Computing the Laplace eigenvalue and level of Maass cusp forms
CCOMPUTING THE LAPLACE EIGENVALUE AND LEVEL OF MAASS CUSPFORMS
Paul Savala
Department of Mathematics, Whittier College, Whittier, California 90608, USAEmail: [email protected]
Abstract.
Let f be a primitive Maass cusp form for a congruence subgroup Γ ( D ) ⊂ SL(2 , Z ) and λ f ( n )its n -th Fourier coefficient. In this paper it is shown that with knowledge of only finitely many λ f ( n ) one canoften solve for the level D , and in some cases, estimate the Laplace eigenvalue to arbitrarily high precision.This is done by analyzing the resonance and rapid decay of smoothly weighted sums of λ f ( n ) e ( αn β ) for X ≤ n ≤ X and any choice of α ∈ R , and β >
0. The methods include the Voronoi summation formula,asymptotic expansions of Bessel functions, weighted stationary phase, and computational software. Thesealgorithms manifest the belief that the resonance and rapid decay nature uniquely characterizes the under-lying cusp form. They also demonstrate that the Fourier coefficients of a cusp form contain all arithmeticinformation of the form. Introduction and Statement of Results
The primary arithmetic information attached to a Maass cusp form is its Laplace eigenvalue. However,in the case of cuspidal Maass forms, the range that these eigenvalues can take is not well-understood. Inparticular it is unknown if, given a real number r , one can prove that there exists a primitive Maass cuspform with Laplace eigenvalue 1 / r (in [10] Hejhal gives a numerical approach which approximates apossible form). Conversely, given the Fourier coefficients of a primitive Maass cusp form f on Γ ( D ), it isnot clear whether or not one can determine its Laplace eigenvalue. In this paper we show that given onlya finite number of Fourier coefficients one can often determine the level D , and then compute the Laplaceeigenvalue to arbitrarily high precision. Doing so requires f to have a spectral parameter r which is not toolarge with respect to the number of known Fourier coefficients of f . This is made precise in the corollaries.The key to our results will be understanding the resonance and rapid decay properties of Maass cuspforms. Let f be a primitive Maass cusp form with Fourier coefficients λ f ( n ). The resonance sum for f (see[17] for background) is given by(1) (cid:88) n ≥ λ f ( n ) φ (cid:16) nX (cid:17) e ( αn β )where φ ∈ C ∞ c ((1 , α ∈ R and β , X > e ( x ) :=exp(2 πix ).Sums of this form were first considered in Iwaniec-Luo-Sarnak [12] for f a normalized Hecke eigenformfor the full modular group with α = 2 √ q for q ∈ Z > and β = 1 /
2. Later in [17] Ren and Ye investigatedthis sum in the case when f was a normalized Hecke eigenform for the full modular group, but with norestrictions on α and β . Sun and Wu did the same in [24] for f a Maass cusp form for the full modular Mathematics Subject Classification.
Key words and phrases.
Maass forms for congruence subgroups, resonance, Voronoi summation formula, Laplace eigenvalue. a r X i v : . [ m a t h . N T ] N ov COMPUTING THE LAPLACE EIGENVALUE AND LEVEL OF MAASS CUSP FORMS group. Ren and Ye then gave resonance results for SL(3 , Z ) Maass cusp forms in [18] and [20]. Next, Renand Ye in [19] and Ernvall-Hyt¨onen-J¨a¨asaari-Vesalainen in [7] considered resonance for SL( n, Z ) Maass cuspforms for n ≥
2. Finally, resonance sums were considered in special cases such as Rankin-Selberg productsin [6], arithemetic functions relating to primes in [23], and used to derive bounds in terms of the spectralparameter r in [22].In this paper we take f to be a primitive Maass cusp form for a congruence subgroup Γ ( D ) ⊂ SL(2 , Z ).Thus our result extends the family of automorphic forms for which their resonance properties are known.Similar analysis and algorithms can be easily implemented for holomorphic cusp forms for Γ ( D ).In all the above cases estimations of (1) were driven by an interest in understanding the resonance andrapid decay of this sum. That is, for which choices of α and β does the sum have a large main term in X , and for which choices is it of rapid decay in X . However, in [21] Ren and Ye proposed that resonancecould be used in an algorithm to detect the presence of automorphic forms. This view of resonance sums isradically different from that which came before. Traditionally resonance sums are estimated roughly to get ageneral picture of their behavior. Yet to use them in a computational algorithm one would need to have veryprecise estimates. The results in this paper grew out of an investigation into the feasibility of implementingthe algorithm suggested by Ren and Ye.The idea of using analytic properties to locate spectral parameters for which Maass cusp forms exist wasfirst considered by Hejhal in [10] (also see [3]). Hejhal’s approach was to use the well-understood asymptoticsof the Bessel functions along with the automorphy properties of Maass forms to single out eigenvalues. In [3]Booker-Str¨ombergsson-Venkatesh use Hejhal’s approach to compute the first ten eigenvalues on PSL(2 , Z ) \H to more than 1000 decimal places, and several hundred of the corresponding Fourier coefficients to more than900 decimal places. Later in [2], Ce Bian showed that one could use a philosophically similar approach tocompute GL(3) automorphic forms. In particular he considered a sum of Fourier coefficients twisted byDirichlet characters, along with appropriate asymptotics, to solve for GL(3) spectral parameters. This wasthe first case where one could (partially) write down a Maass cusp form for GL(3) which did not comefrom the Gelbart-Jacquet lift of a GL(2) form (see [9]). Finally, Farmer-Lemurell [8] used the approximatefunctional equation to construct non-linear systems of equations, and used these to compute more than 2000spectral parameters associated to GL(4) Maass forms.Our goal is to work in the reverse direction. Given only limited information about a primitive Maass cuspform f (in particular a finite list of high Fourier coefficients of f ), we will determine its level and estimateits spectral parameter, and thus its Laplace eigenvalue. The estimate for the Laplace eigenvalue depends onthe eigenvalue not being too large with respect to the level D and a parameter X . Since a priori the spectralparameter is unknown, this presents some uncertainty into the calculations. However, by visual inspection(as demonstrated in Section 4 at the end of this paper) one may still be able to obtain a reliable estimate.Theorem 1 gives the precise form of the resonance sum for f , which is useful for determining computationalprecision. Corollaries 1 and 2 answer the classical resonance questions “for which parameters α and β doesthe sum (1) have resonance and rapid decay?” These two corollaries extend Sun and Wu’s result becausewe allow f to be a form on a congruence subgroup Γ ( D ). Corollary 3 is the result of greatest interest,since it potentially allows one to estimate the spectral parameter r (and thus the Laplace eigenvalue of f ) toarbitrarily high precision, using easily available mathematical software. Corollary 4 gives a computationaltest to determine a range for the level D , and in many cases solve for it explicitly. In Section 4 we givenumerical examples illustrating these ideas. The results are as follows: OMPUTING THE LAPLACE EIGENVALUE AND LEVEL OF MAASS CUSP FORMS 3
Theorem 1.
