓ(k)−1/2, if y k.

Remark 5.1.1. This proposition is implicitly proved in [Xia07].

5.1.2 Pretrace formula for holomorphic cusp forms

Let

X yk 1 h(z, w):= , ( j(γ, z))k (w + γ.z)k γ∈Γ0(N)

96 ! a b where j(γ, z):= cz + d if γ = , and c d

az + b γ.z = . cz + d

We have a pre-trace formula as follows (see [Oli11] Appendix 1 for details )

(−1)k/2π Lemma 5.1.1. Let Ck = 2(k−3)(k−1) . Then

−1 X fi(z) fi(−w) Ck h(z, w) = . h fi, fii fi∈Bk(N)

Deﬁne

Deﬁnition 5.1.1. Atkin-Lehner operators of level N are deﬁned to be the elements in the set ( √ ! ) ra √b A (N):= σ = √ √ r : σ ∈ SL (R), r|N, N|rs, a, b, s, d ∈ Z, (a, s) = 1 . 0 rs rd 2

A well known result is ([HT12])

Lemma 5.1.2. Let F(z) be a holomorphic cusp newform of level N and weight k. Then the

k/2 function | f (z)| = |y F(z)| is A0(N)-invariant.

5.1.3 Ampliﬁcation method

Let Tl be the l-th Hecke operator as deﬁned in (2.1.1). Let L be a positive integer which we will chose later. Let

Λ(L):= p ∈ Z : p prime , (p, N) = 1, L 6 p < 2L , and n o Λ(L)2 := p2 : p ∈ Λ(L) .

97 Deﬁnition 5.1.2. Let ( ! ) a b G (N):= γ = : a, b, c, d ∈ Z, N|c, det(γ) = l . l c d

Let j(γ, z)2(z − γ.z)2 u (z):= . γ 4Im(z)2 det γ Let n o M(z, l, δ):= # γ ∈ Gl(N): uγ(z) 6 δ .

For any ﬁnite sequence of complex numbers {yl}, we have

X X y X yk 1 y T (h(z, w)) = √l (det α)k/2 . l l j(α, z)k (w + α.z)k l l l α∈Gl(N)

Otherwise, by Lemma 5.1.1, we have

X X X Tl ( fi(z)) fi(−w) X X λi(l) fi(z) fi(−w) ylTl (h(z, w)) = Ck yl = Ck yl . h fi, fii h fi, fii l l fi∈Bk(N) l fi∈Bk(N)

Hence, by chosing w = −z, we have

X X fi(z) fi(z) Ck ylλi(l) h fi, fii fi∈Bk(N) l k X yl X k/2 y 1 X yl X −k/2 = √ (det α) = √ uα(z) . j(α, z)k (−z + α.z)k k l l α∈Gl(N) l 2 l α∈Gl(N)

We then establish an “ampliﬁed” version of the formula above. By the multiplicativity of the eigenvalues (2.1.2), for any sequence of complex numbers xl, we get

2 X X 2 X X 2 | fi(z)| | fi(z)| Ck xlλi(l) = Ck xl1 xl2 λi(l1)λi(l2) (5.1.1) h fi, fii h fi, fii fi∈Bk(N) l fi∈Bk(N) l1,l2 2 X X | fi(z)| = Ck ylλi(l) h fi, fii fi∈Bk(N) l X yl X −k/2 = √ uα(z) , k l 2 l α∈Gl(N)

98 where X yl := xl1 xl2 . d|(l1,l2) 2 l=l1l2/d Now, let  2 sign(λi(l)) if l ∈ Λ(L) ∪ Λ(L) xl := . 0 otherwise We therefore have

X x λ (l) L1−. l i l 2 2 Indeed, this follows from the relation λi(l) − λi(l ) = 1, which implies that

n 2 o max |λi(l)| , λi(l ) > 1/2,

and the prime number theorem (Theorem 1.1.1). ! a b As in [HT13], we split the counting of matrices α = as c d

M = M∗ + Mu + Mp

according to whether c , 0 and (a + d)2 , 4l (generic), or c = 0 and a , d (upper-

triangular), or (a + d)2 = 4l (parabolic).

Moreover, we have

Lemma 5.1.3. If δ < 1,M(z, l, δ) = 0.

√ Proof. It suﬃces to show that |uγ(z)| > 1 when γ ∈ Gl(N). When Trace(γ) > 2 l, we have √ 2 |uγ(z)| > |Im uγ(z)| = Trace(γ) /4l > 1. When Trace(γ) < 2 l, let g ∈ SL2(R) be a matrix

such that √ √ ! l cos θ l sin θ g−1γg = √ √ , − l sin θ l cos θ

99 −1 where θ ∈ R. By a direct calculation, we have |ug−1γg(z)| = |uγ(gz)|. Let w = g z = x + iy, then

−2 2 2 2 2 2 |uγ(z)| = |ug−1γg(w)| = y | sin θ(1 + |w| ) + 4y cos θ|/4 > 1.

Remark 5.1.2. A calculation with full details can be found in [Oli11] Appendix B.

By (5.1.1), we have

2 2− | fi(z)| X |yl| X −k/2 CkL √ |uα(z)| (5.1.2) h fi, fii k l 2 l α∈Gl(N) Z ∞ X |yl| X −k/2 X |yl| −k/2 √ |uα(z)| + √ δ d (Mu + M∗) (z, l, δ) k k 0 l 2 l α∈Gl(N) l 2 l α parabolic Z ∞ X |yl| X −k/2 X |yl| (Mu + M∗) (z, l, δ) √ |uα(z)| + k √ dδ, k +1 k k 1 2 l 2 l α∈Gl(N) l 2 l δ α parabolic where the last step follows from integration by parts and Lemma 5.1.3.

The remaining problem is to establish an upper-bound for M∗, Mu and the sum over parabolic matrices.

5.1.4 Counting lattice points

As in [HT13], we estimate the sum of M∗(z, l, δ) and the sum of Mu(z, l, δ) separately.

We state two lemmas in [HT13] below.

Lemma 5.1.4 ([HT13] Lemma 2.1). Let Θ be a Eucilidean lattice of rank 2 and D be a disc of radius R > 0 in Θ ⊗Z R (not necessarily centered at 0). If λ1 6 λ2 are the successive

100 minima of Θ, then

R R2 #(Θ ∩ D) 1 + + . (5.1.3) λ1 λ1λ2

2 Lemma 5.1.5 ([HT12] Lemma 1). Let F(N) A0(N)\H be the fundamental domain. Let z ∈ F(N). Then we have √ 3 Im z > (5.1.4) 2N

and for any (c, d) ∈ Z2 distinct from (0, 0) we have

1 |cz + d|2 > . (5.1.5) N

Remark 5.1.3. This is where the square-free condition comes into play. (5.1.5) is not true √ 2 1 3 when N = q for an integer q. For example, let z = q + i 2q2 , then it is easy to check that z is in the fundamental domain but Lemma 5.1.5 fails.

