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THE SPECTRAL DECOMPOSITION OF θ 2 | | 6 1 PAULD.NELSON 0 2 Abstract. Letθbeanelementarythetafunction,suchastheclassicalJacobi v thetafunction. Weestablishaspectraldecompositionandsurprisinglystrong o asymptoticformulasforh|θ|2,ϕiasϕtraversesasequenceofHecke-translates N ofaniceenoughfixedfunction. Thesubtletyisthattypically|θ|2∈/L2. 3 1 1. Introduction ] T Let Γ be a congruencesubgroupofthe modulargroupSL (Z), andlet H denote N 2 the upper half-plane. The Plancherel formula for Γ H, developed by Selberg in . \ h the course of proving his famous trace formula, decomposes the Petersson inner t product a m 1 dxdy ϕ ,ϕ := ϕ (z)ϕ (z) , (1) [ h 1 2i vol(Γ\H)Zz∈Γ\H 1 2 y2 of a pair of square-integrable automorphic functions ϕ ,ϕ L2(Γ H) in terms of 2 1 2 ∈ \ v theirinnerproductsagainsttheconstantfunction1,theelementsofanorthonormal 9 basis cusp for the space of cusp forms, and the unitary Eisenstein series Ea,1/2+it B 2 attached to the various cusps a of Γ; with suitable normalization, it reads (see for 5 instance [9], [10, 15], [6] for details) 2 § 0 ϕ ,ϕ = ϕ ,1 1,ϕ + ϕ ,ϕ ϕ,ϕ 1 2 1 2 1 2 . h i h ih i h ih i 1 ϕ∈XBcusp (2) 0 dt 16 + a Zt∈Rhϕ1,Ea,1/2+itihEa,1/2+it,ϕ2i4π. X : v This formula may be used to establish (among other things) the equidistribution i of the Hecke correspondences T on Γ H through estimates such as X n \ r ϕ1,Tnϕ2 = ϕ1,1 1,ϕ2 +O(τ(n)n−1/2+ϑ) (3) a h i h ih i for unit vectors ϕ ,ϕ L2(Γ H). Here the Hecke operator T is normalized so 1 2 n ∈ \ that T 1 = 1 (thus for Γ = SL (Z), T f(z) is the average of f((az +b)/d) over n 2 n all factorizations n = ad and integers 0 6 b < d) and ϑ [0,7/64] quantifies ∈ the strongest known bound [13] for the Hecke eigenvalues of the cusp forms and Eisenstein series occurring in (2). Given such a bound for Hecke eigenvalues, the estimate (3) follows from (2) and the Cauchy–Schwarzinequality. Variationson(3)inwhichϕ ,ϕ arenot bothsquare-integrableturnouttoplay 1 2 a fundamental role in analytic number theory, extending far beyond the evident application to Hecke equidistribution. For instance, in the periods-basedapproach tothesubconvexityproblemonGL followingVenkatesh[27]andMichel–Venkatesh 2 [18], the basic quantitative input is an analogue of (3) for 2010 Mathematics Subject Classification. Primary11F27;Secondary 11F37,11F72. 1 THE SPECTRAL DECOMPOSITION OF |θ|2 2 ϕ = E 2 the squared magnitude of a unitary Eisenstein series E and 1 • | | ϕ = Φ2 that of a cusp form Φ, 2 • | | so that ϕ is rapidly-decaying but ϕ fails to belong to L2, or even to L1. The 2 1 inner products E 2, Φ2 arise naturally after applying Cauchy–Schwarz to the h| | | | i integralrepresentationL(Ψ Φ,1/2)= ΨΦ,E fortheRankin–SelbergL-function × h i attached to a pair cusp forms Ψ,Φ; the magnitudes of such L-functions are in turn related to fundamental arithmetic equidistribution problems, such as those concerning the distribution of solutions to x2+y2+z2 =n (see for instance [17]). The standard Plancherel formula does not apply to E 2, Φ2 , and indeed, its h| | | | i “formalapplication”givesthewronganswer. Therearisestheneedforaregularized Plancherel formula which the authorsof [18, 4.3.8]developin generalitysufficient § for their purposes. This paper is concerned with another such variation. We will prove analogues (in fact, counterintuitive strengthenings) of the spectral decomposition(2) and the asymptotic formula (3) for ϕ := θ 2 the squared magnitude of an elementary theta function, such as 1 • | | the Jacobi theta function θ(z):=y1/4 exp(2πin2z), z =x+iy (4) n∈Z X on Γ (4) Γ, and 0 ⊇ ϕ =:ϕofsufficientlymildgrowththattheinnerproduct