Let f be a primitive Maass cusp form for a Hecke congruence subgroup Γ ( D ) of SL(2 , Z ) withLaplace eigenvalue / r , and λ f ( n ) its n -th Fourier coefficient. Let φ ∈ C ∞ c ((1 , be a smooth cutofffunction and α ∈ R , β > , X > be real numbers. Then for any positive integer N , (cid:88) n ≥ λ f ( n ) e ( αn β ) φ (cid:16) nX (cid:17) = i − Dλ f ( D ) (cid:88) n< b ∗ λ f D ( n ) N − (cid:88) k =0 C r,k X / − k × (cid:16) nD (cid:17) − / − k P + α,β,X (cid:32) − sgn ( α )2 (cid:114) nXD , k (cid:33) − i πDλ f ( D ) (cid:88) n< b ∗ λ f D ( n ) N − (cid:88) k =0 C r,k d r,k X / − k × (cid:16) nD (cid:17) − / − k P − α,β,X (cid:32) − sgn ( α )2 (cid:114) nXD , k (cid:33) + E N ( X, r ) , where b ∗ := ( | α | β ) X β − D min { , − β } , (2) C r,k := ( − k Γ(2 ir + 2 k + 1 / k − (4 π ) k +1 (2 k )!Γ(2 ir − k + 1 /
2) = O k (cid:0) r k (cid:1) , (3) P ± α,β,X ( w, k ) := (cid:90) √ t ± / − k φ ( t ) e ( αX β t β + wt ) dt, (4) d r,k := − r + (2 k + 1 / k + 1) = O k (cid:0) r (cid:1) , (5) λ f D ( n ) := (cid:40) λ f ( n ) if ( n, D ) = 1; λ f ( n ) if ( n, D ) > , and the error term E N ( X, r ) satisfies E N ( X, r ) (cid:28) φ,β,N λ f ( D ) (cid:34) e − π √ X/D (cid:18) XD (cid:19) + r N (cid:18) XD (cid:19) − N + (cid:18) XD (cid:19) − N (1 + r ) X (cid:104)(cid:16) r DX (cid:17) N − (cid:105) r D − X (cid:35) . Set Q = min t ∈ [1 , √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | α | βX β t β − − sgn ( α ) (cid:114) nXD (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . If Q (cid:54) = 0 then P ± α,β,X (cid:18) − sgn ( α )2 (cid:114) nXD , k (cid:19) = O β,k,φ (cid:18) αX β Q + 1 Q (cid:19) . In particular, by trivial estimation P ± α,β,X ( w, k ) (cid:28) φ regardless of the value of Q . Corollaries 1 and 2 simplify Theorem 1 by preserving only the largest terms. Corollary 1 gives conditionsfor rapid decay, while Corollary 2 gives conditions for a main term. Comparing Corollary 2 to the result of
COMPUTING THE LAPLACE EIGENVALUE AND LEVEL OF MAASS CUSP FORMS
Sun and Wu [24] one can see that our result shows the same main term of size X / , however our corollaryalso shows the role that the level D plays. Corollary 1.
With notations as in Theorem 1, if for some (cid:15) > one has r D (cid:28) X − (cid:15) and | α | βX β min { , − β } < (cid:114) XD , then (cid:88) n ≥ λ f ( n ) φ (cid:16) nX (cid:17) e ( αn β ) (cid:28) X − M for all M > . The implied constant may depend on α , β , r , D , M , (cid:15) and φ , but not on X . Corollary 2 arises from substituting N = 1 into Theorem 1, fixing specific choices of α and β , and groupingmore terms into the error term for a simpler expression. In addition, the reader should note that the errorterm in Corollary 2 can potentially dominate the main term, depending on the relationship between r , D and X . In order to obtain maximum precision we leave the error term as is in Corollary 2, and consider aspecial case in Corollary 3 with a more controlled error term. Corollary 2.