Then, we have

Lemma 5.1.6. For any z = x + iy ∈ F(N), 1 6 Λ 6 NO(1), and δ > 1, ! X Λδ (Λδ)3/2 (Λδ)2 M (z, l, δ) + + N, (5.1.6) ∗ Ny N1/2 N 16l6Λ

! X (Λδ)1/2 Λδ (Λδ)3/2 M (z, l, δ) + + N. (5.1.7) ∗ Ny N1/2 N 16l6Λ l square

O(1) For 1 6 l1 6 Λ 6 N ,

 1/2 3/2  X  Λ3δ 3 Λ3δ  2  Λ δ  M∗(z, l1l , δ)  + +  N . (5.1.8)  Ny N1/2 N  16l6Λ  

101 ! a b Proof. By the deﬁnition of M , we count the number of matrices α = ∈ G (N) ∗ c d v

such that v is an integer and |uα(z)| 6 δ as follows. Since

1 1 |u (z)| = |az + b − z(cz + d)|2 = |v + |cz + d|2 − (cz + d)(a + d)|2 , (5.1.9) α 4vy2 4vc2y2 by considering the imaginary part above, we obtain

|a + d| 6 2 (vδ)1/2 .

By considering the real part, we obtain

|v + |cz + d|2 − (cx + d)(a + d)| 6 2 (vδ)1/2 |cy|.

We therefore have

|v + |cz + d|2| 6 2 (vδ)1/2 (|cy| + |cx + d|) 6 4 (vδ)1/2 |cz + d|.

Since v > 0, we obtain that

|cz + d| (vδ)1/2 .

Furthermore, by the inequalities above, we get |cy| (vδ)1/2.

By (5.1.9),

|az + b − z(cz + d)| = |(a − d)z + b − cz2 + (cz + d)(z − z)| 6 2 (vδ)1/2 y,

which implies that

|(a − d)z + b − cz2| (vδ)1/2 y. (5.1.10)

Consider the lattice h1, zi inside C. Its covolume equals y. By (5.1.5), the shortest distance between two diﬀerent points in the lattice is at least N−1/2. In (5.1.10), we are

102 counting lattice points (a − d, b) in a disc of volume vδy2 centered at cz2. Thus, by

1/2 2 (vδ) y vδy − (5.1.3), there are 1 + N−1/2 + y possible pairs (a d, b) for each c.

When v = l is an integer, since |a+d| (vδ)1/2, we have (Λδ)1/2 many possible a+d

for a given triple (a − d, b, c).

Now, consider

(a − d)2 + 4bc = (a + d)2 − 4v. (5.1.11)

When v = l is a square, for any given triple (a − d, b, c), the number of pairs (a + d, l)

satisfying (5.1.11) is N.

2 When v = l1l2 and l1 is square-free, (5.1.11) becomes a Pell equation. So the solution √ 1+ 5 is a power of a fundamental unit which is always greater than 2 . Therefore, the number

of pairs (a + d, l2) satisfying (5.1.11) is N .

Finally, since c (vδ)1/2 /y and N|c, we have (vδ)1/2 /Ny possible values for c for

1/2 2 (vδ) y vδy − all three cases above. For each c, we have 1 + N−1/2 + y possible pairs (a d, b). For each (a − d, b, c), we have (vδ)1/2 possible (a + d, l) for the case in (5.1.6). And for the

cases in (5.1.7) and (5.1.8), we have N possible (a+d, l). This completes the proof.

Lemma 5.1.7. For any z = x + iy ∈ F(N) and 1 6 Λ 6 NO(1), the following estimations

hold true when l1, and l2 run over primes.

X 1/2 Mu(z, l1, δ) 1 + (ΛδN) y + Λδy N , (5.1.12)

16l16Λ

103 X 2 1/2 3 Mu(z, l1l2, δ) Λ + Λ (δN) y + Λ δy N , (5.1.13)

16l1l26Λ

X 2 5/2 1/2 4 Mu(z, l1l2, δ) Λ + Λ (δN) y + Λ δy N , (5.1.14) 16l1l26Λ

X 2 2 2 1/2 4 Mu(z, l1l2, δ) 1 + Λ (δN) y + Λ δy N . (5.1.15) 16l1l26Λ ! a b Proof. By (5.1.10), we need to count the number of matrices α = such that ad = v, 0 d and

|(a − d)z + b| (vδ)1/2 y

2 2 2 for all the cases such that v = l1, v = l1l2, v = l1l2 and v = l1l2.

We again consider the lattice h1, zi of covolume y and shortest length at least N−1/2 in

1/2 2 (vδ) y vδy − C. By (5.1.3), in each case, we have 1 + N−1/2 + y possible values of (a d, b). In the

ﬁrst case, we have either a = 1 or d = 1 since ad = l1, which gives rise of O(1) possible

matrices. In the next two cases, we have O(Λ) possible values of d because ad = l1l2

2 and ad = l1l2 respectively. In the last case, since both l1, l2 are prime, we have either

2 2 2 2 2 (a = 1, d = l1l2), or (a = l1, d = l1l2), or (a = l1, d = l2), or equivalent conﬁgurations. In each conﬁguration, and for a given value a − d, there are N many pairs of (a, d).

Therefore, the proof is completed.

5.1.5 The estimation of parabolic matrices

In this section, we establish an upper bound for the sum over parabolic matrices. The

−k/2 treatment in [HT13] does not apply to this case, since |uα(z)| decays much slower than the geometric side of the pre-trace formula in Maaß form case. We need a more careful

104 discussion here.

2 Denote by F(N) A0(N)\H the fundamental domain of Atkin-Lehner operators.

Lemma 5.1.8. Let z ∈ F(N),N−O(1) y 1 and k > 2, we have

X −k/2 1/2 −1/2 −1/3 1/3 −5/3 −4/3 −1 |uα(z)| θ(l)l l + y + N y + N y + N N ,

α∈Gl(N) α parabolic where θ(l) = 1 when l is a perfect square and θ(l) = 0 otherwise. Furthermore, the implied

constant does not depend on k.

Proof. When l is not a square, there are no parabolic matrices by deﬁnition (i.e. matrices ! a b such that (a + d)2 = 4l ), so that θ(l) = 0 in this case. Let l be a square integer. c d Let α be a matrix in the sum above. Since α is parabolic, there is a cusp a ∈ P1(Q) which

a is ﬁxed by α (see [Oli11] for example ). Moreover, one can assume that a = c for some

a, c ∈ Z. Furthermore, when a, c , 0, we can assume that (a, c) = 1. Let σa be a 2-by-2 matrix such that σa.∞ = a and ! a b σ = ∈ SL (Z), a c d 2 for some b, d ∈ Z.

