aa r X i v : . [ m a t h . N T ] A ug ANALYTIC TWISTS OF MODULAR FORMS
ALEXANDRE PEYROT
Abstract.
We investigate non-correlation of Fourier coefficients of Maassforms against a class of real oscillatory functions, in analogy to known re-sults with Frobenius trace functions. We also establish an equidistributionresult for twisted horocycles as a consequence of our non-correlation result.
Contents
1. Introduction 12. Stationary phase integrals 53. Analysis of K t
84. Proof of Theorem 1 95. Examples 256. Horocycle twists 28References 301.
Introduction
In this paper we are interested in sums of Fourier coefficients of GL Maassforms against a certain class of oscillatory functions. The type of oscillatory func-tions we consider can be thought as archimedean analogs of trace functions studiedby Fouvry, Kowalski and Michel in [4]. Our main result gives a non-correlationstatement between Fourier coefficients of Maass forms against a family of func-tions, K t : R > → C , depending on a large real parameter t .1.1. Setup.
We let throughout f be a fixed cuspidal Maass Hecke eigenform forSL ( Z ), and denote by 1 / t f the associated eigenvalue of the Laplacian. Theform f admits a Fourier expansion f ( z ) = X n =0 ρ f ( n ) | n | − / W it f (4 π | n | y ) e ( nx ) , where W ν is a Whittaker function, W it ( y ) = e − y/ Γ (cid:0) + it (cid:1) Z ∞ e − x x it − (cid:18) xy (cid:19) it − d x. The Fourier coefficients, ρ f ( n ), are normalized so that by Rankin-Selberg,(1) X n ≤ X | ρ f ( n ) | ≍ X. We moreover know that the Fourier coefficients oscillate substantially. For example,the following estimate(2) X n ≤ x ρ f ( n ) e ( αn ) ≪ f x / ǫ holds for any ǫ > α ∈ R (see [5] theorem 8.1). In order tounderstand better the oscillatory nature of the Fourier coefficients, we make thefollowing definition. Definition 1.
Let ( K ( n )) n ∈ N be a bounded sequence of complex numbers. We saythat ( K ( n )) does not correlate with ( ρ f ( n )) if we have X n ≤ x ρ f ( n ) K ( n ) ≪ f,A x (log x ) − A , for all A ≥ , x > . Non-correlation statements are therefore a way to measure the extent to whichthe oscillations of a given sequence “lines up” with the oscillations of the Fouriercoefficients. For example, (2) gives a non-correlation statement for the additivetwist K ( n ) = e ( αn ) with a power saving of 1 / − ǫ . Another important example ofnon-correlation arises when K ( n ) = µ ( n ), the M¨obius function, in which case non-correlation is an incarnation of the Prime Number Theorem (see [3] for a generalresult combining this and additive twists). Obtaining power saving statementsagainst the M¨obius function would be equivalent to proving a strong zero-free regiontowards the Riemann Hypothesis for the L -function attached to f . We give herea final example, which will be the main motivation for our work: let p be a primenumber and let K be an isotypic trace function of conductor p , then [4] gives anon-correlation result for ( K ( n )) with a power saving of 1 / − ǫ .We will study non-correlation against a family of functions ( K t ) t ∈ R , K t : R > → C , where t is a parameter which we will let grow to infinity. Definition 2.