With notations as in Theorem 1, let q < X/D be a positive integer, set α = 2 (cid:112) q/D and β = 1 / . Then (cid:88) n ≥ λ f ( n ) φ (cid:16) nX (cid:17) e (cid:18) (cid:114) qnD (cid:19) = c + q λ f ( D ) (cid:18) XD (cid:19) λ f D ( q )+ c − d r, q λ f ( D ) (cid:18) XD (cid:19) λ f D ( q )+ E (cid:48) ( X, r ) where c + := i − π (cid:90) √ t φ ( t ) dt, c − := − i + 18 π (cid:90) √ t − φ ( t ) dt,d r, = − r − , and E (cid:48) ( X, r ) (cid:28) φ λ f ( D ) (cid:34) e − π √ XD (cid:18) XD (cid:19) + r (cid:18) XD (cid:19) − + (1 + r ) (cid:18) XD (cid:19) − + qX (cid:32) r (cid:18) XD (cid:19) − (cid:33)(cid:35) . In Corollary 3 we see that by carefully keeping track of all constants in Theorem 1, one can use theresonance properties of f to solve for the spectral parameter r . Doing so requires knowing the level D of f ,which is handled in Corollary 4. The error term in Corollary 3 comes from the error term in Corollary 2.However the error in Corollary 2 can dominate the main term, depending on the relationship between r, D and X . By imposing the condition r D (cid:28) X − (cid:15) we guarantee that the error term is of decay in X , thusincreasing the accuracy as X tends to infinity. OMPUTING THE LAPLACE EIGENVALUE AND LEVEL OF MAASS CUSP FORMS 5
Corollary 3.
With notation as in Theorem 1, recall that f has Laplace eigenvalue + r . If for some (cid:15) > one has r D (cid:28) X − (cid:15) , then r = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ f ( D )2 c − (cid:18) XD (cid:19) − / (cid:32) (cid:88) X ≤ n ≤ X λ f ( n ) φ (cid:16) nX (cid:17) e (cid:18) (cid:114) nD (cid:19) − c + λ f ( D ) (cid:18) XD (cid:19) / (cid:33) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O N,φ (cid:18) λ f ( D ) (cid:18) XD (cid:19) − (cid:15) (cid:19) , where c ± are as in Corollary 2.In Corollary 4 the parameter c plays the role of a “guess” at the level D . Indeed, if c = D , then α (cid:15) willsatisfy the rapid decay conditions of Corollary 1, and α q will satisfy the resonance conditions on Corollary2. Thus Corollary 4 shows that if the c behaves sufficiently like the level D , then in fact the two are close.Numerical examples demonstrating the ideas in Corollaries 3 and 4 are given in Section 4. Corollary 4.
With notation as in Theorem 1, for a fixed choice of q, c ∈ Z > and < (cid:15) < , define α (cid:15) ( c ) = (cid:15) √ c , α q ( c ) = 2 (cid:114) qc . Suppose that for some Maass cusp form f as in Theorem 1 and r D (cid:28) X − (cid:15) , (cid:88) n ≥ λ f ( n ) φ (cid:16) nX (cid:17) e (cid:0) α (cid:15) ( c ) √ n (cid:1) (cid:28) X − M for all M > as X → ∞ , and (cid:88) n ≥ λ f ( n ) φ (cid:16) nX (cid:17) e (cid:0) α q ( c ) √ n (cid:1) = O (cid:0) X δ (cid:1) for some δ > as X → ∞ . Then we have the inequalities c (cid:0) √ q + (cid:112) c X (cid:1) < D < c(cid:15) . Since D is an integer, if one can choose (cid:15) and q to make this range small enough that it only contains asingle integer, then one has solved for D . Note that as X → ∞ the range for D becomes c/q ≤ D ≤ c/(cid:15) .Thus unless some computational reason prohibits it, choosing q = 1 and (cid:15) close to 1 is optimal.Since our approach allows one to ascertain properties of a given Maass form, one may wonder where Maassforms show up in the larger theory. A Maass form can be lifted to an automorphic cuspidal representation π = ⊗ ν ≤∞ π ν of GL(2) over the adelic ring A Q of Q (see [4] Section 3.2). Our analysis and algorithms showthat the non-Archimedean local representations π p , p < ∞ , or a finite list of them, can be used to uniquelydetermine the Archimedean local representation π ∞ and the global conductor. This can be regarded as anew type of strong multiplicity one theorem. The Langlands program (see [15]) predicts that all L-functionscan be expressed as products of automorphic L-functions for cuspidal representations of GL(n, A Q ). Ourresults offer a possible new approach to this conjecture when an otherwise defined L-function is only knownto match finitely many local components and L-factors of an automorphic L-function. COMPUTING THE LAPLACE EIGENVALUE AND LEVEL OF MAASS CUSP FORMS
