Subconvexity for G L 3 (R) L -Functions via Integral Representations
SSubconvexity for GL ( R ) L -Functions via Integral Representations Raphael SchumacherDepartment of MathematicsETH ZurichR¨amistrasse 1018092 ZurichSwitzerland [email protected]
Abstract
We study the subconvexity problem for GL ( R ) L -functions in the t-aspect usingintegral representations by combining techniques employed by Michel–Venkatesh intheir study of the corresponding problem for GL with ideas from recent works ofMunshi, Holowinsky–Nelson and Lin. Our main objective is to explain in detail theorigin of the “key identity” arising in the latter series of works from the perspective ofintegral representations of L -functions and automorphic representation theory. The subconvexity problem for standard automorphic L -functions on GL over arbitrarynumber fields was completely solved in the paper of Philippe Michel and Akshay Venkatesh[14]. They used suitably truncated integral representations of the corresponding L -functions[14, Lemma 5.1.4, 254–256], [20, Lemma 11.9., 1088–1089] and dynamical arguments, specificto GL , to obtain their main theorem. The starting point of their argument, specialized tothe case of automorphic forms on GL ( Z ) \ GL ( R ), is the global zeta integral Z ( ϕ, + iT )given by Z ( ϕ, + iT ) = (cid:90) y ∈ R × / Z × ϕ ( a ( y )) y iT d × y and a basic unfolding principle relating global and local zeta functionals with their associated L -functions. The main step in the proof is the estimation of the global functional from abovefor a suitable choice of vector ϕ for which the local functional has a good lower bound.The first subconvex bounds for GL(1) twists of a fixed Hecke–Maass cusp form on GL(3),which is not necessarily self-dual, were obtained by Munshi in [16, 17]. His technique was1 a r X i v : . [ m a t h . N T ] A p r ubsequently simplified by Holowinsky–Nelson [5] for the q -aspect and Lin [13] for the t -aspect. The authors of those papers discovered, through a careful study of Munshi’s work(see [5, App. B]), a key identity implicit in his papers underlying the success of his method.By extracting that key identity, which amounts to the Poisson summation formula appliedto an incomplete Gauss sum, they were able to streamline the method and improve theexponent. For the t -aspect as addressed by Lin [13], the relevant key identity is ∞ (cid:88) r =1 r − iT e (cid:16) − nr (cid:17) V (cid:18) rN/T (cid:19) = NT / (cid:18) πT (cid:19) − iT e (cid:18) − T π (cid:19) n − iT V A (cid:18) πnN (cid:19) + O (cid:18) NT A (cid:19) + (cid:18) NT (cid:19) − iT (cid:88) r ∈ Z r (cid:54) =0 (cid:90) R x − iT e (cid:18) − nTN x (cid:19) V ( x ) e (cid:18) − rN xT (cid:19) dx. (1)Here V and V A are fixed smooth cutoffs; see [13] for details. As explained in [5, 13], thekey identity leads quickly to a subconvex bound after an amplification step and some fairlystandard manipulations.The authors of these works left open the question of whether there might be a naturalway to discover the usefulness of the key identity. It is also natural to ask whether the ap-plicability of such identities to the subconvexity problem is an “accident” specific to the pairof groups (GL(3) , GL(1)) or whether it extends, e.g., to (GL( n ) , GL( m )). One might hopethat the theory of integral representations of L -functions could offer some useful perspective.In this paper, we explain the key identity (1) and its application to subconvexity from theperspective of integral representations of the L -function attached to automorphic represen-tations π on GL ( R ). Our hope – not yet realized – is that the insight gained by doing so willbe useful for extending the method of Munshi and its simplifications by Holowinsky–Nelsonand Lin to more general pairs (GL( n ) , GL( m )).Our starting point, as in the work of Michel–Venkatesh, is a truncated integral represen-tation for the global zeta integral Z ( ϕ, + iT ) on GL ( R ), using a carefully chosen vector ϕ ∈ π obtained roughly as a large unipotent translate of a vector whose Whittaker func-tion W ϕ is a smooth bump function of suitable support. After localizing the zeta integral Z ( ϕ, + iT ) to a suitable bounded interval I ⊂ R × + , we then extend the integral by zeroto the positive real line R + and approximate after unfolding the corresponding local zetaintegral Z ( W ϕ , + iT ) by a Riemann sum. This approximation step is seen to have the sameeffect as the key identity (1). Thus, the key identity can be seen as the replacement of thelocal zeta integral Z ( W ϕ , + iT ) by its discrete version via a Riemann sum. In view of thisquantization, it is not just the phase n − iT for which the key identity is a substitute as in[13], but it is really the whole local zeta integral which the key identity replaces. The keyidentity has now become more comprehensible, because there are no longer additional errorterms.This method will also work for the q -aspect, where it reproduces the key identity of [5].Therefore, the above mentioned Whittaker function W ϕ , which relies on basic automorphic2rinciples, is able to predict the key identity relevant for the GL subconvexity problem inall known