0 −1 0 0 Consider α = σa ασa. We have that α .∞ = ∞. This shows that α is an upper- triangular matrix. Since it is parabolic with determinant l, it must be of the form √ ! l t α0 = ± √ . 0 l

For each α, we have found an upper-triangular matrix α0 through the adjoint action of

0 σa. Then we count the sum over α’s by parameterizing them as pairs (α , σa).

105 0 −1 From the equation α = σaα σa , we obtain that √ ! l − act a2t α = √ . −c2t l + act

2 Since α ∈ Gl(N), we have N|c t. Furthermore, since N is square-free, we have r, s ∈ Z such

that rs = N, and s|c,(c, r) = 1 and r|t. √ ! l 0 When t = 0, all α = ± √ are the same. When t , 0 and c = 0, we set a = 1. 0 l √ 2 2 −1 −2 When t , 0 and a = 0, we set c = 1. Moreover, |uα(z)| = 2 lyi + t|cz − a| (4l) y .

Therefore, we have

X −k/2 |uα(z)| (5.1.16)

α∈Gl(N) α parabolic √ √ √ X (2 ly)k X (2 ly)k X (2 ly)k 1 + + + √ k √ k √ k 2 2 t,0 2 lyi + t t,0 2 lyi + t|z| a,c,t,0 2 lyi + t|cz − a| N|t s.t. α∈G (N) √ √ l √ X (2 ly)k X (2 ly)k X (2 ly)k 1 + + + , √ kα √ kα √ kα kβ 2 kβ 2kβ t,0 2 ly |t| t,0 2 ly |t|z| | a,c,t,0 2 ly t|cz − a| N|t s.t. α∈Gl(N) by the Arithmetic-Geometric Mean Inequality for some positive α, β such that α + β = 1

(e.g. for any A, B > 0, AαBα 6 αA + βB ). Moreover, the implied constant is absolute and independent of k.

1 | |2 Now let kβ = 1 + for some positive < 2 . By noticing that z > 1/N when z is in

the fundamental domain, the sum of the ﬁrst three terms is easy to obtain. Let t = rt1 and

c = sc1 in the fourth sum, then (sc1, ra) = 1 by the choices of a, c, r, s. Then (5.1.16) is

bounded by

1 1+ √ l 2 y X X 1 + ly(yl) + . 21+ rs=N c1,a r|sc1z − a| (sc1,ra)=1

106 Let 1 6 R 6 N. Break the r, s sum apart as     1 1+ X X  X l 2 y  +  .   21+ rs=N rs=N  c1,a r|sc1z − a| r>R s>N/R (sc1,ra)=1 First consider the case that r > R. Since z is in the fundamental domain, there are

integers b0 and d0 such that √ √ ! ! ra b0/ r y Im √ √ .z y, 0 = 2 6 rsc1 rd r|sc1z − a|

2 which implies that r|sc1z − a| > 1. Applying Lemmas 5.1.4, 5.1.5 to the lattice h1, zi, we

2 consider the value of |sc1z − a| dyadically to obtain

1 1+ l 2 y 1/2 ! X X N 1 1/2 1+ N 1 + + (l y) . 21+ R1/2 Ry rs=N c1,a r|sc1z − a| r>R (sc1,ra)=1 Next consider the case that s > N/R. We open the norm square in the denominator to

We then choose R = N5/3y4/3 to complete the proof.

5.2 The proof of Theorem 3.2.1

√ 3 By (5.1.4), it suﬃces to consider the case that y > 2N . By Proposition 5.1.1, when

−2/3 1 + − 1 + 1/2 z ∈ F(N) and Im z > N we have | f (z)| k 4 N 6 h f, f i . Thus, we only need to

107 √ ∈ F 3 −1 Im −2/3 show the sup-norm bound (3.2.1) when z (N) and 2 N 6 (z) 6 N .

In (5.1.2), one has  L, l = 1,   2 2 2 |yl| 1, l = l1 or l1l2 or l1l or l l with L < l1, l2 < 2L primes,  2 1 2 0, otherwise.

Next, we consider the contribution of upper-triangular, parabolic and generic matrices √ separately on the right hand side of (5.1.2). Since δ is always larger than 2 l, all k-aspect

implied constants below are 2−k.

Upper-triangular matrices

When l = 1, we choose Λ = 1 in (5.1.12), then this part contributes

N L 1 + N1/2y + y .

When l = l1, via (5.1.12) again, the upper bound is

N L−1/2 1 + L1/2N1/2y + Ly .

When l = l1l2, via (5.1.13), the upper bound is

N L−1 L + L2N1/2y + L3y .

2 −3/2 5/2 1/2 4 When l = l1l2, via (5.1.14) the upper bound is N L L + L N y + L y . When

2 2 −2 2 1/2 4 l = l1l2, via (5.1.15) the upper bound is N L 1 + L N y + L y . Therefore, the total contribution is N L + LN1/2y + L5/2y . Notice that k > 3, so every integral is

convergent.

108 Parabolic matrices

2 From Lemma 5.1.8, we know that when l = 1, l1, the upper bound is LN + L y + N−1/3y1/3 + N−5/3y−4/3 N,

2 2 and when l = l1l2, the upper bound is N + L2 y + N−1/3y1/3 + N−5/3y−4/3 N.

When l is not a square, there is no contribution from the parabolic case. Hence the total contribution from the parabolic case is LN + L2 y + N−1/3y1/3 + N−5/3y−4/3 N.

Generic matrices

When l = 1, via (5.1.6), the upper bound is N L (Ny)−1 + N−1/2 + N−1 . When

−1/2 −1 3/2 −1/2 2 −1 l = l1, via (5.1.6), the upper bound is N L L(Ny) + L N + L N . When

−1 2 −1 3 −1/2 4 −1 l = l1l2, via (5.1.6), the upper bound is N L L (Ny) + L N + L N .

2 When l = l1l2, via (5.1.8), the upper bound is N L−3/2 L3/2(Ny)−1 + L3N−1/2 + L9/2N−1 .

2 2 −2 2 −1 4 −1/2 6 −1 When l = l1l2, via (5.1.7), the upper bound is N L L (Ny) + L N + L N . Hence the total contribution from the generic case is N L(Ny)−1 + L2N−1/2 + L4N−1 .

For the convergence, we need to use Lemma 5.1.7 when δ is suﬃciently large.

Therefore, we choose L = N1/3 in (5.1.2) to obtain

| f (z)|2 i kN−1/3+, h fi, fii which implies Theorem 3.2.1.