The fact that resonance and rapid decay of sums of Fourier coefficients of f can be used to determine thelevel D and Laplace eigenvalue 1 / r supports the belief that these resonance and rapid decay propertiescan be used to characterize the underlying Maass form. This valuable insight allows us to understand moreabout the oscillatory nature of Maass forms.2. Proof of Theorem 1
Let f be a primitive Maass cusp form for Γ ( D ) with Laplace eigenvalue 1 / r . Then f has Fourierexpansion (see [4] Section 1.9) f ( z ) = (cid:88) n (cid:54) =0 λ f ( n ) √ yK ir (2 π | n | y ) e ( nx ) . Here K ir is the modified Bessel function of rapid decay (see [25] p. 181). If Φ ∈ C ∞ ( R > ) vanishes in aneighborhood of zero and is rapidly decreasing, then we have the Voronoi summation formula (see Kowalski-Michel-VanderKam [14] Appendix A) (cid:88) n ≥ λ f ( n )Φ( n ) = 1 Dλ f ( D ) (cid:88) n ≥ λ f D ( n ) (cid:90) ∞ Φ( x ) J f (cid:18) π (cid:114) nxD (cid:19) dx (6) + (cid:15) f Dλ f ( D ) (cid:88) n ≥ λ f D ( n ) (cid:90) ∞ Φ( x ) K f (cid:18) π (cid:114) nxD (cid:19) dx, where J f ( z ) := − π sin( πir ) (cid:16) J ir ( z ) − J − ir ( z ) (cid:17) , K f ( z ) := 4 (cid:15) f cosh( πr ) K ir ( z ) ,λ f D ( n ) = (cid:40) λ f ( n ) if ( n, D ) = 1; λ f ( n ) if ( n, D ) > . Here J ± ir is the Bessel function of the first kind (see [25] p. 181), and (cid:15) f = ± f isan even or odd Maass form respectively. In our case we set(7) Φ( n ) = φ (cid:16) nX (cid:17) e ( αn β )where φ ∈ C ∞ c ((1 , J v ( z ) and K v ( z ) for | z | (cid:29) K v ( z ) = (cid:114) π z e − z (cid:32) O (cid:18) ν − z (cid:19)(cid:33) and J ± ν ( z ) = (cid:114) πz cos (cid:16) z ∓ π ν − π (cid:17) (cid:34) N − (cid:88) k =0 ( − k Γ( ν + 2 k + 1 / z ) k (2 k )!Γ( ν − k + 1 /
2) + R ( N ) (cid:35) − (cid:114) πz sin (cid:16) z ∓ π ν − π (cid:17) (cid:34) N − (cid:88) k =0 ( − k Γ( ν + 2 k + 3 / z ) k +1 (2 k + 1)!Γ( ν − k − /
2) + R ( N ) (cid:35) , where | R ( N ) | < (cid:12)(cid:12)(cid:12)(cid:12) Γ( ν + 2 N + 1 / z ) N (2 N )!Γ( ν − N + 1 / (cid:12)(cid:12)(cid:12)(cid:12) = O N (cid:0) ν N z − N (cid:1) , | R ( N ) | < (cid:12)(cid:12)(cid:12)(cid:12) Γ( ν + 2 N + 3 / z ) N +1 (2 N + 1)!Γ( ν − N − / (cid:12)(cid:12)(cid:12)(cid:12) = O N (cid:0) ν N +2 z − N − (cid:1) , OMPUTING THE LAPLACE EIGENVALUE AND LEVEL OF MAASS CUSP FORMS 7 for any ν ∈ C . After rearranging we have J f ( z ) = e iz ( i + 1) (cid:114) πz N − (cid:88) k =0 (4 π ) k +1 C r,k z k (cid:16) − d r,k iz (cid:17) (9) − e − iz ( i − (cid:114) πz N − (cid:88) k =0 (4 π ) k +1 C r,k z k (cid:16) d r,k iz (cid:17) + O N,φ (cid:18) F r,N z N +1 / (cid:19) , where d r,k := − r + (2 k + 1 / k + 1) = O k (cid:0) r (cid:1) ,C r,k := ( − k Γ(2 ir + 2 k + 1 / k − (4 π ) k +1 (2 k )!Γ(2 ir − k + 1 /
2) = O k (cid:0) r k (cid:1) ,F r,N := 1(2 N )! N (cid:89) (cid:96) =1 (cid:0) ir + 12 − (cid:96) (cid:1) = O N (cid:0) r N (cid:1) , (10)with C r,k and d r,k first defined in (3) and (5). We note that this definition of C r,k appears somewhatunnatural, since it includes an extra factor 2(4 π ) − k − which is cancelled out in (9). However, this definitionof C r,k will lead to simpler expressions in (14) and (15). Finally, note that the implied constants do not depend on the spectral parameter r or the level D .We first apply the asymptotics of K ir ( z ) from (8) to K f ( z ) := 4 (cid:15) f cosh( πr ) K ir ( z ) appearing in (6), toarrive at (cid:90) ∞ φ (cid:18) xX (cid:19) e ( αx β ) K f (cid:18) π (cid:114) nxD (cid:19) dx (cid:28) cosh( πr ) (cid:18) D π n (cid:19) (cid:90) XX φ (cid:18) xX (cid:19) e ( αx β ) x − e − π √ nx/D × (cid:40) O (cid:18)(cid:16) r + 14 (cid:17)(cid:18) Dnx (cid:19) (cid:19)(cid:41) dx (cid:28) φ cosh( πr ) (cid:18) Dn (cid:19) (cid:40) (cid:18) r + 14 (cid:19)(cid:18) DnX (cid:19) (cid:41) × (cid:90) XX x − e − π √ nx/D dx (cid:28) X (cid:18) Dn (cid:19) e − π √ nX/D cosh( πr ) (cid:40) (cid:18) r + 14 (cid:19)(cid:18) DnX (cid:19) (cid:41) . Using the known bound λ f ( n ) (cid:28) n θ for θ = + (cid:15) (see [13]) we see that E (1) ( X, r ) := 1 Dλ f ( D ) (cid:88) n ≥ λ f D ( n ) (cid:90) ∞ φ (cid:18) xX (cid:19) e ( αx β ) K f (cid:18) π (cid:114) nxD (cid:19) dx (11) (cid:28) Φ λ f ( D ) (cid:18) XD (cid:19) (cid:88) n ≥ n θ − e − π √ nX/D (cid:18) (cid:18) r + 14 (cid:19)(cid:18) DnX (cid:19) (cid:19) (cid:28) λ f ( D ) (cid:18) XD (cid:19) e − π √ X/D (cid:18) r (cid:18) DX (cid:19) (cid:19) . COMPUTING THE LAPLACE EIGENVALUE AND LEVEL OF MAASS CUSP FORMS