θ 2,ϕ converges 2 • h| | i absolutely; for instance, it will suffice to impose the growth condition ϕ(z) ht(z)1/2−δ (5) ≪ for some fixed δ > 0, where the height function ht : Γ H R is defined + \ → by ht(z):=maxγ∈SL2(Z)Im(γz). Unfortunately, θ 2 / L2, so the standard Plancherel formula does not apply to | | ∈ the inner products θ 2,ϕ . Indeed, it is immediate from (1) that a continuous h| | i function f on Γ H satisfying the asymptotic f(z) ht(z)β near the cusps is \ | | ≍ absolutelyintegrableifandonlyifβ <1,whilefrom(4)weseethat θ 4(z) ht(z) | | ≫ near the cusp . In this article, we develop a different, robust technique for ∞ decomposing and estimating such inner products. A special case of the results obtained is the following. Theorem 1. Let θ denote theJacobi theta function (4). Fix a measurablefunction ϕ on Γ H satisfying the growth condition (5). Let n traverse a sequence of integers \ coprime to the level of Γ. Then θ 2,T ϕ = θ 2,1 1,ϕ +O(τ(n)n−1/2). (6) n h| | i h| | ih i Curiously, the new bound (6) is stronger than the more straightforward esti- mate (3) in that n−1/2 replaces n−1/2+ϑ. To put it another way, the strength of (6) is comparable to that of (3) together with the assumption of the (unsolved) Ramanujan conjecture ϑ=0. We will explain this surprise shortly. Elementary theta functions have long been a cornerstone of the analytic theory of L-functions on GL owing to their appearance in the Shimura integral for the 2 symmetric square (see [24], [5]). Our motivation for studying the inner products θ 2,ϕ is that asymptotic formulas such as (6) turn out to be the global quanti- h| | i tative inputs to the method introduced in [19] for attacking the quantum variance THE SPECTRAL DECOMPOSITION OF |θ|2 3 problem. That problem concerns the (initially quite mysterious) sums given by f 2,Ψ Ψ , f 2 (7) 1 2 h| | ih | | i f∈F X forsome“niceenoughfamilyofautomorphicforms” andfixedobservablesΨ ,Ψ , 1 2 F whichwe assume here for simplicity to be cuspidal. The asymptotic determination of (7)when consistsoftheMaassformsofeigenvalueboundedbysomeparameter F T maybeunderstoodasafundamentalprobleminsemiclassicalanalysis(see →∞ for instance [31, 15.6],[20, 4.1.3],[32] [16], [19, 1]). The method of [19] uses the § § § theta correspondence to relate the sums (7) to inner products roughly of the form θ 2,h h , (8) 1 2 h| | i whereh ,h arehalf-integralweightMaass–Shimura–Shintani–Waldspurgerlifts of 1 2 Ψ ,Ψ . The local data underlying the lifts depends rather delicately upon the 1 2 family . It turns out, non-obviously, that when is “nice enough,” the product F F h h is essentially a translate of the product ϕ := h0h0 of some fixed lifts h0,h0. 1 2 1 2 1 2 If that translate is induced by (for instance) the correspondence T , then (8) is of n the form θ 2,T ϕ and so estimates like (6) become relevant for determining the n h| | i asymptotics of sums like (7). Fortheapplicationsmotivatingthiswork,itisindispensibletohavemoreflexible forms of Theorem 1 in which: The squared magnitude θ 2 is generalized to any product θ θ of unary 1 2 • | | theta series θ ,θ , such as those obtained by imposing congruence condi- 1 2 tions in the summation defining θ. One allowsgreatervariationthanϕ T ϕfor n coprime to the levelofΓ. n • 7→ For instance, one would like to consider variation under the diagonal flow on Γ SL (R), or with respect to “Hecke operators”at primes dividing the 2 \ level. The dependence of the error term upon ϕ and θ ,θ is quantified. 