A family of smooth functions ( K t ) t ∈ R , K t : R > → C is called afamily of analytic trace functions if there exist real numbers a < b, b > and afamily of analytic functions ( M t ( s )) t ∈ R in the strip a < ℜ ( s ) < b , such that thefollowing conditions hold.1. The following integral converges for any a < σ < b , (3) 12 πi Z ( σ ) M t ( s ) x − s d s, and is equal to K t ( x ) for all x ∈ R > , t ∈ R .2. There exist constants c , c depending on the family ( K t ) t ∈ R , independent of t ,such that we may write M t ( σ + iν ) = g t ( σ + iν ) e ( f t ( σ + iν )) , in such a way thatfor all x ∈ [ t, t ] , the following (4) g ( j ) t ( σ + iν ) ≪ j ν σ − / − j ∀ j ≥ , holds, as well as the following conditions on f t .(a) Whenever | ν | ≤ c t or | ν | ≥ c t , we have (5) (cid:12)(cid:12)(cid:12)(cid:12) f ′ t ( σ + iν ) − π log( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≫ . (b) When c t ≤ | ν | ≤ c t , either (5) holds, or we have (6) f ′′ t ( σ + iν ) ≫ ν − , while for all ǫ > , j ≥ , (7) f ( j ) t ( σ + iν ) ≪ j,ǫ ν ǫ − j . NALYTIC TWISTS OF MODULAR FORMS 3 (c) Finally, we require that (8) f ′′ t ( σ + iν ) − πν ≫ ν − , whenever c t ≤ | ν | ≤ c t .Remark . Throughout the paper, we will abuse notation and say that K t is ananalytic trace function when it arises as part of such a family. Remark . Conditions (3) - (7) guarantee by means of stationary phase that theintegral representation is concentrated around multiplicative character of conductor t . Condition (8) ensures that we avoid functions such as e ( x ), as motivated inSection 5. Remark . By the properties of the Mellin transform, we note that if K t ( x ) is ananalytic trace function, then for any constant α ∈ R > , we have that K t ( αx ) is alsoan analytic trace function. Remark . We note that in interesting examples, in conjunction with condition (5),we will also have some stationary points in the region c t ≤ | ν | ≤ c t , guaranteeingthat || K t || ∞ ≍ Remark . We note that in practice, we may always ensure that condition (3) holds,by studying K t ( x ) V (cid:0) xt (cid:1) , where V is a smooth compactly supported function in[ , M t ( s ) is given by R ∞ K t ( x ) x s − d x, and the integral in (3)converges absolutely.We give here some examples of analytic trace functions (see Section 5 for proofs). Example 1.
The normalized J -Bessel function of order t , F it ( x ) := t / Γ (cid:18)
12 + it (cid:19) J it ( x ) , is an analytic trace function.This should be thought of as an archimedean analog of Kloosterman sums. Wenow give as a second example that of higher rank Bessel functions as appearing in[9], in analogy to hyper-Kloosterman sums. Example 2.
For any n ≥
3, the n -th rank Bessel function of order t , J n,t := t n − πin Z ( ) Γ (cid:18) s − intn (cid:19) Γ (cid:18) sn + itn − (cid:19) n − e (cid:16) s (cid:17) x − s d s, is an analytic trace function.We will study sums of the shape S ( t ) := X n ρ f ( n ) K t ( n ) V (cid:16) nt (cid:17) , where K t is an analytic trace function and V is a smooth function supported in[1 ,
2] and such that V ( j ) ( x ) ≪ j
1. for convenience we also normalize V so that R V ( y )d y = 1. We will show in Section 3 that any analytic trace function, K t ,satisfies || K t || ∞ ≪
1, so that by Cauchy-Schwarz and (1), we have that S ( t ) ≪ t. Our main result improves on that bound.
Theorem 1.
Let K t : R → C be an analytic trace function. We have S ( t ) ≪ t − / ǫ , where the implicit constant depends only on f, ǫ and on || K t || ∞ . ALEXANDRE PEYROT
Remark . For simplicity we have studied the case where n ≍ t . We note that for N ≤ t , one may study similarly Z ( N ) := X n ρ f ( n ) K t ( n ) V (cid:16) nN (cid:17) . If for x ≍ N , conditions (5) - (8) hold (which is the case in practice), we may showthat Z ( N ) ≪ t / ǫ N / , which improves on the trivial bound so long as N ≫ t / ǫ .Our bound has an application to the geometric question of equidistribution ofhorocycle flows with respect to a twisted signed measure. Let us recall that forevery continuous compactly supported function f on SL ( Z ) \ H , we have Z f ( x + iy )d x → µ (SL ( Z ) \ H ) − Z SL ( Z ) \ H f ( z )d µ ( z ) , as y →
0, where µ ( z ) = d x d yy denotes the hyperbolic measure (see [13]). In [11]Str¨ombergsson gives a similar result by restricting to subsegments of hyperboliclength y − / − δ , i.e. that for any δ > f as above,1 β − α Z βα f ( x + iy )d x → µ (SL ( Z ) \ H ) − Z SL ( Z ) \ H f ( z )d µ ( z ) , uniformly as y → β − α remains bigger than y / − δ . We use Theorem 1to give the following twisted version of Str¨ombergsson’s result, which is analogousto what is proven in [4] for horocycles twisted by Frobenius trace functions. Theorem 2.