Thus the term involving the integral transform of K f is of rapid decay in X , and so will be part of the errorterm.Next we use the asymptotics for J f from (9). To simplify the presentation we write1 Dλ f ( D ) (cid:88) n ≥ λ f D ( n ) (cid:90) ∞ φ (cid:18) xX (cid:19) e ( αx β ) J f (cid:18) π (cid:114) nxD (cid:19) dx = 1 Dλ f ( D ) (cid:88) n ≥ λ f D ( n ) G + N ( n ) + 1 Dλ f ( D ) (cid:88) n ≥ λ f D ( n ) G − N ( n ) + E (2) ( X, r ) , (12)where G + N ( n ) comes from substituting the first sum in (9), G − N ( n ) from substituting the second, E (2) ( X, r ) = O F r,N Dλ f ( D ) (cid:88) n ≥ λ f D ( n ) n − N − / (cid:90) ∞ φ (cid:18) xX (cid:19) e ( αx β ) (cid:18) Dx (cid:19) N +1 / dx comes from the error term in (9), and F r,n is defined in (10). Recall that N ≥
1, and thus the sum in the errorterm is absolutely convergent. In addition, the function Φ( y ) = φ ( y/X ) e ( αy β ) defined in (7) has compactsupport in ( X, X ) ⊂ R , and thus the integral in the error term is also absolutely convergent. In particularthe integral (estimated trivially) is (cid:28) φ X ( D/X ) N +1 / . Using the bound λ f ( n ) (cid:28) n θ for θ = 7 /
64 + (cid:15) thesum in E (2) ( X ) is (cid:28) N,φ N ≥
1. Thus(13) E (2) ( X, r ) (cid:28) N,φ r N λ f ( D ) (cid:18) XD (cid:19) − N . We now return to estimating the sums involving G ± N . After making the change of variables x = Xt wearrive at G + N ( n ) = (1 + i ) N − (cid:88) k =0 C r,k X − k (cid:16) nD (cid:17) − − k P + α,β,X (cid:32) (cid:114) nXD , k (cid:33) (14) + i − π N − (cid:88) k =0 C r,k d r,k X − k (cid:16) nD (cid:17) − − k P − α,β,X (cid:32) (cid:114) nXD , k (cid:33) and G − N ( n ) = ( i − N − (cid:88) k =0 C r,k X − k (cid:16) nD (cid:17) − − k P + α,β,X (cid:32) − (cid:114) nXD , k (cid:33) (15) − i π N − (cid:88) k =0 C r,k d r,k X − k (cid:16) nD (cid:17) − − k P − α,β,X (cid:32) − (cid:114) nXD , k (cid:33) , where P ± α,β,X ( w, k ) = (cid:90) ∞ t ± / − k φ ( t ) e ( αX β t β + wt ) dt, as defined in (4). It is helpful to note that the superscript in G ± N matches the sign of the term ± (cid:112) nX/D in the oscillatory integral P α,β,X , as this sign will play an important role in the size of these oscillatoryintegrals.A similar situation arises in [21] in the proof of Theorem 4, however with N = 1 and with the termsappearing in the SL(3 , Z ) case. Nonetheless the techniques are the same, and so we use the analogoustechniques for our situation. We will now summarize that approach. OMPUTING THE LAPLACE EIGENVALUE AND LEVEL OF MAASS CUSP FORMS 9
Let w ∈ R , k ∈ Z ≥ , α ∈ R and β ∈ R > . By repeated integration by parts we have P ± α,β,X ( w, k ) = (cid:90) √ g ± s ( t ; k ) e ( ψ ( t )) dt, where ψ ( t ) := αX β t β + wt is the phase function, and g ± ( t ; k ) = t ± / − k φ ( t ) , g ± s ( t ; k ) = (cid:32) g ± s − ( t ; k )2 πiψ (cid:48) ( t ) (cid:33) (cid:48) for s ≥ . Since α, w ∈ R the phase function is real. Suppose that | ψ (cid:48) ( t ) | (cid:29) Q = Q ( w ) >
0. Then by the arguments in[21] p. 13 we have(16) P ± α,β,X ( w, k ) (cid:28) φ,β,s (cid:88) ≤ m ≤ s ( | α | βX β ) m Q ( w ) m + s . If sgn ( α ) = sgn ( w ) then the phase function ψ ( t ) = αX β t β + wt = sgn ( α ) (cid:0) | α | X β t β + | w | t (cid:1) has no critical points, provided the terms | α | β and w are not both zero, since | ψ (cid:48) ( t ) | (cid:29) | α | βX β + | w | . Now, set w = ± (cid:114) nXD with n ≥
1, as arising in (14) and (15). For this choice (up to sign) of w we may choose Q (cid:18) ± (cid:114) nXD (cid:19) = Q = | α | X β + 2 (cid:114) nXD . Thus when sgn ( α ) = sgn ( w ), by (16) we obtain P ± α,β,X (cid:18) sgn ( α )2 (cid:114) nXD , k (cid:19) (cid:28) φ,β,s (cid:18) nXD (cid:19) − s/ for all s ≥