1 2 • The main purpose of this article is to supply such flexible forms. Our results, to be formulated precisely in 2, are natural completions of Theorem 1: working § over a fixed global field, we prove that the translates under the metaplectic group of a product of two elementary theta functions equidistribute with an essentially optimal rate and polynomial dependence upon all parameters. In quantifying the latter we make systematic use of the adelic Sobolev norms developed in [18, 2]. § Beyondthemotivatingapplicationtothequantumvarianceprobleminitsmany unexplored aspects, the expansions and estimates established in this article seem ripe for further exploitation, owing for instance to the appearance of θ in the Shimura integral. They also appear to be of intrinsic interest. These consider- ations inspired the separate and focused discussion afforded here. The regularized Plancherel formula of Michel–Venkatesh, which builds on a methodofZagier[30],doesnotseemtoapplytotheinnerproducts θ 2,ϕ consid- h| | i eredhere: thatmethodwouldinvolvefindinganEisensteinseries withparameter E to the right of the unitary axis for which θ 2 L2, but it is easy to see from | | −E ∈ the expansion θ 2(z) y1/2+ near the cusp that such an does not exist. | | ∼ ··· ∞ E The singularnatureofthe parameter1/2presents further difficulties if one tries to adapt that method. We are not aware of an adequate formal reduction (e.g., via a simple approximationargumentortruncation) by which one may deduce flexible THE SPECTRAL DECOMPOSITION OF |θ|2 4 forms of estimates such as (6) directly from (2). The technique developed here is more direct, and specific to θ 2; it was found after some searching(see Remark 5). | | We illustrate it now briefly in the context of Theorem 1: (1) Some careful but elementary manipulations (change of variables, Poisson summation, folding up, Mellin inversion, contour shift) to be explained in 3 give an (apparently new) pointwise expansion § dt θ 2 =1+ c(a,t)E∗ (9) | | a Zt∈R a,1/2+it 2π X where a traverses the cusps of Γ (4), 0 E∗ :=2ξ(1+2it)E , ξ(s):=π−s/2Γ(s/2)ζ(s), a,1/2+it a,1/2+it andthecomplexcoefficientsc(a,t)areexplicitanduniformlybounded(and best described in the language of Theorem 2 below; see also [19, 9]). § It seems worth recalling here that the standard unitary Eisenstein series E , as given in the simplest case a= ,Γ=SL (Z) by a,1/2+it 2 ∞ 1 ys ξ(2s 1) E (z):= =ys+ − y1−s+ (10) s 2 cz+d2s ξ(2s) ··· (c,d)∈ZX2−{(0,0)}:| | gcd(c,d)=1 for Re(s) > 1, vanishes like O(t) as t 0, while ξ(1+2it) has a simple → pole att=0,sothe normalizedvariantE∗ is well-definedandE∗ a,1/2+it a,1/2 is not identically zero. (2) From the expansion (9) we deduce first that θ 2,1 =1 and then that h| | i dt θ 2,ϕ = θ 2,1 1,ϕ + c(a,t) E∗ ,ϕ . (11) h| | i h| | ih i a Zt∈R h a,1/2+it i2π X (3) We deduce Theorem 1 in the expected way by replacing ϕ with T ϕ and n appealing to the consequence E∗ ,T ϕ (1 + t)−100τ(n)n−1/2 h a,1/2+it n i ≪ | | of standard bounds for unitary Eisenstein series and the rapid decay of ξ(1+2it). The mainanalyticdifference betweentheintegrandsonthe RHSof (2)and(11) is in their t 0 behavior: for nice enough ϕ,ϕ ,ϕ , the typical magnitude of the 1 2 → integrand in (2) is t while that in (11) is 1. The more glaring difference ≍ | | ≍ between the two expansions is that (11) contains no cuspidal contribution. The improvement of (6) compared to (3) is now explained by noting as above that the Hecke eigenvalues of unitary Eisenstein series (unlike those of cusp forms) are known to satisfy bounds consistent with the Ramanujan conjecture ϑ=0. Remark 1. As in [18, 4.3.8] or [30], one can define a regularized inner product § θ 2,ϕ whenever ϕ admits an asymptotic expansionnear each cusp in terms of reg h| | i finitefunctionsyβlog(y)m