Let ( K t ) t ∈ R be a family of analytic trace functions. Let f be a Maassform on SL ( Z ) \ H , and V be a smooth real valued function with compact supportin [ , ] such that V ( j ) ( x ) ≪ , for all j ≥ . We then have for any δ > , β − α Z βα f ( x + iy ) K /y (cid:18) xy (cid:19) V ( x )d x → , uniformly as y → so long as β − α remains bigger than y / − δ . Outline of proof of Theorem 1.
We will show in Section 3 that our defini-tion of analytic trace function implies that we may essentially write K t ( x ) = 12 π Z ν ≍ t g t ( σ + iν ) e ( f t ( σ + iν )) x − σ − iν d ν. Interchanging order of summation and integration, we may therefore write S ( t ) = 12 π Z ν ≍ t g t ( σ + iν ) e ( f t ( σ + iν )) ∞ X n =1 ρ f ( n ) n − σ − iν V (cid:16) nt (cid:17) d ν. We then adapt the circle method of Munshi, as in [8], allowing us to write the innersum essentially as1 K Z KK X q ≍ Q X a ≍ Q ( a,q )=1 aq X n ≍ t ρ f ( n ) n iv e (cid:18) n ¯ aq − nxaq (cid:19) X m ≍ t m − i ( ν + v ) e (cid:18) − m ¯ aq + mxaq (cid:19) d v, where K ≤ t is a parameter that will ultimately be chosen optimally to be K = t / ,and Q = ( t/K ) / . We may now apply Poisson summation to the m -sum, and NALYTIC TWISTS OF MODULAR FORMS 5
Voronoi summation to the n -sum to arrive at the following expression for S ( t ), X n ≪ K ρ f ( n ) √ n X q ≍ Q X ( m,q )=11 ≤| m |≪ q e (cid:18) n ¯ mq (cid:19) Z K − K Z ν ≍ t n − iτ/ g ( q, m, τ, ν ) e ( f ( q, m, τ, ν ))d ν d τ, where g is a non-oscillatory amplitude function of size K and f is a well under-stood phase. In particular, we note that (8) implies that f ′′ ( q, m, τ, ν ) ≫ | ν | − ,so that we may use second derivative bounds for multivariable integrals and savein the integral. Applying the Cauchy-Schwarz inequality to get rid of the Fouriercoefficients, and using the second derivative bound to save ( Kt ) / in the integral,we arrive at S ( t ) ≪ Kt / X q,q ′ ≍ Q X m,m ′ ≍ Q Q − K / + X n ≍ tn ≡ qm ′ − q ′ m mod qq ′ K / | n | / / ≪ K / t / + tK / , which upon taking K = t / gives the desired result.1.3. Notations.
Throughout the paper, we will let f ( x ) ≪ g ( x ) , f ( x ) ≫ g ( x ) and f ( x ) = O ( g ( x )) denote the usual Vinogradov symbols. The notation f ( x ) ≍ g ( x )will be used to mean that both f ( x ) ≪ g ( x ) and g ( x ) ≪ f ( x ) hold. Moreover, anysubscript in these notations will be taken to mean that the implied constants areallowed to depend on those parameters. The notation ¯ a (mod q ) will always be usedto denote the multiplicative inverse of a modulo q .1.4. Acknowledgements.