0. On the other hand, if sgn ( α ) = − sgn ( w ) we set b ∗ = ( | α | β ) X β − D min { , − β } , as defined in (2). For n ≥ b ∗ one has | ψ (cid:48) ( t ) | = (cid:12)(cid:12)(cid:12)(cid:12) | α | βX β t β − − (cid:114) nXD (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) Q := (cid:114) nXD (cid:29) √ b ∗ X (cid:29) | α | βX β . Thus when sgn ( α ) = − sgn ( w ), by (16) we have P ± α,β,X (cid:18) − sgn ( α )2 (cid:114) nXD , k (cid:19) (cid:28) φ,β,s (cid:18) nXD (cid:19) − s/ for all s ≥
0. We therefore rewrite (12) as1 Dλ f ( D ) (cid:88) n< b ∗ λ f D ( n ) G − sgn ( α ) N ( n ) + E (2) ( X, r ) + E (3) ( X, r ) where E (3) ( X, r ) := 1 Dλ f ( D ) (cid:88) n ≥ λ f D ( n ) G sgn ( α ) N ( n ) + 1 Dλ f ( D ) (cid:88) n ≥ b ∗ λ f D ( n ) G − sgn ( α ) N ( n ) , and E (2) is given in (13). We will show that the terms appearing in E (3) ( X, r ) can be bounded sufficientlyfor our purposes using the above analysis. Indeed, when sgn ( α ) = sgn ( w ) (and n ≥
1) or sgn ( α ) = − sgn ( w )(and n ≥ b ∗ ), by the analysis above as well the asymptotics for C r,k and d r,k given in (3) and (5), we have G ± sgn ( α ) N ( n ) (cid:28) φ,β,N,s X N − (cid:88) k =0 C r,k (cid:18) nXD (cid:19) − k − s − (1 + d r,k ) (cid:28) X (cid:18) nXD (cid:19) − − s (1 + r ) N − (cid:88) k =0 (cid:18) r DX (cid:19) k = X (cid:18) nXD (cid:19) − − s (1 + r ) X (cid:104)(cid:16) r DX (cid:17) N − (cid:105) r D − X .
Set s = 4 N . Then by the above analysis and the bound λ f ( n ) (cid:28) n θ with θ = 7 /
64 + (cid:15) , we have(17) E (3) ( X, r ) (cid:28) φ,β,N λ f ( D ) (1 + r ) (cid:18) XD (cid:19) − N X (cid:104)(cid:16) r DX (cid:17) N − (cid:105) r D − X for all N ≥ (cid:88) n ≥ λ f ( n ) e ( αn β ) φ (cid:16) nX (cid:17) = 1 Dλ f ( D ) (cid:88) n< b ∗ λ f D ( n ) G − sgn ( α ) N ( n ) + E N ( X, r )for all N ≥
1, where E N ( X, r ) = E (1) ( X, r ) + E (2) ( X, r ) + E (3) ( X, r ). By combining the estimates for E ( i ) ( X, r ) given in (11), (13) and (17) we have E N ( X, r ) (cid:28) φ,β,N λ f ( D ) (cid:34) e − π √ X/D (cid:18) XD (cid:19) + r N (cid:18) XD (cid:19) − N (19) + (1 + r ) (cid:18) XD (cid:19) − N X (cid:104)(cid:16) r DX (cid:17) N − (cid:105) r D − X (cid:35) . We note that the implied constants in the bound on E N ( X, r ) do not depend on the spectral parameter r or the level D , as this fact will be important in the corollaries. In addition, we note that if one substitutesthe definition of G − sgn ( α ) N ( n ) given in (14) and (15) into (18), then this gives the estimate for the resonancesum in Theorem 1.Next we estimate the integral P ± α,β,X ( w, k ) = (cid:90) √ t ± − k φ ( t ) e ( αX β t β + wt ) dt, as defined in (4). In particular we will estimate the integral when α ∈ R , w as in (17) and sgn ( α ) = − sgn ( w ).We use the weighted first derivative test from Huxley [11], Lemma 5.5.5. OMPUTING THE LAPLACE EIGENVALUE AND LEVEL OF MAASS CUSP FORMS 11
Set g ± ( t ) = t ± / − k φ ( t ) ,f ( t ) = αX β t β − sgn ( α )2 t (cid:114) nXD . (20)Then P ± α,β,X (cid:18) − sgn ( α )2 (cid:114) nXD , k (cid:19) = (cid:90) √ g ± ( t ) e ( f ( t )) dt. Following the notation of [11] the integral will be estimated in terms of the real parameters satisfying | f ( r ) ( t ) | ≤ C r TM r , | g ( s ) ± ( t ) | ≤ C s UN r , for r = 2 , s = 0 , ,
2, where f ( r ) denotes the r -th derivative of f , and similarly for g ( s ) ± . Since φ ( t ) is aSchwartz function and t ∈ [1 , √
2] we have | g ( s ) ± ( t ) | ≤ C s for some constant C s depending only on φ . In addition, f ( r ) ( t ) = 2 β (2 β − · · · (2 β − r + 1) αX β t β − r for all r ≥
2. Set C r = | β (2 β − · · · (2 β − r + 1) | max { , β − r } ,T = | α | X β ,M = 1 , and for g ± we set U = N = 1. Finally, we set Q := min t ∈ [1 , √ | f (cid:48) ( t ) | , as defined in (6). If f (cid:48) ( t ) is not identically zero, then applying the weighted first derivative test we have(21) P ± α,β,X (cid:32) − sgn ( α )2 (cid:114) nXD , k (cid:33) = O β,k,φ (cid:18) | α | X β Q + 1 Q (cid:19) . These estimates conclude the proof of Theorem 1. (cid:3) Proof of Corollaries
Proof of Corollary 1.