aspects. Indeed, the method should apply in the adelic setting as well, leading toa uniform in all aspects bound as in the work of Michel–Venkatesh; we will return to thispoint in a future paper and we hope that the structural perspective suggested here may beuseful in identifying analogous phenomena for GL n ( R ).This paper is organized as follows: Section 2 introduces important definitions and basicfacts, which will be needed through the paper. In section 3 we present and prove a newformula for the global zeta integral for GL ( R ), which relies on a stationary phase compu-tation for the special choice of Whittaker function in the subsections 3.1 and 3.2. This newidentity reduces the subconvexity problem for GL to the problem of estimating a periodintegral of a well-chosen vector in an automorphic representation and is the heart of section3. In section 4 we explain the genesis of the key identity and in the last part of the paper,we use Lin’s results [13] to conclude. As usual we denote exp(2 πix ) by e ( x ). We will use the variable ε > A (cid:28) B will meanthat | A | ≤ C | B | for some constant C . The notation A (cid:16) B will mean that B/T ε (cid:28) A (cid:28) BT ε . We will also use the space R × + := (0 , ∞ ) ∼ = R × / Z × with the corresponding measure d × y := dy | y | .We define the two brackets ( a, b ) and [ a, b ] by ( a, b ) := gcd( a, b ) and [ a, b ] := lcm( a, b ). Wedenote by S ( a, b ; c ) the Kloosterman sum modulo c [5, 13].Let π be an automorphic cuspidal representation of GL ( R ) and let W ( π ) and K ( π ) be itsunique Whittaker and Kirillov models with respect to the additive character ψ : x z y (cid:55)→ e ( x + y ) . Therefore, π ⊂ L (GL ( Z ) \ GL ( R )) is a fixed Hecke-Maass cusp form on GL ( Z ).We define the matrix element a ( y ) by a ( y ) : = y ∈ GL ( R ) . The following definitions and theorems are sometimes modified versions of the definitionsand theorems given in the corresponding references. The little modifications are necessaryto make the following computations in this article as natural as possible.3 efinition 1. (The Whittaker function for GL ( R )) [7, pages 180–181], [8, pages 235–236]Let ϕ ∈ π . We define the Whittaker function W ϕ corresponding to ϕ ∈ π by W ϕ ( g ) : = (cid:90) (cid:90) (cid:90) ϕ x z y g e ( − x − y ) dxdydz. Definition 2. (The first projection ϕ of ϕ ) [2, pages 63–72], [3], [9], [15]We define the first projection ϕ of the automorphic form ϕ by ϕ ( g ) : = (cid:90) u ∈ R / Z (cid:90) v ∈ R / Z ϕ u v g e ( − v ) dudv. We have the Fourier-Whittaker expansion ϕ ( g ) = ∞ (cid:88) n = −∞ n (cid:54) =0 a ( | n | , | n | W ϕ n g , with Fourier-Whittaker coefficients a ( | n | , ∈ C .In particular, by setting g := a ( y ), we obtain the expansion ϕ ( a ( y )) = ∞ (cid:88) n = −∞ n (cid:54) =0 a ( | n | , | n | W ϕ ( a ( ny )) . Definition 3. (The dual automorphic form (cid:101) ϕ ) [7, pages 180–181], [8, pages 235–236]We define the automorphic form (cid:101) ϕ , which is dual to the automorphic form ϕ by the expression (cid:101) ϕ ( g ) : = ϕ (cid:0) t g − (cid:1) = ϕ (cid:0) w · t g − (cid:1) , where w : = − ∈ SL ( R ) ⊂ GL ( R ) . Definition 4. (The dual Whittaker function (cid:102) W ϕ belonging to (cid:101) ϕ ) [7, page 181], [8, 235–236]We define the dual Whittaker function (cid:102) W ϕ corresponding to the automorphic form (cid:101) ϕ by (cid:102) W ϕ ( g ) : = W ϕ ( w · t g − ) = W (cid:101) ϕ ( g ) , where the matrix element w is as above.This means that we have the Fourier-Whittaker expansion (cid:101) ϕ ( g ) = ∞ (cid:88) n = −∞ n (cid:54) =0 a (1 , | n | ) | n | (cid:102) W ϕ n g . heorem 5. (The GL ( R ) projection identity) [8, page 238]Let ϕ ∈ π be an automorphic form on GL ( R ) . We have ϕ ( g ) = (cid:90) x ∈ R (cid:101) ϕ x · w (cid:48) · t g − dx, where w (cid:48) is given by w (cid:48) : = − −
10 1 0 ∈ GL ( R ) . Definition 6. (The global Zeta integrals for GL ( R )) [7, page 171], [8, pages 234–241], [15]We define the two global zeta integrals Z ( ϕ, s ) and (cid:101) Z ( (cid:101) ϕ, s ) by Z ( ϕ, s ) : = (cid:90) R × / Z × ϕ y y s − d × y, (cid:101) Z ( (cid:101) ϕ, s ) : = (cid:90) R × / Z × (cid:90) x ∈ R (cid:101) ϕ y x · w (cid:48) y s − dxd × y. Definition 7. (The local Zeta integrals for GL ( R )) [2, page 137], [8, page 223], [15]The two local zeta integrals Z ( W ϕ , s ) and (cid:101) Z ( (cid:102) W ϕ , s ) for GL ( R ) are given by Z ( W ϕ , s ) : = (cid:90) R × W ϕ y y s − d × y, (cid:101) Z ( (cid:102) W ϕ , s ) : = (cid:90) R × (cid:90) x ∈ R (cid:102) W ϕ y x · w (cid:48) y s − dxd × y. Definition 8. (The L -function for a representation π on GL ( R )) [4, 174, 279], [9], [15]We set L ( π, s ) : = ∞ (cid:88) n =1 a ( n, n s . Theorem 9. (The relation between Z ( ϕ, s ) and Z ( W ϕ , s ) ) [8, pages 234–248], [15]Let ϕ ∈ π be the automorphic function corresponding to the Whittaker function W ϕ .We have Z ( ϕ, s ) = L ( π, s ) Z ( W ϕ , s ) . roof. This is given in the adelic language in [7, 8]. We recall the basic unfolding calculationfor completeness.We calculate Z ( ϕ, s ) = (cid:90) R × / Z × ϕ y y s − d × y = (cid:90) R × / Z × ∞ (cid:88) n = −∞ n (cid:54) =0 a ( | n | , | n | W ϕ ( a ( ny )) y s − d × y = ∞ (cid:88) n = −∞ n (cid:54) =0 a ( | n | , | n | (cid:90) R × / Z × W ϕ ( a ( ny )) y s − d × y = ∞ (cid:88) n =1 a ( n, n (cid:90) R × W ϕ ( a ( ny )) y s − d × y = ∞ (cid:88) n =1 a ( n, n s (cid:90) R × W ϕ ( a ( y )) y s − d × y = L ( π, s ) Z ( W ϕ , s ) . Theorem 10. (The global functional equation for GL ( R ) ) [8, 234–248],[15]Let ϕ ∈ π be the automorphic function corresponding to the Whittaker function W ϕ .We have Z ( ϕ, s ) = (cid:101) Z ( (cid:101) ϕ, − s ) = L ( (cid:101) π, − s ) (cid:101) Z ( (cid:102) W ϕ , − s ) . Theorem 11. (The local functional equation for GL ( R ) ) [2, 133–142], [8, pages 223–224]We have the local functional equation (cid:101) Z ( (cid:102) W ϕ , − s ) = γ ( π, s ) Z ( W ϕ , s ) , where the gamma factor γ ( π, s ) is defined by γ ( π, s ) : = L ( π, s ) L ( (cid:101) π, − s ) = ε ∞ ( π, s ) L ∞ ( (cid:101) π, − s ) L ∞ ( π, s ) = π s − Γ (cid:0) − s + α (cid:1) Γ (cid:0) − s + α (cid:1) Γ (cid:0) − s + α (cid:1) Γ (cid:0) s − α (cid:1) Γ (cid:0) s − α (cid:1) Γ (cid:0) s − α (cid:1) . In the above equation, the three constants α , α , α ∈ C are complex Langlands parameters,which satisfy | Re( α i ) | < for i = 1 , , and depend on the representation π of GL ( R ) .Therefore, the gamma factor γ ( π, s ) has no poles for Re( s ) ≤ . Theorem 12. (Substructure of the Kirillov model of π ) [10], [11, Theorem 1]Let U n ( R ) be the subgroup of upper triangular unipotent matrices in GL n ( R ) with ’s onthe diagonal and real entries above the diagonal and denote by θ n : U n ( R ) → C its uniquemultiplicative character such that θ n ( u · v ) = θ n ( u ) θ n ( v ) for all u, v ∈ U n ( R ) .Let π be a generic unitary irreducible representation of GL n ( R ) and denote by C ∞ c ( θ n − , GL n − ( R )) the space of smooth and compactly supported modulo U n − ( R ) functions f : GL n − ( R ) → C such that f ( ug ) = θ n − ( u ) f ( g ) for all u ∈ U n − ( R ) , g ∈ GL n − ( R ) .Given a function φ ∈ C ∞ c ( θ n − , GL n − ( R )) there is a unique Whittaker function W ∈ K ( π ) such that, for all g ∈ GL n − ( R ) , W (cid:20)(cid:18) g
00 1 (cid:19)(cid:21) = φ ( g ) . The Geometric Approximate Functional Equationfor GL ( R ) In this section, we construct a test vector ϕ ∈ π coming from a carefully chosen element W ϕ in the Whittaker model of π , such that the local zeta integral Z ( W ϕ , + s + iT ) is ofsize T s/ − / for all s ∈ C with − ≤ Re( s ) ≤ and Im( s ) ∈ (cid:2) − T, C T (cid:3) for some fixedconstant C > ϕ enables us to write the global zeta integral Z ( ϕ, + iT ) for GL ( R ) as a truncatedglobal zeta integral with a small error term of size O ( T / − κ + ε ) for a nonnegative number κ .We will call this truncated integral representation of the global zeta integral Z ( ϕ, + iT ) forGL ( R ) the geometric approximate functional equation for GL ( R ) as in [14]. The geometricapproximate functional equation for GL ( R ) [14, Lemma 5.1.4, 254–256], namely Z ( ϕ, + iT ) = (cid:90) y ∈ R × / Z × ϕ ( a ( y )) y iT d × y = (cid:90) y ∈ R × + ϕ ( a ( y )) y iT (cid:16) h (cid:16) yT κ (cid:17) − h (cid:16) yT − κ (cid:17)(cid:17) d × y + O (cid:0) T − κ/ (cid:1) , where ϕ ( a ( y )) : = ∞ (cid:88) n = −∞ n (cid:54) =0 a ( | n | ) (cid:112) | n | W ϕ ( a ( ny ))with W ϕ ( a ( y )) := W ϕ (cid:20)(cid:18) y
00 1 (cid:19)(cid:21) := e ( − y ) W (cid:16) yT (cid:17) and κ ≥ , was deduced in [14] and explained to us in detail by Prof. Dr. Nelson with all the proofsover R .There is a strong relation between the geometric approximate functional equation and theusual approximate functional equation, which we will discuss at the end of this section. In this section, we construct a Whittaker function whose GL(1) Mellin transform localizesto frequency of size approximately T .We fix once and for all a compactly supported nonzero smooth function V ∈ C ∞ c ( R × + ) whichis nonnegative and such that the support of V is the interval [ π , C π ], and let T ∈ R + be7 large positive real parameter.We define the special Whittaker function W ϕ for GL ( R ) by W ϕ (cid:18) y x (cid:19) · O ( R ) · (cid:18) z z (cid:19)
00 1 : = T / V (cid:16) yT / (cid:17) e (cid:18) − y √ T (cid:19) V ( z ) e ( x )for y > x ∈ R and z ∈ R × .By Theorem 12 and the 1-periodicity of the function W ϕ in the x -variable, this Whittakerfunction W ϕ initially defined on (cid:18) GL ( R ) 00 1 (cid:19) , extends uniquely to a Whittaker function onall of GL ( R ), which we also denote by W ϕ and it is this extension, which we denote by W ϕ .Because the representation space π is isomorphic to the Whittaker model W ( π ) attachedto π , this Whittaker function W ϕ gives by the Fourier-Whittaker expansion rise to a vector ϕ ∈ π inside the automorphic representation π . GL ( R ) Let W ϕ be the Whittaker function constructed in § s ∈ C with − ≤ Re( s ) ≤ and Im( s ) ∈ (cid:2) − T, C T (cid:3) for some fixed constant C >