I would like to thank Philippe Michel for suggestingthis problem to me and for the guidance received throughout this project. I am alsovery grateful for the numerous enlightening conversations with Ian Petrow. Thispaper benefited from suggestions and comments from Pierre Le Boudec, Ramon M.Nunes and Paul Nelson.2.
Stationary phase integrals
Throughout the paper, we will need several stationary phase lemmas to estimateoscillatory integrals. In particular, we will regularly be faced with a special kindof oscillatory integral which we now define. Let W be any smooth real valuedfunction, with support in [ a, b ] ⊂ (0 , ∞ ), and such that W ( j ) ( x ) ≪ a,b,j
1. We thendefine W † ( r, s ) := Z ∞ W ( x ) e ( − rx ) x s − d x, where r ∈ R and s ∈ C . Munshi gives in [8] estimations and asymptotics for W † ,however we will also need a slightly more precise version of this asymptotic. Tothis purpose, we quote from [1] a version of the stationary lemma. Lemma 1.
Let < δ < / , and X, Y, V, V , Q > , Z := Q + X + Y + V + 1 ,and assume that Y ≥ Z δ , V ≥ V ≥ QZ δ/ Y / . Suppose that w is a smooth function on R with support on an interval [ a, b ] of finitelength V , satisfying w ( j ) ( t ) ≪ j XV − j , ALEXANDRE PEYROT for all j ≥ . Suppose that h is a smooth function on [ a, b ] , such that there exists aunique point t in the interval such that h ′ ( t ) = 0 , and furthermore that h ′′ ( t ) ≫ YQ , h ( j ) ( t ) ≪ j YQ j , for j = 1 , , , · · · , t ∈ [ a, b ] . Then, the integral defined by I := Z ∞−∞ w ( t ) e ih ( t ) d t has an asymptotic expansion of the form I = e ih ( t ) p h ′′ ( t ) X n ≤ δ − A p n ( t ) + O A,δ ( Z − A ) , and (9) p n ( t ) := √ πe πi/ n ! (cid:18) i h ′′ ( t ) (cid:19) n G (2 n ) ( t ) , where A is arbitrary, and (10) G ( t ) := w ( t ) e iH ( t ) ; H ( t ) = h ( t ) − h ( t ) − h ′′ ( t )( t − t ) . Furthermore, each p n is a rational function in h ′ , h ′′ , · · · , satisfying (11) d j d t j p n ( t ) ≪ j,n X (cid:0) V − j + Q − j (cid:1) (cid:16) ( V Y /Q ) − n + Y − n/ (cid:17) . We want to extract the first five terms in the asymptotic expansion, in orderto have a small enough error term that will be easy to deal with. We thereforecompute p ( t ) = √ πe (1 / w ( t ) , and G ′ ( t ) = w ′ ( t ) e iH ( t ) + iw ( t ) H ′ ( t ) e iH ( t ) ,G ′′ ( t ) = e iH ( t ) ( w ′′ ( t ) + 2 iw ′ ( t ) H ′ ( t ) + iw ( t ) H ′′ ( t ) − w ( t ) H ′ ( t ) ) . We now see that H ( t ) = 0 , while H ′ ( t ) = h ′ ( t ) − h ′′ ( t )( t − t ) , and H ′′ ( t ) = h ′′ ( t ) − h ′′ ( t ) . Hence, we see that also H ′ ( t ) , H ′′ ( t ) = 0. We therefore have p ( t ) = √ πe (1 / i h ′′ ( t ) w ′′ ( t ) . Noting that only the terms that don’t contain H ( i ) for i = 0 , , H ( j ) ( t ) = h ( j ) ( t ) for j ≥
3, we have G (4) ( t ) = w (4) ( t ) + 4 iw ′ ( t ) h (3) ( t ) + iw ( t ) h (4) ( t ) , and thus p ( t ) = − √ πe (cid:0) (cid:1) h ′′ ( t ) ( w (4) ( t ) + 4 iw ′ ( t ) h (3) ( t ) + iw ( t ) h (4) ( t )) . In general, G (2 n ) ( t ) is a linear combination of terms of the form w ( ν ) ( t ) H ( ν ) ( t ) · · · H ( ν l ) , NALYTIC TWISTS OF MODULAR FORMS 7 where ν + · · · + ν l = 2 n .We now wish to use these in the context of the study of W † ( r, s ) , where we write s = σ + iβ ∈ C . We may thus use the lemma above with w ( x ) = W ( x ) x σ − , and h ( x ) = − πrx + β log x. Then,(12) h ′ ( x ) = − πr + βx , and h ( j ) ( x ) = ( − j − ( j − βx j , for j ≥
2. The unique stationary point is given by x = β πr . We now let ˇ W ( x ) := x − σ X n =0 p n ( x ) , and claim it is non-oscillatory in the following sense. Claim 1.