withexponentofrealpartRe(β)=1/2. Theregulariza- 6 tion takes the form θ 2,ϕ := θ 2,ϕ for an auxiliary Eisenstein series . reg h| | i h| | −Ei E The difference ϕ satisfies the growth condition (5), so the results of this paper −E apply directly to such regularized inner products. Remark 2. It follows in particular from (9) that θ 2 is orthogonal to every cusp form, (12) | | THE SPECTRAL DECOMPOSITION OF |θ|2 5 as does not appear to be widely known. By taking a residue in Shimura’s integral [24], (12) is equivalent to the well-known fact that L(adϕ,s) is holomorphic at s = 1 for every cusp form ϕ; compare also with [12]. The proof given here is not directly dependent on such considerations. Remark 3. In more high-level terms, our argument amounts to viewing θ 2 as the | | restriction to the first factor of a theta kernel for (SL ,O ), where O denotes the 2 2 2 orthogonalgroupofthesplitbinaryquadraticform(x,y) x2 y2;theexpansion 7→ − (9) then amounts to the (regularized) decomposition of that kernel with respect to the O -action. We may then understand (12) as asserting that cusp forms do 2 not participate in the global theta correspondence with the split O , as should be 2 well-known to experts. This paper is organized as follows. In 2, we formulate our main results. In 3, § § wesketchadirectproofofthesimplestspecialcasestatedabove(Theorem1). The general case is treated in 6 after some local ( 4) and global ( 5) preliminaries. § § § Acknowledgements. WegratefullyacknowledgethesupportofNSFgrantOISE- 1064866and SNF grant SNF-137488 during the work leading to this paper. 2. Statements of main results We refer to 5 for detailed definitions in what follows and to [29, 25, 8, 7] for § general background. Letkbeaglobalfieldofcharacteristic=2withadeleringA. Letψ :A/k C(1) 6 → be a non-trivial additive character. Denote by Mp (A) the metaplectic double 2 cover of SL (A), (SL ) (resp. (Mp )) the space of automorphic forms on 2 A 2 A 2 SL (k) SL (A) (resp. SL (k) Mp (A)), ω the Weil representation of Mp (A) 2 \ 2 2 \ 2 2 attached to ψ and the dual pair (Mp ,O(1)), ω φ θ (Mp ) the 2 ψ ∋ 7→ ψ,φ ∈ A 2 standard theta intertwiner parametrizing the elementary theta functions, (χ) I the unitary induction to SL (A) of a unitary character χ : A×/k× C(1), and 2 → Eis : (χ) (SL ) the standard Eisenstein intertwiner defined by averaging 2 I → A over (∗∗) SL (k) and analytic continuation along flat sections. Choose a Haar { ∗ }\ 2 measure d×y on A×, hence on A×/k×. Usingd×y,wedefinein 5.11forallnontrivialunitarycharactersχofA×/k× an § Mp (A)-equivariant map I : ω ω (χ); it may be regarded as a restricted 2 χ ψ ⊗ ψ → I tensor product of local maps, regularized by the local factors of L(χ,1). As we explain in 5.11, the composition Eis I makes sense for all unitary χ. χ § ◦ Denote by the integral over unitary characters χ of A×/k× with respect to (0) the measure dual to d×y. Write simply for an integraloverSL (k) SL (A) with R 2 \ 2 respect to Tamagawa measure, or equivalently, the probability Haar. R The map ω φ θ has kernel given by the subspace of odd functions, so ψ ψ,φ ∋ 7→ we restrict attention to the even subspace ω(+) = φ ω :φ(x)=φ( x) . (That ψ { ∈ ψ − } subspace is reducible, but its further reduction is unimportant for us.) Theorem 2. For φ ,φ ω(+) and σ Mp (A), we have the pointwise expansion 1 2 ∈ ψ ∈ 2 θ (σ)θ (σ)= θ θ + Eis(I (φ ,φ ))(σ) (13) ψ,φ1 ψ,φ2 ψ,φ1 ψ,φ2 χ 1 2 Z Z(0) THE SPECTRAL DECOMPOSITION OF |θ|2 6 Moreover, we have the inner product formula θ θ =2 φ φ . (14) ψ,φ1 ψ,φ2 1 2 A Z Z Finally, for ϕ (SL ) satisfying the growth condition (28) analogous to (5), we 2 ∈ A have the