0. We have the following stationaryphase computation, namely Z ( W ϕ , + s + iT ) = (cid:90) R × W ϕ ( a ( y )) y iT − / s d × y = (cid:90) R × + T / V (cid:16) yT / (cid:17) e (cid:18) − y √ T (cid:19) y iT − / s d × y = T / T / · ( iT − / s ) (cid:90) R × + V ( z ) e ( − T z ) z iT − / s d × z (cid:28) T s/ (cid:90) R × + V ( z ) e (cid:18) − T z + T + Im( s )2 π log( z ) (cid:19) d × z (cid:124) (cid:123)(cid:122) (cid:125) (cid:28) T − / (cid:28) T s/ − / as T → ∞ , where s ∈ C with Re( s ) ∈ [ − , ] and Im( s ) ∈ (cid:2) − T, C T (cid:3) .In the above calculation, we have used the results of the stationary phase analysis in [14,page 209].If s ∈ C with − ≤ Re( s ) ≤ and it does not hold that Im( s ) ∈ (cid:2) − T, C T (cid:3) , then we haveagain by [14, page 209] that Z ( W ϕ , + s + iT ) is negligible, i.e., of size O( T − N ) for eachfixed N .We fix once and for all a smooth function h ∈ C ∞ ( R × + ) with values in [0 ,
1] that is identically1 in the interval (0 ,
1] and falls off to zero rapidly outside this interval, such that it is zeroon [2 , ∞ ). 8ith this choice of the Whittaker function W ϕ , we obtain the following result. Theorem 13. (The geometric approximate functional equation for GL ( R ) )Fix κ ≥ . Let ϕ ∈ π be the automorphic function corresponding to the Whittaker function W ϕ ∈ W ( π ) constructed in § Z ( ϕ, + iT ) = (cid:90) R × / Z × ϕ ( a ( y )) y iT − / d × y = (cid:90) R × + ϕ ( a ( y )) (cid:16) h (cid:16) yT κ (cid:17) − h (cid:16) yT − κ (cid:17)(cid:17) y iT − / d × y + O (cid:0) T / − κ/ ε (cid:1) = (cid:90) T − κ Having made the above choices, the proof is essentially identical to that given in [14,Lemma 5.1.4, 254–256]. We record the proof for completeness. In what follows, we use thenotation (cid:104) s (cid:105) := (1 + | s | ) / .We have (cid:90) R × + ϕ ( a ( y )) y iT − / d × y = (cid:90) R × + ϕ ( a ( y )) y iT − / (cid:16) h (cid:16) yT κ (cid:17) − h (cid:16) yT − κ (cid:17)(cid:17) d × y + (cid:90) R × + ϕ ( a ( y )) y iT − / (cid:16) − h (cid:16) yT κ (cid:17)(cid:17) d × y + (cid:90) R × + ϕ ( a ( y )) y iT − / h (cid:16) yT − κ (cid:17) d × y. We estimate the two integrals (cid:90) R × + ϕ ( a ( y )) y iT − / (cid:16) − h (cid:16) yT κ (cid:17)(cid:17) d × y and (cid:90) R × + ϕ ( a ( y )) y iT − / h (cid:16) yT − κ (cid:17) d × y separately and show that both integrals are O (cid:0) T / − κ/ ε (cid:1) .First, we estimate the second integral. By the Mellin inversion formula, we can write h (cid:16) yT − κ (cid:17) = 12 πi (cid:90) − c + i ∞− c − i ∞ H ( − s ) T κs y s ds for any real number c > , where H ( s ) := (cid:90) ∞ h ( y ) y s d × y 9s the Mellin transform of the function h ( y ).Substituting the above expression for h (cid:0) yT − κ (cid:1) into the second integral (cid:82) R × + ϕ ( a ( y )) y iT − / h (cid:0) yT − κ (cid:1) d × y and interchanging the two integration processes, we calculate (cid:90) R × + ϕ ( a ( y )) y iT − / h (cid:16) yT − κ (cid:17) d × y = (cid:90) R × + ϕ ( a ( y )) y iT − / πi (cid:90) − c + i ∞− c − i ∞ H ( − s ) T κs y s dsd × y = 12 πi (cid:90) − c + i ∞− c − i ∞ H ( − s ) T κs (cid:90) R × + ϕ ( a ( y )) y s + iT − / d × yds = 12 πi (cid:90) − c + i ∞− c − i ∞ H ( − s ) T κs L ( π, + s + iT ) Z ( W ϕ , + s + iT ) ds. In the above calculation, we have also used that Z ( ϕ, + s ) = (cid:90) R × + ϕ ( a ( y )) y s − / d × y = L ( π, + s ) Z ( W ϕ , + s ) , from Theorem 9 before.For Re( s ) = − , we can estimate L ( π, + s + iT ) Z ( W ϕ , + s + iT ) (cid:28) (cid:104) s (cid:105) O (1) T / ε T / · ( − / − / (cid:28) (cid:104) s (cid:105) O (1) T / − / − / ε (cid:28) (cid:104) s (cid:105) O (1) T / ε , where we have used the stationary phase analysis from the beginning of section § s ) = − , we see that (cid:90) R × + ϕ ( a ( y )) y iT − / h (cid:16) yT − κ (cid:17) d × y = 12 πi (cid:90) − c + i ∞− c − i ∞ H ( − s ) T κs L ( π, + s + iT ) Z ( W ϕ , + s + iT ) ds (cid:28) T / − κ/ ε . A similar calculation shows that the first integral also satisfies (cid:90) R × + ϕ ( a ( y )) y iT − / (cid:16) − h (cid:16) yT κ (cid:17)(cid:17) d × y (cid:28) T / − κ/ ε and the theorem is proved. Z ( ϕ, + iT ) In the next section, we provide an approximate functional equation for Z ( ϕ, + iT ), whichwill be used in the deduction of a subconvex bound for GL ( R ) using the integral represen-tation of the corresponding L -function. 10 emma 14. (The shape of the local zeta integral Z ( W ϕ , + iT ) and its truncation)Let W ϕ ∈ W ( π ) be the Whittaker function constructed in § f ∈ C ∞ ( R × ) be a fixedsmooth function on R × . Let n ∈ [ T / − κ , T / ε ] .It holds that (cid:90) R × W ϕ ( a ( y )) f (cid:16) yn (cid:17) y iT − / d × y = C T · T − / · f (cid:18) T / πn (cid:19) + O ( T − / κ + ε ) as T → ∞ and in particular that Z ( W ϕ , + iT ) = C T · T − / + O ( T − / ) as T → ∞ , where the quantity C T is given by C T : = (2 π ) − iT e − πi T iT e (cid:18) − T π (cid:19) V (cid:18) π (cid:19) . Proof. By an extended analysis of the beginning of section § (cid:90) R × W ϕ ( a ( y )) f (cid:16) yn (cid:17) y iT − / d × y = T iT (cid:90) R × V ( z ) e ( − T z ) f (cid:18) T / zn (cid:19) z iT − / d × z = (2 π ) − iT e − πi T iT e (cid:18) − T π (cid:19) T − / f (cid:18) T / πn (cid:19) V (cid:18) π (cid:19) + O ( T − / κ + ε ) . This proves the first formula of Lemma 14. To obtain the second identity, we use [19,Lemma 2.8, page 7] to calculate that (cid:90) R × W ϕ ( a ( y )) y iT − / d × y = T iT (cid:90) R × V ( z ) e ( − T z ) z iT − / d × z = (2 π ) − iT e − πi T iT e (cid:18) − T π (cid:19) T − / V (cid:18) π (cid:19) + O ( T − / ) . Lemma 15. (Integral to Sum and Sum to Integral Transformation)Let f ( y ) be a smooth function on R × + depending on T such that f ( y ) (cid:28) T − N for all N ∈ N and y ∈ [0 , T − ε ] . Let g ∈ C ∞ c ([ π , π ]) be a fixed compactly supported smooth function.Let ϕ ∈ π be the automorphic form corresponding to the Whittaker function W ϕ ∈ W ( π ) constructed in § Y ∈ [ T − ε , T κ ] the transformation formula S f ( Y ) : = (cid:90) R × + ϕ ( a ( y )) f ( y ) g (cid:16) yY (cid:17) y iT − / d × y = C T · T − / ∞ (cid:88) n =1 a ( n, n / iT f (cid:18) T / πn (cid:19) g (cid:18) T / πnY (cid:19) + O ( T − / κ + ε ) . roof. Expanding the above integral S f ( Y ) by employing the Fourier-Whittaker expansionof ϕ ( a ( y )) from Definition 2, we get that S f ( Y ) = ∞ (cid:88) n = −∞ n (cid:54) =0 a ( | n | , | n | (cid:90) y ∈ R × + W ϕ ( a ( ny )) f ( y ) g (cid:16) yY (cid:17) y iT − / d × y. The integral in the above expression restricts the n -sum to the range where n ≥ S f ( Y ) = ∞ (cid:88) n =1 a ( n, n / iT (cid:90) y ∈ R × + W ϕ ( a ( y )) f (cid:16) yn (cid:17) g (cid:16) ynY (cid:17) y iT − / d × y. Applying the first identity from Lemma 14 with the allowed function f ( y ) := f ( y ) g (cid:0) yY (cid:1) , weobtain by using the Rankin-Selberg estimate (cid:80) n ≤ X | a ( n, | (cid:28) X ε [13, page 2] that S f ( Y ) = C T · T − / ∞ (cid:88) n =1 a ( n, n / iT f (cid:18) T / πn (cid:19) g (cid:18) T / πnY (cid:19) + O ( T − / κ + ε ) . We obtain the following approximate functional equation. Theorem 16. (An Approximate Functional Equation for Z ( ϕ, + iT ) )Let k ( y ) := h (cid:0) yT κ (cid:1) − h (cid:0) yT − κ (cid:1) and let c > . We define the smooth functions h ( y ) : = h (cid:16) yT − ε (cid:17) − h (cid:16) yT − κ (cid:17) and h ( y ) := h (cid:16) yT κ (cid:17) − h (cid:16) yT − ε (cid:17) ,F ( s ) : = (cid:90) ∞ h ( y ) y s d × y and G (cid:18) T π ny (cid:19) := 12 πi (cid:90) − c + i ∞− c − i ∞ F ( − s ) n s γ ( π, + s + iT ) y s ds,h (cid:18) T π ny (cid:19) : = Re (cid:18) G (cid:18) T π ny (cid:19)(cid:19) and h (cid:18) T π ny (cid:19) := Im (cid:18) G (cid:18) T π ny (cid:19)(cid:19) and let g ∈ C ∞ c ([ π , π ]) be a fixed compactly supported smooth function.All the above functions are uniformly bounded on R × + by some constant C > .We have for any κ ∈ [0 , ] and ϕ ∈ π corresponding to the Whittaker function W ϕ ∈ W ( π ) constructed in § Z ( ϕ, + iT ) (cid:28) T ε (cid:88) f ∈{ h ,h ,h } sup T − ε ≤ Y ≤ T κ {| S f ( Y ) |} + O (cid:0) T / − κ/ ε (cid:1) , where S f ( Y ) : = (cid:90) R × + ϕ ( a ( y )) f ( y ) g (cid:16) yY (cid:17) y iT − / d × y. roof. By making a smooth dyadic subdivision of the truncated global zeta integral (cid:90) R × + ϕ ( a ( y )) k ( y ) y iT − / d × y = (cid:90) T − κ ≤ y ≤ T κ ϕ ( a ( y )) k ( y ) y iT − / d × y, we get by the geometric approximate functional equation that Z ( ϕ, + iT ) (cid:28) T ε sup T − κ ≤ Y ≤ T κ (cid:40)(cid:90) R × + ϕ ( a ( y )) k ( y ) g (cid:16) yY (cid:17) y iT − / d × y (cid:41) + O (cid:0) T / − κ/ ε (cid:1) . The idea to truncate the Y -range after a smooth dyadic subdivision further to T − ε ≤ Y ≤ T κ and to project the contribution in the supremum coming from the terms with T − κ ≤ Y ≤ T − ε onto the terms with T − ε ≤ Y ≤ T κ by using the local functional equation from Theorem 11was proposed to us by Prof. Dr. Nelson. This is also the strategy which we follow below.Define the integrals I m ( y ) : = (cid:90) R × + ϕ ( a ( y )) h m ( y ) y iT − / d × y for m = 0 , , , . We start with the observation that (cid:90) R × + ϕ ( a ( y )) k ( y ) y iT − / d × y = (cid:90) R × + ϕ ( a ( y )) h ( y ) y iT − / d × y + (cid:90) R × + ϕ ( a ( y )) h ( y ) y iT − / d × y and make the change of variables y ↔ y in the first integral I ( y ) to get that I ( y ) equals I ( y ) : = (cid:90) y ∈ R × + ϕ ( a ( y )) h ( y ) y iT − / d × y = (cid:90) y ∈ R × + ϕ ( a (1 /y )) h (1 /y ) y / − iT d × y. Using the GL ( R ) projection identity from Theorem 5, the above transforms to I ( y ) = (cid:90) y ∈ R × + (cid:90) x ∈ R (cid:101) ϕ y xy · w (cid:48) h (1 /y ) y / − iT dxd × y. Changing variables and using the Fourier-Whittaker expansion of (cid:101) ϕ ( g ) from Definition 4,we obtain I ( y ) = (cid:90) y ∈ R × + (cid:90) x ∈ R (cid:101) ϕ y x · w (cid:48) h (1 /y ) y − / − iT dxd × y = ∞ (cid:88) n = −∞ n (cid:54) =0 a (1 , | n | ) | n | (cid:90) y ∈ R × + (cid:90) x ∈ R (cid:102) W ϕ ny x · w (cid:48) h (1 /y ) y − / − iT dxd × y. n into positive and negative contributions, we get I ( y ) = ∞ (cid:88) n =1 a (1 , n ) n (cid:90) y ∈ R × + (cid:90) x ∈ R (cid:102) W ϕ ny x · w (cid:48) h (1 /y ) y − / − iT dxd × y + ∞ (cid:88) n =1 a (1 , n ) n (cid:90) y ∈ R × + (cid:90) x ∈ R (cid:102) W ϕ − ny x · w (cid:48) h (1 /y ) y − / − iT dxd × y. Changing variables again, we get that I ( y ) = ∞ (cid:88) n =1 a (1 , n ) n (cid:90) y ∈ R × (cid:90) x ∈ R (cid:102) W ϕ ny x · w (cid:48) h (1 /y ) y − / − iT dxd × y = ∞ (cid:88) n =1 a (1 , n ) n / − iT (cid:90) y ∈ R × (cid:90) x ∈ R (cid:102) W ϕ y x · w (cid:48) h ( n/y ) y − / − iT dxd × y. Substituting into this expression the inverse Mellin transformation formula h (cid:18) ny (cid:19) = 12 πi (cid:90) − c + i ∞− c − i ∞ F ( − s ) n s y − s ds for any real number c > h (cid:16) ny (cid:17) , we obtain I ( y ) = ∞ (cid:88) n =1 a (1 , n ) n / − iT πi (cid:90) − c + i ∞− c − i ∞ F ( − s ) n s (cid:101) Z ( (cid:102) W ϕ , − s − iT ) ds. Using the local functional equation (cid:101) Z ( (cid:102) W ϕ , − s − iT ) = γ ( π, + s + iT ) Z ( W ϕ , + s + iT )from Theorem 11, we get that I ( y ) = ∞ (cid:88) n =1 a (1 , n ) n / − iT πi (cid:90) − c + i ∞− c − i ∞ F ( − s ) n s γ ( π, + s + iT ) Z ( W ϕ , + s + iT ) ds = ∞ (cid:88) n =1 a (1 , n ) n / − iT (cid:90) y ∈ R × + W ϕ ( a ( y )) G (cid:18) T π ny (cid:19) y iT − / d × y, where the function G (cid:16) T π ny (cid:17) is defined by G (cid:18) T π ny (cid:19) : = 12 πi (cid:90) − c + i ∞− c − i ∞ F ( − s ) n s γ ( π, + s + iT ) y s ds. 14e have for m = 2 , h m ( z ) (cid:28) T − N for all N ∈ N is negligible for z ∈ [0 , T − ε ],because this can be seen by shifting the contour of the definition for G (cid:16) T π ny (cid:17) to minusinfinity and using the fact that γ ( π, + s + iT ) (cid:28) T − s [6, Formula (5.115), page 151], [18,Formula (1.20), page 9]. Moreover, shifting the contour to 0 ± i ∞ , we get that | h m ( z ) | (cid:28) C := (cid:82) i ∞− i ∞ | F ( − s ) | ds < ∞ for m = 2 , z ∈ R × + , because there are no poles of γ ( π, + s + iT ) for s ∈ C with Re( s ) ≤ 0. It follows also that h (cid:48) m ( z ) and h (cid:48)(cid:48) m ( z ) are uniformlybounded.By the local functional equation from Theorem 11 and the beginning of section § (cid:101) Z ( (cid:102) W ϕ , − s − iT ) = γ ( π, + s + iT ) Z ( W ϕ , + s + iT ) (cid:28) T − s T s/ − / = T − s/ − / . Shifting the contour to 0 ± i ∞ , we therefore see that (cid:90) y ∈ R × + W ϕ ( a ( y )) G (cid:18) T π ny (cid:19) y iT − / d × y = 12 πi (cid:90) i ∞− i ∞ F ( − s ) n s (cid:101) Z ( (cid:102) W ϕ , − s − iT ) ds (cid:28) T − / . Defining the two sums S m : = ∞ (cid:88) n =1 a (1 , n ) n / − iT (cid:90) y ∈ R × + W ϕ ( a ( y )) h m (cid:18) T π ny (cid:19) y iT − / d × y for m = 2 , , we get that I ( y ) = S + iS . By a smooth dyadic subdivision with the function g ( y ), we get by using the above remarkson h m ( z ) for m = 2 , W ϕ ( a ( y )) in § S m (cid:28) T ε sup T − ε ≤ Y ≤ T / (cid:40) ∞ (cid:88) n =1 a (1 , n ) n / − iT g (cid:18) T / πnY (cid:19) (cid:90) y ∈ R × + W ϕ ( a ( y )) h m (cid:18) T π ny (cid:19) y iT − / d × y (cid:41) , which can, by the Rankin-Selberg bound (cid:80) n ≤ X | a ( n, | (cid:28) X ε [13, page 2] and the factthat we have (cid:82) y ∈ R × + W ϕ ( a ( y )) h m (cid:16) T π ny (cid:17) y iT − / d × y (cid:28) T − / , be further simplified to S m (cid:28) T ε sup T − ε ≤ Y ≤ T κ (cid:40) ∞ (cid:88) n =1 a (1 , n ) n / − iT g (cid:18) T / πnY (cid:19) (cid:90) y ∈ R × + W ϕ ( a ( y )) h m (cid:18) T π ny (cid:19) y iT − / d × y (cid:41) + O ( T / − κ/ ε ) . Using a similar stationary phase analysis as in the proof of Lemma 14, the above two supremafor m = 2 , S m (cid:28) T ε sup T − ε ≤ Y ≤ T κ (cid:40) C T · T − / ∞ (cid:88) n =1 a (1 , n ) n / − iT g (cid:18) T / πnY (cid:19) h m (cid:18) T / πn (cid:19)(cid:41) + O ( T / − κ/ ε ) , O ( T / − κ/ ε ) dominates the other error term O ( T − / κ + ε ) comingfrom Lemma 14 as we have assumed κ ≤ .Moreover, by using that a (1 , n ) = a ( n, n / − iT = n / iT and that the h m ( y )’s, as well as g ( y ) are real valued functions, we conclude that for m = 2 , S m (cid:28) T ε sup T − ε ≤ Y ≤ T κ (cid:40) C T · T − / ∞ (cid:88) n =1 a ( n, n / iT h m (cid:18) T / πn (cid:19) g (cid:18) T / πnY (cid:19)(cid:41) + O ( T / − κ/ ε ) . From this expression, we conclude via Lemma 15 and f ( y ) := h m ( y ) that for m = 2 , 3, wehave S m (cid:28) T ε sup T − ε ≤ Y ≤ T κ (cid:40)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R × + ϕ ( a ( y )) h m ( y ) g (cid:16) yY (cid:17) y iT − / d × y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:41) + O ( T / − κ/ ε )to conclude finally by a smooth dyadic subdivision of the I ( y ) integral with the same function g ( y ) that Z ( ϕ, + iT ) (cid:28) T ε sup T − ε ≤ Y ≤ T κ (cid:40)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R × + ϕ ( a ( y )) h ( y ) g (cid:16) yY (cid:17) y iT − / d × y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:41) + O ( T / − κ/ ε )+ T ε (cid:88) f ∈{ h ,h } sup T − ε ≤ Y ≤ T κ (cid:40)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R × + ϕ ( a ( y )) f ( y ) g (cid:16) yY (cid:17) y iT − / d × y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:41) . This is the claimed formula. Remark . Taking the supremum over the full range T − κ ≤ Y ≤ T κ in Theorem 16 is alsoenough to obtain a subconvex bound, because we would get a saving of . Shrinking therange to T − ε ≤ Y ≤ T κ has only the effect of optimizing the saving to Lin’s [13]. GL ( R ) L -Functions via IntegralRepresentations Let W ϕ be the Whittaker function constructed in § ϕ ∈ π the corresponding auto-morphic form.In this section, we will bound the integral S f ( Y ) by two terms F