Let β ≫ . Then for all j ≥ , and x ∈ [ a, b ] , ˇ W j ( x ) ≪ σ,j,a,b . Proof.
We computeˇ W ( j ) ( x ) = j X l =0 (cid:18) jl (cid:19) ( x − σ ) ( j − l ) 5 X n =0 p ( l ) n ( x ) . Now, it is clear that ( x − σ ) ( j − l ) ≪ j,σ,a,b , and so we just need to control thederivatives of each p n . Since w is a product of a power of x with W and W ( j ) ( x ) ≪ j
1, we can easily see that p ( x ) ≪ j,σ,a,b
1. Now h ′′ ( x ) = − βx j , and since β ≫ , by the same argument as for p , it is clear that p ( x ) ≪
1. Wemay apply the same reasoning for p , and more generally for any p n , since (12)implies the higher derivatives of h don’t grow compared to the powers of h ′′ in thedenominator. (cid:3) We may now give the following result for W † ( r, s ). Lemma 2.
Let r ∈ R and s = σ + iβ ∈ C , such that x = β πr ∈ [ a/ , b ] . Then, W † ( r, s ) = √ πe (1 / √− β (cid:18) β πr (cid:19) σ (cid:18) β πer (cid:19) iβ ˇ W (cid:18) β πr (cid:19) + O (min {| β | − / , | r | − / } ) . Proof.
This is a direct application of Lemma 1 with X = V = Q = 1 , Y =max {| β | , | r |} , V = b − a, using the above computations as well as (11). (cid:3) We also quote from [8] the following lemma.
Lemma 3. W † ( r, s ) = O a,b,σ,j min ((cid:18) | β || r | (cid:19) j , (cid:18) | r || b | (cid:19) j )! . ALEXANDRE PEYROT Analysis of K t In this section, we analyse further the integral representation of K t . We makea partition of unity in the integral: let I = { } ∪ j ≥ {± (cid:0) (cid:1) j } , such that for each l ∈ I , we take a smooth function W l ( x ) supported in [ l , l ] for l = 0 and such that x k W ( k ) l ( x ) ≪ k , for all k ≥
0. for l = 0, take W ( x ) supported in [ − ,
2] with W ( k )0 ( x ) ≪ l
1. andsuch that 1 = P l ∈I W l ( x ) . We then let for any i ∈ I , I l,t ( x ) := 12 π Z R g t ( σ + iν ) e ( f t ( σ + iν )) x − σ − iν W l ( ν )d ν. We prove the following result.
Lemma 4.
Let K t be an analytic trace function. We have, for x ∈ [ t, t ] , and any ǫ > , K t ( x ) = X Supp ( W l ) ⊂ [ ± t − ǫ , ± t ǫ ] ∪ [ − t ǫ ,t ǫ ] I l,t ( x ) + O ( t − ) . Moreover, we also have max x ∈ [ t, t ] | K t ( x ) | ≪ . Proof.