This is the case of rapid decay. There will be no main terms precisely when thesum in the right-hand side of (18) vanishes, which is when 4 b ∗ <
1. Rearranging this we see that there willbe no main terms if | α | βX β min { , − β } < (cid:114) XD .
The condition r D (cid:28) X − (cid:15) is needed to ensure that (19) is of rapid decay in X . (cid:3) Proof of Corollary 2.
Corollary 2 covers the case of a single Fourier coefficient λ f D ( n ) appearing on theright-hand side (besides the term λ f ( D ) appearing as a coefficient). This gives a simpler asymptotic for the resonance sum, at the expense of an error term which is not of rapid decay in X . Moreover, Ren and Ye[17] showed that resonance for SL(2 , Z ) holomorphic forms occurs at α = ± √ q and β = 1 /
2. Sun and Wu[24] showed the same result, but for Maass cusp forms for the full modular group. In this corollary we dothe same, but modify α to reflect the dependence on the level (which for the previously mentioned paperswas D = 1, since they only considered the full modular group).We first we consider a special case of the asymptotic (21). For a fixed choice of n ∈ Z > we set β = 12 , α = 2 (cid:114) qD , w = − sgn ( α )2 (cid:114) nXD , in (21). If n = q , then the phase function f (as defined in (20)) satisfies f (cid:48) ( t ) ≡
0, and thus P ± √ qD , ,X (cid:32) − (cid:114) qXD , k (cid:33) = (cid:90) √ t ± − k φ ( t ) dt in fact has no dependence on D , q or X . If n (cid:54) = q , then | f (cid:48) ( t ) | = (cid:12)(cid:12) √ q − √ n (cid:12)(cid:12)(cid:114) XD , and thus(22) P ± √ qD , ,X (cid:32) − (cid:114) nXD , k (cid:33) = O k,φ (cid:32) DX |√ n − √ q | (cid:18) √ q |√ n − √ q | (cid:19)(cid:33) . We can use the asymptotics given in (22) for the integral P ± α, / ,X ( w, k ) defined in (4) to calculate thecontribution for n (cid:54) = q in the resonance sum estimate (18). This gives1 Dλ f ( D ) (cid:88) n< b ∗ n (cid:54) = q λ f D ( n ) G − N ( n )(23) (cid:28) √ qXλ f ( D ) N − (cid:88) k =0 r k (cid:18) XD (cid:19) − k (cid:88) ≤ n< b ∗ n (cid:54) = q n θ − − k |√ n − √ q | (cid:18) √ q |√ n − √ q | (cid:19)(cid:32) d r,k (cid:18) nXD (cid:19) − (cid:33) (cid:28) qXλ f ( D ) (cid:18) r (cid:18) XD (cid:19) − (cid:19) N − (cid:88) k =0 (cid:18) r DX (cid:19) k = qXλ f ( D ) (cid:18) r (cid:18) XD (cid:19) − (cid:19) X (cid:104)(cid:16) r DX (cid:17) N − (cid:105) r D − X .
Combining this estimate with those in (19) we arrive at(24) (cid:88) n ≥ λ f ( n ) e (cid:18) (cid:114) qnD (cid:19) φ (cid:16) nX (cid:17) = λ f D ( q ) Dλ f ( D ) G − N ( q ) + E (cid:48) N ( X, r ) OMPUTING THE LAPLACE EIGENVALUE AND LEVEL OF MAASS CUSP FORMS 13 for any integer N ≥
1, where E (cid:48) N ( X, r ) is E N ( X, r ) plus the contribution for n (cid:54) = q calculated in (23). Thus E (cid:48) N ( X, r ) (cid:28) φ,β,N λ f ( D ) (cid:34) e − π √ XD (cid:18) XD (cid:19) + r N (cid:18) XD (cid:19) − N (25) + (1 + r ) (cid:18) XD (cid:19) − N X (cid:104)(cid:16) r DX (cid:17) N − (cid:105) r D − X + qX (cid:18) r (cid:18) XD (cid:19) − (cid:19) X (cid:104)(cid:16) r DX (cid:17) N − (cid:105) r D − X (cid:35) . To allow easier comparison to similar results for holomorphic cusp forms and Maass cusp forms for thefull modular group we set N = 1 and substitute the definition of G − N ( q ) given in (12) to arrive at (cid:88) n ≥ λ f ( n ) φ (cid:16) nX (cid:17) e (cid:18) (cid:114) qnD (cid:19) = i − π λ f D ( q ) Dλ f ( D ) X (cid:16) qD (cid:17) − P +2 √ qD , ,X (cid:32) − (cid:114) qXD , (cid:33) − i π λ f D ( q ) Dλ f ( D ) d r, X (cid:16) qD (cid:17) − P − √ qD , ,X (cid:32) − (cid:114) qXD , (cid:33) + E (cid:48) ( X, r ) . where E (cid:48) ( X, r ) (cid:28) φ λ f ( D ) (cid:34) e − π √ XD (cid:18) XD (cid:19) + r (cid:18) XD (cid:19) − + (1 + r ) (cid:18) XD (cid:19) − + qX (cid:18) r (cid:18) XD (cid:19) − (cid:19)(cid:35) . Note that some of these error terms can be larger or smaller than the others depending on the relationshipbetween r, X, D and q . Thus for the time being we preserve all terms to allow maximum accuracy andflexibility in the application of this corollary. In Corollary 3 we will impose a relationship on these variablesto arrive at a simpler error term.Set c + := i − π P +2 √ qD , ,X (cid:32) − (cid:114) qXD , (cid:33) = i − π (cid:90) √ t φ ( t ) dt, (26) c − := − i + 18 π P − √ qD , ,X (cid:32) − (cid:114) qXD , (cid:33) = − i + 18 π (cid:90) √ t − φ ( t ) dt. Then (26) becomes (cid:88) n ≥ λ f ( n ) φ (cid:16) nX (cid:17) e (cid:18) (cid:114) qnD (cid:19) = c + q λ f ( D ) (cid:18) XD (cid:19) λ f D ( q )(27) + c − d r, q λ f ( D ) (cid:18) XD (cid:19) λ f D ( q )+ E (cid:48) ( X, r ) . This gives Corollary 2. (cid:3)
Proof of Corollary 3.