and O , such that S f ( Y ) = (cid:90) R × / Z × ϕ ( a ( y )) f ( y ) g (cid:16) yY (cid:17) y iT − / d × y (cid:28) F + O . These two terms F and O will be independent from the function f ∈ { h , h , h } .16 .1 The Analysis of the Integral S f ( Y ) Let T − ε ≤ Y ≤ T κ and let f ∈ { h , h , h } . Assume that κ ≤ to use Lin’s results later.We have by the definition of the suitable Whittaker function W ϕ in § S f ( Y ) = (cid:90) R × + ϕ ( a ( y )) f ( y ) g (cid:16) yY (cid:17) y iT − / d × y = ∞ (cid:88) n =1 a ( n, n (cid:90) R × W ϕ ( a ( ny )) f ( y ) g (cid:16) yY (cid:17) y iT − / d × y = T / ∞ (cid:88) n =1 a ( n, n (cid:90) ∞ e (cid:18) − ny √ T (cid:19) y iT V (cid:16) nyT / (cid:17) g (cid:16) yY (cid:17) f ( y ) √ y d × y = T / Y / − iT ∞ (cid:88) n =1 a ( n, n (cid:90) ∞ e (cid:18) − nY y √ T (cid:19) y iT V (cid:18) nY yT / (cid:19) g ( y ) f ( Y y ) √ y d × y. Define the variable N := T / /Y , set S ( N ) := S f ( Y ) and let f n ( y ) := V (cid:0) nyN (cid:1) g ( y ) f ( Y y ) y √ y toget that S ( N ) (cid:28) √ N ∞ (cid:88) n =1 a ( n, n (cid:90) ∞ y iT e (cid:18) − nT yN (cid:19) f n ( y ) dy. After the change of variables y := x and the definition V n ( x ) := f n ( x ) x = V (cid:0) nNx (cid:1) g (cid:0) x (cid:1) f (cid:0) Yx (cid:1) √ x ,this is equal to S ( N ) (cid:28) √ N ∞ (cid:88) n =1 a ( n, n (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V n ( x ) dx. Let V ( x ) := V (cid:0) x (cid:1) f (cid:0) Yx (cid:1) √ x be a smooth and compactly supported bump function on R × + independent of n . By the stationary phase method [16, Lemma 3., page 919] there exists asmooth and compactly supported function, for example w ( z ) := V ( π ) V ( πz ) g (cid:0) πz (cid:1) ∈ C ∞ c ([1 , (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V n ( x ) dx = w (cid:16) nN (cid:17) (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V ( x ) dx + O ( T − ε ) , because we have (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V n ( x ) dx = c T · T − / V (cid:18) π (cid:19) g (cid:18) N πn (cid:19) f (cid:18) N Y πn (cid:19) (cid:114) N πn + O ( T − / κ + ε ) , (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V ( x ) dx = c T · T − / V (cid:18) N πn (cid:19) f (cid:18) N Y πn (cid:19) (cid:114) N πn + O ( T − / κ + ε ) , c T ∈ C is given by c T := √ πe − πi e (cid:18) − T π (cid:19) (cid:18) πnN (cid:19) − iT . In the above two asymptotic formulas, the implied constant of the error term is uniform,because it depends continuously on nN , which varies in the compact set [1 , S ( N ) (cid:28) √ N ∞ (cid:88) n =1 a ( n, n w (cid:16) nN (cid:17) (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V ( x ) dx. Absorbing the fraction n into the weight function w by defining w ( x ) := w ( x ) x ∈ C ∞ c ([1 , S ( N ) (cid:28) √ N ∞ (cid:88) n =1 a ( n, w (cid:16) nN (cid:17) (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V ( x ) dx. We have therefore to study the main integral (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V ( x ) dx, which is exactly the integral appearing in [13, page 6] in the proof of Lin’s key identity.This key observation tells us that Lin’s key identity can be understood as replacing theabove integral (cid:82) R × + x − iT e (cid:0) − nTNx (cid:1) V ( x ) dx by the corresponding Riemann lattice sum plus itsoscillations (error terms) around the exact value of the integral. This also explains why thereare two terms, namely F and O present in this method. We will follow closely the work [13]in the end of our argument. We use the letters p and (cid:96) to denote prime numbers. Let P and L be two large parameters,which will be specified later as small powers of the parameter T ∈ R + . The notations p ∼ P and (cid:96) ∼ L are used to denote prime numbers in the two dyadic segments [ P, P ] and [ L, L ]respectively. We also assume that [ P, P ] ∩ [ L, L ] = ∅ . The sums (cid:80) p ∼ P and (cid:80) (cid:96) ∼ L describesums over all the prime numbers p ∈ [ P, P ] and (cid:96) ∈ [ L, L ].We have the following Lemma 18. (Another form of Lin’s key identity)[13]We have (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V ( x ) dx = (cid:18) (cid:96)TN p (cid:19) − iT ∞ (cid:88) r =1 r − iT e (cid:16) − np(cid:96)r (cid:17) V (cid:18) rN p/(cid:96)T (cid:19) − (cid:88) r ∈ Z r (cid:54) =0 J iT (cid:16) n, rp(cid:96) (cid:17) , ith J iT (cid:16) n, rp(cid:96) (cid:17) : = (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V ( x ) e (cid:18) − rN p(cid:96)T x (cid:19) dx. Proof. We can calculate using the Poisson summation formula that ∞ (cid:88) r =1 r − iT e (cid:16) − np(cid:96)r (cid:17) V (cid:18) rN p/(cid:96)T (cid:19) = (cid:90) R × + z − iT e (cid:16) − np(cid:96)z (cid:17) V (cid:18) zN p/(cid:96)T (cid:19) dz + (cid:88) r ∈ Z r (cid:54) =0 (cid:90) R × + z − iT e (cid:16) − np(cid:96)z (cid:17) V (cid:18) zN p/(cid:96)T (cid:19) e ( − rz ) dz = (cid:18) N p(cid:96)T (cid:19) − iT (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V ( x ) dx + (cid:18) N p(cid:96)T (cid:19) − iT (cid:88) r ∈ Z r (cid:54) =0 (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V ( x ) e (cid:18) − rN p(cid:96)T x (cid:19) dx. In the above calculation, we