109 Chapter 6: The Proof of the Distribution of Inverses mod c

6.1 Wilton’s bound for Eisenstein series

In order to prove Theorem 3.3.1, we want to relate the problem to a sum of Kloosterman sums ﬁrst. Then, we apply Kuznietsov trace formula to obtain sums of coeﬃcients of cusp forms and Eisenstein series. In order to estimate these sums, we will need Wilton’s type of bounds (Lemma 2.1.1).

2 The Wilton’s bound of cuspidal automorphic forms on SL2 (Z) \H are stated in (6.1.7) and (6.1.8). For Eisenstein series, one may expect a similar upper bound. In order to study this case, we need a Voronoi type of duality formula. Since the coeﬃcients of standard

2 Eisenstein series on SL2 (Z) \H are proportional to the divisor function τ(n, v) (see [DI83]

Section 3 ), it suﬃces to establish a duality formula for τ(n, v).

v Lemma 6.1.1. Let τ (n, v) = P d1 . Let h (x) be a compactly supported smooth d1d2=n d2 function on R+. Then we have a summation formula for any positive integer c as

X an 1 Z τ(n, v)e h (n) = h(x) ζ(1 + 2v)c−2v xv + ζ(1 − 2v)c2v x−v dx (6.1.1) c c n ∞ ∞ √ ! 1 X X an¯ Z 4π nx + τ(n, v)e ∓ h(x)J± dx, c c 2v c ± n=1 0 where

−π J+ (z) = (J (z) − J (z)) and J− (z) = 4 cos(πv)K (z). 2v sin πv 2v −2v 2v 2v

110 Remark 6.1.1. The function ζ(1 + 2v)c−2v xv + ζ(1 − 2v)c2v x−v in the integral is analytic for

all v ∈ C, since the pole at v = 0 of ζ(1 + 2v) and ζ(1 − 2v) can be cancelled.

± Proof. Since J2v(z) are both analytic as functions of v when z , 0, and both sides of (6.1.1) 1 Re are meromorphic functions, it suﬃces to prove this Lemma for all v such that 2 > (v) > 0.

Suppose g is a compactly supported function of class C1 on R2. Let a, c be two coprime integers anda ¯ be the inverse of a mod c, e.g. aa¯ ≡ 1(mod c). Let τg(m):= P m1m2=m g(m1, m2). Then by [IK04] Prop. 4.11, we have

X ma X ma¯ τ (m)e = τ (n)e − , (6.1.2) g c f c m∈Z n∈Z

where f is given by the Fourier transform of g, namely

1 x y f (x, y):= gˆ , . c c c

Let η(x) be a smooth function such that η(x) = 1 when x > and η(x) = 0 when x < /2

with 0 < < 1. We deﬁne

!v x g (x, y):= η(x)η(y)h (xy) , 0 y

2 so that g0 is compactly supported on R .

Applying (6.1.2) with g = g0, we obtain

X ma X na¯ τ (m)e = τ (0) + τ (n)e − . g0 c f0 f0 c m∈Z n,0

We arrange the ﬁrst term as

1 1 X n 1 X n τ (0) = − gˆ (0, 0) + gˆ 0, + gˆ , 0 . f0 c 0 c 0 c c 0 c n∈Z n∈Z

111 As in the proof of [IK04] Thm. 4.10, after both applying Possion summation formula

and noticing that η(x) = 0 when x < 0, we get   1 X n Z ∞ X  Z ∞ X x v gˆ 0, =  g (x, cm) dx = η(x)η(cm)h (cmx) dx c 0 c  0  cm n∈Z 0 m∈Z 0 m>1 Z ∞ X h (cmx) = x2vη(x) dx. (6.1.3) (cmx)v 0 m>1

n Similarly, we execute the summation ofg ˆ0 c , 0 to obtain 1 1 X n − gˆ (0, 0) + gˆ , 0 c 0 c 0 c n∈Z   Z ∞ X h (cmx) 1 Z ∞ y h(y)  = x−2vη(x)  − η dy dx. (6.1.4)  (cmx)−v cx x y−v  0 m>1 0 By Section 2.11 [Tit51], for any compactly supported function F(x) on R+, we have Z ∞ X∞ Z ∞ xs−1 F(mx)dx = ζ(s) ys−1F(y)dy, (6.1.5) 0 m=1 0 for any Re (s) > 1, and

 ∞  Z ∞ X 1 Z ∞  Z ∞ xs−1  F(mx) − F(y)dy dx = ζ(s) ys−1F(y)dy, (6.1.6)  x  0 m=1 0 0 for any 0 < Re (s) < 1.

Let F(x) = h (cx) (cx)−v and s = 2v+1 in (6.1.5). When Re (v) > 0, (6.1.3) is absolutely

convergent and uniformly bounded as → 0.

Let F(x) = h (cx) (cx)v and s = −2v + 1 in (6.1.6). From Section 2.11 [Tit51], we have

X∞ 1 Z ∞ F(mx) = F(y)dy + O(1). x m=1 0 1 Re As a result, one can verify that when 2 > (v) > 0, (6.1.4) is absolutely convergent and uniformly bounded as → 0.

Thus, we let → 0 to obtain

1 Z τ (0) = h(x) ζ(1 + 2v)c−2v xv + ζ(1 − 2v)c2v x−v dx. f0 c

112 Next, we compute τ f0 (n) for n , 0.

When n > 0,

X τ f0 (n) = ( f0(n1, n2) + f0(−n1, −n2)) n1,n2>0 n1n2=n !v X 2 x n x + n y = η(x)η(y)h(xy) cos 2π 1 2 dxdy. c y c n1,n2>0 n n =n 1 2 " √ √ −1 Changing the variables (x, y) → (u vn2/n2, u vn1/n2), and letting → 0, we get

!v Z ∞ Z ∞ √ −1 ! ! X n2 2 2π vn1n2(u + u ) τ f0 (n) = h(v) cos du dv. n1 c 0 0 c n1,n2>0 n1n2=n By [IK04] (4.112)–(4.117), we have

Z ∞ √ ! + 4π nx τ f0 (n) = τ(n, v) h(x)J2v dx. 0 c

When n < 0, the proof is similar.