inner product expansion θ θ ϕ= θ θ ϕ+ Eis(I (φ ,φ ))ϕ. (15) ψ,φ1 ψ,φ2 ψ,φ1 ψ,φ2 χ 1 2 Z Z Z Z(0)Z Theorem 2 specializes to (9) upon taking k = Q and φ ,φ as in the example 1 2 of 5.4. A special case of Theorem 2 was proved by us in [19, 9]; the proofs are § § presented differently, and their comparison may be instructive. The contribution from the trivial character χ to the RHS of (15) should be compared with the case of Siegel–Weil discussed in [4, 7.2]. § More generally, suppose φ ω(+), φ ω(+) for some nontrivial characters 1 ∈ ψ 2 ∈ ψ′ ψ,ψ′ of A/k. One can write ψ′(x) = ψ(ax) for some a k×. If a k×2, then ∈ ∈ ωψ ∼= ωψ′, and so one can study θψ,φ1θψ′,φ2 using Theorem 2. If a ∈/ k×2, one can prove (more easily) an analogue of Theorem 2 involving dihedral forms for the R quadratic space k2 (x,y) x2 ay2; see 5.12. One finds in particular that ∋ 7→ − § θψ,φ1θψ′,φ2 =0. (16) Z Following [18, 2], we employ unitary Sobolev norms on ω and automorphic d Sobolev norms X§ on (SL ) (see 4.6, 5.3). Denote bSy Ξ : SL (A) R the Sd A 2 § § 2 → >0 Harish–Chandra spherical function ( 5.13). § Theorem 3. There exists an integer d depending only upon the degree of k so that for nontrivial characters ψ,ψ′ of A/k and φ ω(+),φ ω(+), ϕ (SL ) and 1 ∈ ψ 2 ∈ ψ′ ∈ A 2 σ SL (A), the right translate σϕ(x):=ϕ(xσ) satisfies 2 ∈ X θψ,φ1θψ′,φ2 ·σϕ= θψ,φ1θψ′,φ2 ϕ+O(Ξ(σ)Sd(φ1)Sd(φ2)Sd (ϕ)). (17) Z Z Z The implied constant depends at most upon (k,ψ,ψ′). OneshouldunderstandtheconclusionofTheorem3asfollows: ifθ ,θ areapair 1 2 ofessentiallyfixedelementarythetafunctions,ϕisanessentiallyfixedautomorphic formonSL ofsufficientdecay,andσ SL (A)traversesasequencethateventually 2 2 ∈ escapes any fixed compact, then θ θ ,σϕ tends to θ θ ,1 1,ϕ as rapidly as 1 2 1 2 h i h ih i the Ramanujan conjecture would predict if θ θ were square-integrable, and with 1 2 polynomialdependence on the parametersof the “essentially fixed” quantities. An inspection of the proofalso revealspolynomialdependence upon the heights of the charactersψ,ψ′. Theestimate(17)canbesharpenedabitatthecostoflengthening theargument(seee.g. Remark6),butalreadysufficesforourintendedapplications; the groundwork has been laid here for the pursuit of more specialized refinements should motivation arise. Remark 4. We have alreadyindicated that Theorem1 followsby specializing The- orem 2 to (11). Alternatively, the T case of Theorem 1 may be recovered from n2 Theorem3bytakingk =Qandforσthefinite-adelicmatrixdiag(n,1/n). Onecan deducefrom(15)amoregeneralformof (17)involvinganextensionofθ θ tothe φ1 φ2 similitude groupPGL (A)whichthenspecializestothe generalcaseofTheorem1; 2 THE SPECTRAL DECOMPOSITION OF |θ|2 7 it is not clear to us how best to formulate such an extension, and our immediate applications do not require it, so we omit it. 3. Sketch of proof in the simplest case We sketch here the proof of the expansion (9) underlying the proof of Theorem 1. Letθ be asin(4). Setz :=x+iy,e(z):=e2πiz. Considerthe Fourierexpansion θ 2(z)=y1/2 e(m2z)e(n2z) | | m,n∈Z X =y1/2 e((m2 n2)x)exp( 2π(m2+n2)y). − − m,n∈Z X Change variables to µ := m n and ν := m + n, so that m2 n2 = µν and − − m2+n2 =(µ2+ν2)/2: θ 2(z)=y1/2 e(µνx)exp( π(µ2+ν2)y)1 . µ≡ν(2) | | − µ,ν∈Z X Detectthecondition1 as 1 ( 1)ξµeπiξν andapplyPoissonsummation µ≡ν(2) 2 ξ=0,1 − to (say) the ν sum: P 1 θ 2(z)= ( 1)ξµexp( π((µx+ ξ +ν)2/y+µ2y)) (18) | | 2 − − 2 ξ=0,1µ,ν∈Z X X 1 = ( 1)µνexp( π((µx+ν)2/y+µ2y)). (19) 2 − − µ∈ZX,ν∈12Z For simplicity, we now consider instead of θ 2(z) the closely related sum | | E(z):= exp( π((µx+ν)2/y+µ2y)) (20) − µ,ν∈Z X obtained by stripping (19) of its 2-adic factors. By isolating the