From Corollary 2 we see that it is simple to solve for d r, = − r − , and thus for r . Indeed, one can rearrange (27) to solve for r for any value of q . However, it is desirable to simultaneously maximize the main term and minimize the error term in (27). This is accomplished when q = 1, and thusthis is the case we use. Numerical computations show that once q gets larger one needs to choose significantlylarger X to achieve similar accuracy. The condition r D (cid:28) X − (cid:15) gives the desired decay of the error term,increasing accuracy as X → ∞ . Finally, note that all constants in the corollary are nonzero. (cid:3) It is interesting to ask whether one can improve the error term in Corollary 3. The obvious way to dothis is to use N ≥ N grows. However when N = 2, ratherthan having a quadratic polynomial in r (as in the case for N = 1), one has a degree 6 polynomial. Whilethis cannot be solved by hand, it can be numerically solved. If one first estimates the eigenvalue with theequation in Corollary 3, then it is feasible to improve the precision of r (without needing to know moreFourier coefficients) by using N = 2 and throwing away the extraneous solutions. Indeed, if one only hasvery limited knowledge of the Fourier coefficients then this approach may be useful. Proof of Corollary 4.
To prove Corollary 4 we first consider α (cid:15) . From Corollary 1 we see that theresonance sum will be of rapid decay if and only if αβX β min { , − β } < (cid:114) XD Setting β = 1 / α = α (cid:15) the assumption of rapid decay means that (cid:15) √ c < √ D Solving for D this becomes(28) D < c(cid:15) Using Corollary 2 we see that the resonance sum will not be of rapid decay when α = α q . Then setting β = 1 / α = α q the assumption of a main term at some q means that (cid:12)(cid:12)(cid:12)(cid:12)(cid:114) qc − (cid:114) qD (cid:12)(cid:12)(cid:12)(cid:12) < X − Solving this for D yields(29) 4 cqX (cid:0) √ qX + √ c (cid:1) < D < cqX (cid:0) √ qX − √ c (cid:1) Using q ≥ c (cid:16) (cid:113) c qX (cid:17) < D < c(cid:15) Note that as X → ∞ this bound on D becomes c ≤ D ≤ c(cid:15) . (cid:3) OMPUTING THE LAPLACE EIGENVALUE AND LEVEL OF MAASS CUSP FORMS 15 Numerical examples
In this section we illustrate the above ideas with a concrete example. We take a specific primitive self-dualMaass cusp form f (see [5] for details of this particular form, and [16] for many other examples) and estimateits level and spectral parameter r , and then compare these to the known values.We begin by estimating the level of f . This involves evaluating the sums given in Corollary 4 for variouschoices of c ≥
1. We first evaluate the sum involving α q ( c ) as defined in Corollary 4. To make the range for D given in Corollary 4 as small as possible we choose q = 1. Unless some computational purpose prohibitsit, this choice is optimal. In Figure 1 below we show four graphs illustrating the size of (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) n ≥ λ f ( n ) φ (cid:16) nX (cid:17) e ( α √ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) for α = α q ( c ) as a function of X . We see that at c = 5 the graph grows as a positive power of X . Thuswe evaluate the sum with α = α (cid:15) ( c ) (as defined in Corollary 4) for c = 5 and (cid:15) = 0 .
95. In Figure 2we see that this graph shows rapid decay in X . Thus from Figures 1 and 2 we deduce that D ≈
5. UsingCorollary 4 we can guarantee that 4 . < D < .
54 and thus D = 5. Indeed, f is a Maass cusp form on Γ (5). c=3 c=4c=5 c=6 Figure 1.
Absolute value of the resonance sum (1) with α = α q ( c ) for X from 1000 to2200 and q = 1. Figure 2.
Absolute value of the resonance sum (1) with α = α (cid:15) ( c ) for X from 1000 to2200 with (cid:15) = 0 .
95 and c = 5.It is important to note that neither graph alone can determine the level. In Figure 3 we see that for c = 1and q = 1 the resonance sum with α = α q ( c ) has a main term, and thus would suggest D ≈
1. However, thesum with α = α (cid:15) (1) does not show rapid decay, and thus D is in fact not near 1. α q ( c ) with q = 1 and c = 1 α (cid:15) ( c ) with (cid:15) = 0 .
95 and c = 1 Figure 3.
A main term with α = α q ( c ) for q = c = 1, but no rapid decay for α (cid:15) ( c ) . Now that we have located the level, we will use this knowledge to compute the eigenvalue. All termsin Corollary 3 are easily computed. Recall that the constants c ± both involve integrals coming from (4),however the integrals are of the form (cid:90) √ t ± φ ( t ) dt OMPUTING THE LAPLACE EIGENVALUE AND LEVEL OF MAASS CUSP FORMS 17 and are easily handled by any modern mathematical software. In Figure 4 we see that as X → ∞ the graphseems to converge to a value near 8. Indeed, the true spectral parameter is r ≈ . X = 2200the calculated value is off by only 0.02. In cases where one can easily compute tens of thousands of Fouriercoefficients one could achieve arbitrarily high accuracy. Figure 4.
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