Eventually, we obtain

X an 1 Z τ(n, v)e h(n) = h(x) ζ(1 + 2v)c−2v xv + ζ(1 − 2v)c2v x−v dx c c n √ 1 X X∞ an¯ Z ∞ 4π nx + τ(n, v)e(± ) h(x)J± ( )dx. c c 2v c ± n=1 0

Lemma 6.1.2. Let g(m, n, c) be a compactly supported smooth function with support on

1 × 1 × 1 i j k [ 2 M, 2M] [ 2 N, 2N] [ 2C, 2C] for some M, N, C > 1, such that ∂m∂n∂cg(m, n, c) Zi+ j+k M−iN− jC−k for any nonnegative integers i, j, k. For any real number α (resp. β ),

let q1 (resp. q2 ) be the smallest integer such that there is an integer p1 (resp. p2) satisfying

113 |α − p1 | 6 1 (resp. |β − p2 | 6 1 ). Then, we have q1 M q2 N

X 1 X e (mα) e (nβ) S (m, ±n; c) g (m, n, c) c c m,n  √    5  √ !   MN  6 MN Z   + Z  MN + (MNC) .  C  q1q2 √ Remark 6.1.2. Lemma 6.1.2 is nontrivial when MN < X1+. The large sieve inequality

in [DI83] or Linnik’s conjecture on average only gives trivial bound (MN)1+.

We will need the following estimation in our proof.

(i) i Lemma 6.1.3. Let f be a function with support on [1/2, 5/2] such that f i Z for some

± Z > 1 and any nonnegative integer i. Let Jv (·) be the function deﬁned in Lemma 6.1.1. Let

t > 0, A > 1 be both real numbers. Then

Z ∞ j ± Z f (x)J2it(Ax)dx , 0 A

where the implied constant only depends on f and j.

Proof. By [Pit13] Lemma 2.1, we have integral representations as

Z ∞ + −π J2it(Ax) = (J2it(Ax) − J−2it(Ax)) = −4i cos(Ax cosh ξ) cos 2tξdξ, sinh πt 0 Z ∞ − J2it(Ax) = 4 cosh(πt)K2it(Ax) = 4 cos(Ax sinh ξ) cos 2tξdξ. 0

Thus, integration by parts gives

Z ∞ Z ∞ Z ∞ + f (x)J2it(Ax)dx = −4i cos 2tξ f (x) cos(Ax cosh ξ)dxdξ 0 0 0 −4i Z ∞ cos 2tξ Z ∞ Z Z ∞ = f 0(x) sin(Ax cosh ξ)dxdξ e−ξdξ. A 0 cosh ξ 0 A 0

− We complete the proof by repeating integration by parts. The case for J2it is similar.

114 Proof of Lemma 6.1.2. We only prove the case with a + sign, the other case will be similar.

× 1 × 1 Let f (x, y, z) be a smooth function with support on [Y, 2Y] [ 2 M, 2M] [ 2 N, 2N] such that f (i, j,k)(x, y, z) Zi+ j+kY−i M− jN−k for some Y > 0, M > 1, N > 1, Z > 1 and any

nonnegative integers i, j, k. Then by Lemma 2.4.3 with s = r = 1, we have √ ! X 1 X 4π mn e (mα) e (nβ) S (m, n; c) f , m, n c c c m,n X X ˆ = 4π ρ j(m)ρ j(n)e (mα) e (nβ) fm,n(t j) j>1 m,n X X X ˜ + 4 ψ jk(m)ψ jk(n)e (mα) e (nβ) fm,n(k − 1) k=0(2) j m,n ∞ ! ! X Z 1 1 + ϕ m, + it ϕ n, + it e (mα) e (nβ) fˆ (t)dt, 2 2 m,n m,n −∞ where

Z ∞ Z ∞ l dy i dx f˜m,n(l) = i Jl(y) f (y, m, n) , fˆm,n(t) = (J2it(x) − J−2it(x)) f (x, m, n) . 0 y 2 sinh πt 0 x

We state Wilton’s bounds as follows. When f is a Maaß newform with spectral param-

eter t f and level D, we have ([HM13] Proposition 2.4)

X 2 1 S (x, α; f ):= λ f (n)e(nα) D 1 + |t f | x 2 D 1 + |t f | x . (6.1.7) n6x

When f is a holomorphic newform with weight k and level D, we have ([HM13] Proposi-

tion 2.5)

X 3 1 S (x, α; f ):= λ f (n)e(nα) Dk 2 x 2 (Dkx) , (6.1.8) n6x where the implied constant only depends on .

115 In particular, when the level D = 1, every Hecke eigenform f j is a newform and, for any positive integer n, its n-th Fourier coeﬃcient ρ j(n) satisﬁes (by (2.1.2))

ρ j(n) = λ j(n)ρ j(1).

Therefore, we have X X ˆ ρ j(m)ρ j(n)e (mα) e (nβ) fm,n(t j) j>1 m,n X X 2 = |ρ j(1)| λ j(m)λ j(n)e (mα) e (nβ) fˆm,n(t j) j>1 m,n ∞ X 2 = |ρ j(1)| S (x, α; f j)S (y, −β; f j)∂x∂y fˆx,y(t j)dxdy j>1 −∞ ∞ X "4 √ 2 ˆ 1 + |t j| |ρ j(1)| 1 + |t j| MN xy ∂x∂y fx,y(t j) dxdy. j>1 −∞ ∞ X 4 " √ 2 ˆ 1 + |t j| |ρ j(1)| 1 + |t j| MN xy ∂x∂y fx,y(t j) dxdy −∞ |t j|6max{2Y,1} ∞ X 4 " √ 2 ˆ + 1 + |t j| |ρ j(1)| 1 + |t j| MN xy ∂x∂y fx,y(t j) dxdy. −∞ |t j|>max{2Y,1} " For |t j| 6 max{2Y, 1}, by Lemma 2.4.4 part (a), we have ∞ √ | |! √ ˆ 1 + log(Y/Z) xy ∂x∂y fx,y(t j) dxdy MN. −∞ 1 + Y/Z " For |t j| > max{2Y, 1}, using Lemma 2.4.4 part (b), we have

∞ !6 ! √ Z 1 Y √ xy ∂ ∂ fˆ t dxdy |t| MN. x y x,y( j) 1/2 + (1 + log ) −∞ |t| |t| |t| " Moreover, applying Lemma 2.4.5 with N = 1, we have

X 2 2 |ρ j(1)| T + 1.

|t j|6T Gathering the results above, and also noticing Z > 1, we have X X ˆ ρ j(m)ρ j(n)e (mα) e (nβ) fm,n(t j) (6.1.9) j 1 m,n > √ 5 6 ZY (1 + log |Y/Z|) + Z MN(MN) .

116 For the sum of coeﬃcients of holomorphic modular forms, we have a similar bound 1 X X X ψ (m)ψ (n)e (mα) e (nβ) f˜ (k − 1) (6.1.10) 2π jk jk m,n k=0(2) j m,n √ 5 6 ZY (1 + log |Y/Z|) + Z MN(MN) .

The proof follows along the same line.