Condition (3) implies that we may write(13) K t ( x ) = 12 πi Z ( σ ) M t ( s ) x − s d s = 12 π Z R g t ( σ + iν ) e ( f t ( σ + iν )) x − σ − iν d ν, for any σ ∈ [ a, b ]. We now wish to run a stationary phase argument to localise theintegral around the points without too much oscillation. If l ≪ t ǫ for some small0 < ǫ < σ/ (1 / σ ), then I i,t ( x ) ≪ t ǫ + ǫ ( σ − / − σ = o (1) , as long as we take σ >
0. We now fix such an ǫ and look at l such that Supp( W l ) ⊂ [ ± t ǫ , ±∞ ), and look at x σ I l,t ( x ) = Z R g t ( σ + iν ) W l ( ν ) e (cid:16) f t ( σ + iν ) − ν π log( x ) (cid:17) d ν, for x ∈ [ t, t ]. We now compute a few derivatives, in order to apply stationaryphase arguments. We have by (4)( g t ( σ + iν ) W l ( ν )) ( j ) ( ν ) ≪ j i σ − / − j , ∀ j ≥ , while by (5) f ′ t ( σ + iν ) − log( x )2 π ≫ , if ν t and by (7) f ( j ) t ( σ + iν ) ≪ l ǫ/ − j . Therefore, in the case that ν t , we may use Lemma 1 (with X = l σ − / , U = l, β − α = 3 l/ , R = 1 , Y = l ǫ/ and Q = l ), to deduce that I l,t ( x ) ≪ A l − A , for any A > ν ≍ t , we use the second derivative bound for oscillatory integralsalong with (6) to deduce that I l,t ( x ) ≪ . (cid:3) NALYTIC TWISTS OF MODULAR FORMS 9
To conclude this section we note that the case where Supp( W l ) ⊂ [ − t ǫ , t ǫ ] can behandled as follows. Since V is a smooth compactly supported function, it admits aMellin transform, ˜ V ( s ) = Z ∞ V ( x ) x s − d x, that decays very rapidly in vertical strips. One can thus write for any α ∈ R ,V ( x ) = Z ( α ) ˜ V ( s ) x − s d s. Using this, we write for any σ ≥ ∞ X n =1 ρ f ( n ) I l,t ( n ) V (cid:16) nt (cid:17) = Z R M t ( σ + iν ) W l ( ν ) ∞ X n =1 ρ f ( n ) n − σ − iν V (cid:16) nt (cid:17) d ν = Z R Z ( α ) M t ( σ + iν ) W l ( ν ) ˜ V ( s ) t s L ( f, σ + iν + s )d s d ν ≪ t / ǫ , by the rapid decay of ˜ V .We will therefore only focus on the cases where the support of W l is close to t .This may be interpreted as the fact that the spectral decomposition of any analytictrace function, K t , concentrates around multiplicative characters of conductor t .4. Proof of Theorem 1
Following Munshi [8] we adapt Kloosterman’s version of the circle method alongwith a conductor dropping mechanism. We quote here the following proposition in[6].
Proposition 1.
Let δ ( n ) = (cid:26) if n = 0;0 otherwise.Then, for any real number Q ≥ , we have δ ( n ) = 2 ℜ Z X ∗ ≤ q ≤ Q
0) is a parameter to be chosen optimally later. We let S l ( t ) := ∞ X n =1 ρ f ( n ) I l,t ( n ) V (cid:16) nt (cid:17) , and note that in order to bound non-trivially S ( t ), it is sufficient to do so for S l ( t ),for i such that Supp W l ⊂ [ ± t − ǫ , ± t ǫ ], as follows from the previous section. Wemay thus write S l ( t ) = ∞ X n =1 ρ f ( n ) I l,t ( n ) V (cid:16) nt (cid:17) = 1 K Z R V (cid:16) vK (cid:17) ∞ X n,m =1 n = m ρ f ( n ) I l,t ( m ) (cid:16) nm (cid:17) iv V (cid:16) nt (cid:17) U (cid:16) mt (cid:17) d v = S + l ( t ) + S − l ( t ) , where U is a smooth functions supported in [1 / , / U ( x ) = 1 for x ∈ Supp( V ) and U ( j ) ≪ j