have made the change of variables z := Np(cid:96)T x .Solving the above expression for (cid:82) R × + x − iT e (cid:0) − nTNx (cid:1) V ( x ) dx implies the claimed identity.We have the following Lemma 19. (Amplification identity)[13]It holds that log( P ) log( L ) P L (cid:88) p ∼ P (cid:88) (cid:96) ∼ L (cid:16) . Proof. We have log( P ) log( L ) P L (cid:88) p ∼ P (cid:88) (cid:96) ∼ L (cid:16) log( P ) log( L ) P L P log( P ) L log( L ) (cid:16) . This implies the above statement.This implies the following Lemma 20. (The amplified key identity)[13]We have (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V ( x ) dx (cid:16) T ε N P (cid:88) p ∼ P p iT (cid:88) (cid:96) ∼ L (cid:96) − iT ∞ (cid:88) r =1 r − iT e (cid:16) − np(cid:96)r (cid:17) V (cid:18) rN p/(cid:96)T (cid:19) − T ε P L (cid:88) p ∼ P (cid:88) (cid:96) ∼ L (cid:88) r ∈ Z r (cid:54) =0 J iT (cid:16) n, rp(cid:96) (cid:17) . roof. Using the above two Lemmas 18 and 19, we can calculate that (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V ( x ) dx (cid:16) log( P ) log( L ) P L (cid:88) p ∼ P (cid:88) (cid:96) ∼ L (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V ( x ) dx = log( P ) log( L ) P L (cid:88) p ∼ P (cid:88) (cid:96) ∼ L (cid:34) (cid:18) (cid:96)TN p (cid:19) − iT ∞ (cid:88) r =1 r − iT e (cid:16) − np(cid:96)r (cid:17) V (cid:18) rN p/(cid:96)T (cid:19) − (cid:88) r ∈ Z r (cid:54) =0 (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V ( x ) e (cid:18) − rN p(cid:96)T x (cid:19) dx (cid:35) = T ε N P (cid:88) p ∼ P p iT (cid:88) (cid:96) ∼ L (cid:96) − iT ∞ (cid:88) r =1 r − iT e (cid:16) − np(cid:96)r (cid:17) V (cid:18) rN p/(cid:96)T (cid:19) − T ε P L (cid:88) p ∼ P (cid:88) (cid:96) ∼ L (cid:88) r ∈ Z r (cid:54) =0 (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V ( x ) e (cid:18) − rN p(cid:96)T x (cid:19) dx. This is the claimed formula. S ( N ) in terms of F and O Using Lemma 20, we can conclude from the formula S ( N ) (cid:28) √ N ∞ (cid:88) n =1 a ( n, w (cid:16) nN (cid:17) (cid:90) R × + x − iT e (cid:18) − nTN x (cid:19) V ( x ) dx, which we obtained at the end of § S ( N ) (cid:28) T ε N / P (cid:88) p ∼ P p iT (cid:88) (cid:96) ∼ L (cid:96) − iT ∞ (cid:88) r =1 r − iT V (cid:18) rN p/(cid:96)T (cid:19) ∞ (cid:88) n =1 a ( n, e (cid:16) − np(cid:96)r (cid:17) w (cid:16) nN (cid:17) + T ε √ N P L ∞ (cid:88) n =1 a ( n, w (cid:16) nN (cid:17) (cid:88) p ∼ P (cid:88) (cid:96) ∼ L (cid:88) r ∈ Z r (cid:54) =0 J iT (cid:16) n, rp(cid:96) (cid:17) . Therefore, we get for all T / − κ ≤ N ≤ T / ε that S ( N ) (cid:28) F + O , where F : = T ε N / P (cid:88) p ∼ P p iT (cid:88) (cid:96) ∼ L (cid:96) − iT ∞ (cid:88) r =1 r − iT V (cid:18) rN p/(cid:96)T (cid:19) ∞ (cid:88) n =1 a ( n, e (cid:16) − np(cid:96)r (cid:17) w (cid:16) nN (cid:17) O : = T ε √ N P L ∞ (cid:88) n =1 a ( n, w (cid:16) nN (cid:17) (cid:88) p ∼ P (cid:88) (cid:96) ∼ L (cid:88) r ∈ Z r (cid:54) =0 J iT (cid:16) n, rp(cid:96) (cid:17) . F and O We get from the above expression for F by following Lin [13] that for any ε > F (cid:28) N ε PT / L / + N / ε (cid:18) P LT (cid:19) / . Similarly, we get again by following Lin [13] that for any ε > O (cid:28) T / ε P + T ε LN / P , by using the above expression for O . L ( π, + iT ) Setting κ := and the two variables P and L as in [13] to P : = T / and L := T / , we obtain using Theorem 16 with S f ( Y ) = S ( N ) and following [13] that Z ( ϕ, + iT ) (cid:28) T ε sup T / − κ ≤ N ≤ T / ε {| S ( N ) |} + T / − κ/ ε (cid:28) (cid:32) T / ε PT / L / + T / ε (cid:18) P LT (cid:19) / (cid:33) + (cid:18) T / ε P + T κ/ ε LT / P (cid:19) + T / − κ/ ε (cid:28) T / − + ε . Finally, because it holds according to Lemma 14 that Z ( W ϕ , + iT ) = C T · T − / + O ( T − / ),we have by Theorem 9 that L ( π, + iT ) = Z ( ϕ, + iT ) Z ( W ϕ , + iT ) (cid:28) Z ( ϕ, + iT ) T / (cid:28) T / − / ε . The saving is not the best currently known, because Munshi [16] obtained a saving of and Aggarwal [1] got a saving of . 21 Acknowledgement We would like to thank our advisor Professor Dr. Paul D. Nelson for entrusting us with thisinteresting project. He introduced us carefully to the representation theory of automorphicforms and the geometric approximate functional equation for GL ( R ).Beginning from a well chosen starting point, we gained very interesting insights into thekey identity of the GL ( R )-problem and the various findings of our research have become aunity. The possible connection of the key identity with automorphic periods and represen-tation theory was suggested to us by Prof. Dr. Nelson. Many thanks to Prof. Dr. Nelsonas well as to Prof. Dr. Roman Holowinsky, Prof. Dr. Matthew Young and Dr. YongxiaoLin for their valuable suggestions which improved this paper.This work was supported by SNSF (Swiss National Science Foundation) under grant 169247. 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Keywords: automorphic form, subconvexity problem for GL ( R ) , key identity, special Whit-taker function, geometric approximate functional equation for GL ( R ) , automorphic repre-sentation theory, integral representation, automorphic repre-sentation theory, integral representation