For the sum of coeﬃcients of Eisenstein series, by [IK04] Section 15, 16 and [DI83]

(1.17) (a diﬀerent normalization is in [DI83] Thm. 1 ), we have π−itτ(n, it) ϕ n, 1 + it = . 2 ζ(1 + 2it)

p1 p2 Let θq = α − and θq = α − . Let 1 q1 2 q2 ! π−it ζ(1 − 2it) L(t, x, q):= q−2it xit + q2it x−it . q ζ(1 + 2it) ˆ We then apply Lemma 6.1.1 with v = it and h(x, y, t) = e(θq1 x)e(θq2 y) fx,y(t) to obtain X 1 1 ˆ ϕ m, 2 + it ϕ n, 2 + it e (mα) e (nβ) fm,n(t) (6.1.11) m,n  ∞ ! √ !  1 X X p¯ n 4π n x   1 ∓ 1 1 J± 1  = h(x, y, t) L(t, x, q1) + ϕ n1, 2 + it e 2it   q1 q1 q1  ± n1=1 " ∞ ! √ !  1 X X p¯ n 4π n2y  ×  1 ∓ 2 2 J±  L(t, y, q2) + ϕ n2, 2 + it e 2 2it  dxdy.  q2 q2 q2  ± n2=1 1 × 1 × One can check that h(x, y, t) is supported on [ 2 M, 2M] [ 2 N, 2N] R. Moreover, by Lemma 2.4.4, !6 ! Z Y(1 + log |t|) h(i, j,0)(x, y, t) |t|−1/2 + (Z + |θ |M + |θ |N)i+ j |t| |t| q1 q2 !6 ! Z Y(1 + log |t|) |t|−1/2 + (Z + 1)i+ j M−iN− j |t| |t| when t > max{2Y, 1}, and 1 + log |Y/Z| h(i, j,0)(x, y, t) (Z + 1 + |θ |M + |θ |N)i+ j 1 + Y/Z q1 q2 Z(1 + log |Y/Z|) (Z + 1)i+ j M−iN− j 1 + Y

117 when t 6 max{2Y, 1}.

2 −1+ Applying Lemma 6.1.3, we have that when n1 > (Z + 1)q1N M (resp. n2 > (Z +

2 −1+ 1)q2N M , the n1-sum (resp. n2 -sum ) in (6.1.11) becomes negligible. Therefore, after truncating both n1 and n2 sums and applying Lemma 2.4.5 with N = 1, we obtain an upper bound for n1 and n2 sums in (6.1.11) as

Z(1 + log |Y/Z|) (Z + 1)2q q (MN) 1 + Y 1 2 when t 6 max{2Y, 1}, and

!6 ! Z Y(1 + log |t|) |t|−1/2 + (Z + 1)2q q (MN) |t| |t| 1 2 when t > max{2Y, 1}.

Next, by [Tit51] Thm 5.17,

L(t, x, q) (1 + |t|).

√ √ Since q1 M and q2 N (Dirichlet’s approximation theorem ), by taking the integral against t, we obtain

∞ ! ! X Z 1 1 ϕ m, + it ϕ n, + it e (mα) e (nβ) fˆ (t)dt (6.1.12) 2 2 m,n m,n −∞ ! MN O ZY5(1 + log |Y/Z|) + Z6 (MN) . q1q2 √ 4π yz Finally, let f (x, y, z) = g x , y, z . We then complete the proof by combining (6.1.9), (6.1.10), and (6.1.12).

118 6.2 The proof of Theorem 3.3.1

Proof of Theorem 3.3.1. Considering the Fourier expansion of h(x, y), we have X X a a¯ c h X − α , X − β g (6.2.1) 1 c 2 c X c>0 00 m,n 1 2 1 2 00 m,n 0 0, T=2i or T=0

FT (x) = 1 when |x| ∈ [T, 2T] and FT (x) = 0 when |x| < [2/3T, 5/2T]. And for T = 0, P F0(x) = 1 − i>0 F2i (x) when x > 0 and F0(x) = 0 when x < −1/2. Moreover, let

( j) − j i FT (x) j T for any T = 2 and nonnegative integer j, where the implied constant depends only on j. Then, denoting by fM,N(m, n, c) = f (m, n, c)FM(m)FN(n) where M, N are dyadic numbers or zeros, we have X X a a¯ c h X − α , X − β g 1 c 2 c X c>0 0 0,

−10 −2 f (m, n, c) (ZX) (M/X1) .

Therefore

X X X X −8 2 2 e (−mα) e (−nβ) S (m, n; c) fM,N(m, n, c) (ZX) X1 X2, M=2i N=2i or 0 c m,n 1+ MZ X1X

119 where we use the trivial bound |S (m, n; c)| 6 c for Kloosterman sums.

1+ When N Z X2X , we have the same upper bound. Thus, by (6.2.1) we have

X X a a¯ c h X − α , X − β g (6.2.2) 1 c 2 c X c>0 0

Next, we need to analyze the trivial cases, i.e. when M = 0 or N = 0.

When M = N = 0, the only (m, n) in the support of f0,0 is (0, 0). S (0, 0, c) = ϕ(c),

1 c f0,0(0, 0, c) = g( ) h(x, y)dxdy. Thus, X1X2 X

∞ P c X c ϕ(c)g( ) e (−mα) e (!−nβ) S (m, n; c) f (m, n, c) = h(x, y)dxdy X . 0,0 X X c,m,n −∞ 1 2 " P n 1 When M = 0, N 0, S (0, n, c) = aµ , f0,N(m, n, c) . Then , a|(n,c) a X1X2

X X e (−mα) e (−nβ) S (m, n; c) f0,N(m, n, c) N=2i c,n,m 1+ N6Z X2X −1 −1 X X X X X1 X2 a N=2i cX nN a|(n,c) 1+ N6Z X2X −1 1+2 −1 −1 1+2 X1 (ZX) 6 (X1 + X2 )(ZX) .

When N = 0, M , 0, we have the same upper bound as above.

120 Finally, by noticing X1, X2 X and applying Lemma 6.1.2,(6.2.2) implies

X X a a¯ c h X − α , X − β g 1 c 2 c X c>0 0

121 Bibliography

[BH10a] Valentin Blomer and Gergely Harcos, Twisted L-functions over number ﬁelds and Hilbert’s eleventh problem, Geom. Funct. Anal. 20 (2010), no. 1, 1–52. MR 2647133 (2011g:11090)

[BH10b] Valentin Blomer and Roman Holowinsky, Bounding sup-norms of cusp forms of large level, Invent. Math. 179 (2010), no. 3, 645–681. MR 2587342 (2011d:11095)

[BHM07a] V. Blomer, G. Harcos, and P. Michel, A Burgess-like subconvex bound for twisted L-functions, Forum Math. 19 (2007), no. 1, 61–105, Appendix 2 by Z. Mao. MR 2296066 (2008i:11067)

[BHM07b] Valentin Blomer, Gergely Harcos, and Philippe Michel, Bounds for modular L- functions in the level aspect, Ann. Sci. Ecole Norm. Sup. (4) 40 (2007), no. 5, 697–740. MR 2382859 (2009g:11058)

1 [BI86] E. Bombieri and H. Iwaniec, On the order of ζ( 2 +it), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), no. 3, 449–472. MR 881101 (88i:11054a)

[Blo04] Valentin Blomer, Shifted convolution sums and subconvexity bounds for automorphic L-functions, Int. Math. Res. Not. (2004), no. 73, 3905–3926. MR 2104288 (2005m:11090)

[BR05] Joseph Bernstein and Andre Reznikov, Periods, subconvexity of L-functions and representation theory, J. Diﬀerential Geom. 70 (2005), no. 1, 129–141. MR 2192063 (2007b:11063)

[Bur62a] D. A. Burgess, On character sums and L-series, Proc. London Math. Soc. (3) 12 (1962), 193–206. MR 0132733 (24 #A2570)

[Bur62b] , On character sums and primitive roots, Proc. London Math. Soc. (3) 12 (1962), 179–192. MR 0132732 (24 #A2569)

[Bur63] , On character sums and L-series. II, Proc. London Math. Soc. (3) 13 (1963), 524–536. MR 0148626 (26 #6133)

122 [Cas73] William Casselman, On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301–314. MR 0337789 (49 #2558)

[Cha11] Tsz Ho Chan, An almost all result on q1q2 ≡ c (mod q), Monatsh. Math. 162 (2011), no. 1, 29–39. MR 2747342 (2011k:11125)

[Cog04] James W. Cogdell, Lectures on L-functions, converse theorems, and functoriality for GLn, Lectures on automorphic L-functions, Fields Inst. Monogr., vol. 20, Amer. Math. Soc., Providence, RI, 2004, pp. 1–96. MR 2071506

[Dav40] H. Davenport, Note on linear fractional substitutions with large determinant, Ann. of Math. (2) 41 (1940), 59–62. MR 0001771 (1,295a)

[DFI93] W. Duke, J. Friedlander, and H. Iwaniec, Bounds for automorphic L-functions, Invent. Math. 112 (1993), no. 1, 1–8. MR 1207474 (94c:11043)

[DFI94] W. Duke, J. B. Friedlander, and H. Iwaniec, Bounds for automorphic L- functions. II, Invent. Math. 115 (1994), no. 2, 219–239. MR 1258904 (95a:11044)

[DFI01] , Bounds for automorphic L-functions. III, Invent. Math. 143 (2001), no. 2, 221–248. MR 1835388 (2003c:11043)

[DFI02] , The subconvexity problem for Artin L-functions, Invent. Math. 149 (2002), no. 3, 489–577. MR 1923476 (2004e:11046)

[DI83] J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coeﬃ- cients of cusp forms, Invent. Math. 70 (1982/83), no. 2, 219–288. MR 684172 (84m:10015)

[Duk88] W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math. 92 (1988), no. 1, 73–90. MR 931205 (89d:11033)

[FI92] J. Friedlander and H. Iwaniec, A mean-value theorem for character sums, Michigan Math. J. 39 (1992), no. 1, 153–159. MR 1137896 (92k:11084)

[FW09] Brooke Feigon and David Whitehouse, Averages of central L-values of Hilbert modular forms with an application to subconvexity, Duke Math. J. 149 (2009), no. 2, 347–410. MR 2541706 (2010m:11067)

[Gar06] M. Z. Garaev, On the logarithmic factor in error term estimates in certain addi- tive congruence problems, Acta Arith. 124 (2006), no. 1, 27–39. MR 2262138 (2007j:11127)

123 [GH11] Dorian Goldfeld and Joseph Hundley, Automorphic representations and L- functions for the general linear group. Volume I, Cambridge Studies in Ad- vanced Mathematics, vol. 129, Cambridge University Press, Cambridge, 2011, With exercises and a preface by Xander Faber. MR 2807433 (2012i:11054)

[GJ79] Stephen Gelbart and Herve Jacquet, Forms of GL(2) from the analytic point of view, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 213–251. MR 546600 (81e:10024)

[Har11] Gergely Harcos, Subconvex bounds for automorphic L-functions and applications, D.Sc. thesis, Hungarian Academy of Sciences, (2011).

[HB09] D. R. Heath-Brown, Convexity bounds for L-functions, Acta Arith. 136 (2009), no. 4, 391–395. MR 2476604 (2010b:11115)

[HL94] Jeffrey Hoffstein and Paul Lockhart, Coefficients of Maass forms and the Siegel zero, Ann. of Math. (2) 140 (1994), no. 1, 161–181, With an appendix by Dorian Goldfeld, Hoffstein and Daniel Lieman. MR 1289494 (95m:11048)

[HM06] Gergely Harcos and Philippe Michel, The subconvexity problem for Rankin- Selberg L-functions and equidistribution of Heegner points. II, Invent. Math. 163 (2006), no. 3, 581–655. MR 2207235 (2007j:11063)

[HM13] Roman Holowinsky and Ritabrata Munshi, Level aspect subconvexity for Rankin-Selberg L-functions, Automorphic representations and L-functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22, Tata Inst. Fund. Res., Mumbai, 2013, pp. 311–334. MR 3156856

[HMQ14] R. Holowinsky, R. Munshi, and Z. Qi, Hybrid subconvexity bounds for L(1/2, S ym2 f × g), Preprint available at arXiv:1401.6695 (2014).

[HS10] Roman Holowinsky and Kannan Soundararajan, Mass equidistribution for Hecke eigenforms, Ann. of Math. (2) 172 (2010), no. 2, 1517–1528. MR 2680499 (2011i:11061)

[HT12] Gergely Harcos and Nicolas Templier, On the sup-norm of Maass cusp forms of large level: II, Int. Math. Res. Not. IMRN (2012), no. 20, 4764–4774. MR 2989618

[HT13] , On the sup-norm of Maass cusp forms of large level. III, Math. Ann. 356 (2013), no. 1, 209–216. MR 3038127

124 [HT14] Roman Holowinsky and Nicolas Templier, First moment of Rankin-Selberg central L-values and subconvexity in the level aspect, Ramanujan J. 33 (2014), no. 1, 131–155. MR 3142436

[Hux02] M. N. Huxley, Integer points, exponential sums and the Riemann zeta function, Number theory for the millennium, II (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, pp. 275–290. MR 1956254 (2004d:11098)

[Hux05] , Exponential sums and the Riemann zeta function. V, Proc. London Math. Soc. (3) 90 (2005), no. 1, 1–41. MR 2107036 (2005h:11180)

[Ich08] Atsushi Ichino, Trilinear forms and the central values of triple product L-functions, Duke Math. J. 145 (2008), no. 2, 281–307. MR 2449948 (2009i:11066)

[IK04] Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathemat- ical Society, Providence, RI, 2004. MR 2061214 (2005h:11005)

[IS95] H. Iwaniec and P. Sarnak, L∞ norms of eigenfunctions of arithmetic surfaces, Ann. of Math. (2) 141 (1995), no. 2, 301–320. MR 1324136 (96d:11060)

[Iwa97] Henryk Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 1474964 (98e:11051)

[JK04] J. Jorgenson and J. Kramer, Bounding the sup-norm of automorphic forms, Geom. Funct. Anal. 14 (2004), no. 6, 1267–1277. MR 2135167 (2005m:11071)

[Kim03] Henry H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2, J. Amer. Math. Soc. 16 (2003), no. 1, 139–183, With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. MR 1937203 (2003k:11083)

[KMV02] E. Kowalski, P. Michel, and J. VanderKam, Rankin-Selberg L-functions in the level aspect, Duke Math. J. 114 (2002), no. 1, 123–191. MR 1915038 (2004c:11070)

[Li11] Xiaoqing Li, Bounds for GL(3) × GL(2) L-functions and GL(3) L-functions, Ann. of Math. (2) 173 (2011), no. 1, 301–336. MR 2753605 (2012g:11092)

[Lin08] E. Lindelof,¨ Quelques remarques sur la croissance de la fonction ζ(s), Bull. Sci. Math. 32 (1908), 341–356.

125 [Lin06] Elon Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. (2) 163 (2006), no. 1, 165–219. MR 2195133 (2007b:11072)

[Mer96] Lo¨ıc Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), no. 1-3, 437–449. MR 1369424 (96i:11057)

[Mic04] P. Michel, The subconvexity problem for Rankin-Selberg L-functions and equidistribution of Heegner points, Ann. of Math. (2) 160 (2004), no. 1, 185– 236. MR 2119720 (2006d:11048)

[MNV] P. Micheal, P. Nelson, and A. Venkatesh, Simultaneous subconvex bounds for product l-functions, In preparation.

[Mol02] Giuseppe Molteni, Upper and lower bounds at s = 1 for certain Dirichlet series with Euler product, Duke Math. J. 111 (2002), no. 1, 133–158. MR 1876443 (2002m:11084)

[MR12] Philippe Michel and Dinakar Ramakrishnan, Consequences of the Gross- Zagier formulae: stability of average L-values, subconvexity, and non- vanishing mod p, Number theory, analysis and geometry, Springer, New York, 2012, pp. 437–459. MR 2867928

[Mun13a] Ritabrata Munshi, The circle method and bounds for L-functions - iii: t-aspect subconvexity for GL(3) L-functions, Preprint available at arXiv:1301.1007 (2013).

[Mun13b] , Shifted convolution sums for GL(3) × GL(2), Duke Math. J. 162 (2013), no. 13, 2345–2362. MR 3127803

[MV10] Philippe Michel and Akshay Venkatesh, The subconvexity problem for GL2, Publ. Math. Inst. Hautes Etudes Sci. (2010), no. 111, 171–271. MR 2653249 (2012c:11111)

[Nel13] Paul D. Nelson, Stable averages of central values of Rankin-Selberg L- functions: some new variants, J. Number Theory 133 (2013), no. 8, 2588– 2615. MR 3045204

[Oli11] Rene Olivetto, On the sup-norm of holomorphic cusp forms, M. S. Thesis. University of Bordeaux. (2011).

[Pit13] Nigel J. E. Pitt, On an analogue of Titchmarsh’s divisor problem for holomorphic cusp forms, J. Amer. Math. Soc. 26 (2013), no. 3, 735–776. MR 3037786

126 [Ram00] Dinakar Ramakrishnan, Modularity of the Rankin-Selberg L-series, and mul- tiplicity one for SL(2), Ann. of Math. (2) 152 (2000), no. 1, 45–111. MR 1792292 (2001g:11077)

[RS94] Zeev Rudnick and Peter Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161 (1994), no. 1, 195–213. MR 1266075 (95m:11052)

[RS98] , The pair correlation function of fractional parts of polynomials, Comm. Math. Phys. 194 (1998), no. 1, 61–70. MR 1628282 (99g:11088)

[Sar01] Peter Sarnak, Estimates for Rankin-Selberg L-functions and quantum unique ergodicity, J. Funct. Anal. 184 (2001), no. 2, 419–453. MR 1851004 (2003c:11050)

[Shp12] Igor E. Shparlinski, Modular hyperbolas, Jpn. J. Math. 7 (2012), no. 2, 235– 294. MR 2995230

[Sou05] David Soudry, On Langlands functoriality from classical groups to GLn, Asterisque (2005), no. 298, 335–390, Automorphic forms. I. MR 2141707 (2006e:11070)

[Sou10a] Kannan Soundararajan, Quantum unique ergodicity for SL2(Z)\H, Ann. of Math. (2) 172 (2010), no. 2, 1529–1538. MR 2680500 (2011j:11098)

[Sou10b] , Weak subconvexity for central values of L-functions, Ann. of Math. (2) 172 (2010), no. 2, 1469–1498. MR 2680497 (2011i:11077)

[Spi71] Robert Spira, Calculation of the gamma function by Stirling’s formula, Math. Comp. 25 (1971), 317–322. MR 0295539 (45 #4605)

[SS89] A. Seeger and C. D. Sogge, Bounds for eigenfunctions of diﬀerential operators, Indiana Univ. Math. J. 38 (1989), no. 3, 669–682. MR 1017329 (91f:58097)

[Tem10] Nicolas Templier, On the sup-norm of Maass cusp forms of large level, Selecta Math. (N.S.) 16 (2010), no. 3, 501–531. MR 2734340 (2012d:11096)

[Tem14] , Large values of modular forms, Preprint available at arXiv:1207.6134 (2014).

[Tit51] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485 (13,741c)

[Wal81] J.-L. Waldspurger, Sur les coeﬃcients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484. MR 646366 (83h:10061)

127 [Wat95] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathe- matical Library, Cambridge University Press, Cambridge, 1995, Reprint of the second (1944) edition. MR 1349110 (96i:33010)

[Wat08] Thomas C. Watson, Rankin triple products and quantum chaos, Preprint available at arXiv:0810.0425 (2008).

[Wey21] Hermann Weyl, Zur Absch¨atzungvon ζ(1 + it), Math. Zeit. 10 (1921), 88–101.

[Wil29] J. R. Wilton, A note on Ramanujan0s arithmetical function τ(n), Proc. Cam- bridge Philos. Soc. 25(II) (1929), 121–129.

[Xia07] Honggang Xia, On L∞ norms of holomorphic cusp forms, J. Number Theory 124 (2007), no. 2, 325–327. MR 2321365 (2008h:11036)

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