contribution of (µ,ν)=(0,0) and writing the remaining pairs (µ,ν) in the form (λc,λd) for some unique up to sign nonzero integers λ,c,d with gcd(c,d)=1, so that (µx+ν)2/y+ µ2y = cz+d2λ2/y, we obtain | | y E(z)=1+ f cz+d2 (c,d)∈(Z2−X{(0,0)})/{±1} (cid:18)| | (cid:19) gcd(c,d)=1 with f(y):= exp( πλ2/y). The function f decays rapidly as y 0 and λ∈Z−{0} − → satisfies f(y) y1/2 as y . Its Mellin transform f(s):= f(y)y−sd×y is P∼ →∞ y∈R×+ given for Re(s)>1/2 by R e f(s)= exp( πλ2/y)y−sd×y =2ξ(2s). Zy∈R×+λ∈XZ−{0} − e ByMellininversion,f(y)= 2ξ(2s)ys ds. ThusE(z)=1+ E∗(z) ds ,where (2) 2πi (2) s 2πi E∗ := 2ξ(2s)E with E as in (10). It is known that E vanishes to order one as s s s R s R THE SPECTRAL DECOMPOSITION OF |θ|2 8 s 1/2,while E∗ isholomorphicforRe(s)>1/2exceptforasimplepole ats=1 → s of constant residue 1. Shifting contours, we obtain ds E(z)=2+ E∗(z) . (21) s 2πi Z(1/2) By standard bounds on E∗(z) that take into account the rapid decay of the factor s Γ(s), ht(z)1/2−δ E∗(z) < . Z(1/2)ZΓ\H | s | ∞ Therefore (21) holds not only pointwise but also weakly when tested against func- tions ϕ satisfying (5). In particular, E,1 = 2,1 =2. h i h i The expansion (9) follows from the above argument applied to θ 2(z) rather | | than E(z), or alternatively, by specializing Theorem 2; see also [19, 9]. § In passing to the general results of 2, we must keep track of how more compli- § cated variants of the 2-adic factors in (19) affect the residue arising in the contour shift. This is ultimately achieved by the inversion formula for an adelic partial Fourier transform. We must also quantify everything; we do so crudely. Remark 5. The proof sketched above and its generalization given below is the third that we have found. The following alternative arguments are possible, but less efficient: (1) Onecanrealizeθastheresidueasε 0ofa1/2-integralweightEisenstein → series E˜ and then subtract off a weight zero Eisenstein series to 3/4+ε 1+ε E regularizetheinnerproduct θE˜ ,ϕ := θE˜ ,ϕ following 3/4+ε reg 3/4+ε 1+ε h i h −E i theschemeof[18, 4.3.5]withsomenecessarymodifications;thehypotheses § donotliterallyapply,butthemethodcanbeadaptedwithsomework. One can then extract the residue of the regularized spectral expansion of this regularized inner product to obtain the required formula for θ 2,ϕ . h| | i (2) As inthe proofofthe standardPlancherelformula(seee.g. [9, 6]), onecan reduce first to understanding θ 2,ϕ when ϕ is an incomplete Eisenstein h| | i series, expand ϕ = c(s)E ds as an integral of spectral Eisenstein se- (2) s 2πi ries E , and then shift contours to the line Re(s) = 1/2. Difficulty arises s R (especiallyinthegeneralityofTheorem2)whenonewishestocomparethe expansion so-obtained to the spectral coefficients E ,ϕ of ϕ, which 1/2+it h i are given by c(1/2+it)+M(1/2+it)c(1/2 it) rather than c(1/2+it) − (here M(s) := ξ(2s)/ξ(2(1 s))). The analogous difficulty in the proof of − the standard Plancherel formula is addressed by the functional equation for the intertwining operators, of which some more complicated variants are required here. The present approachis more direct. 4. Local preliminaries 4.1. Generalities. In this section we work over a local field k of characteristic = 2. Let ψ : k C(1) be a nontrivial character. Equip k with the Haar measure 6 → self-dual for the character ψ defined by ψ (x):=ψ(2x). 2 2 When k is non-archimedean, we denote by o its ring of integers, p its maximal ideal, and q :=#o/p the cardinality of its residue field. 4.2. “The unramified case”. We use this phrase to mean specifically that k is non-archimedean, ψ is unramified, and the residue characteristic of k is =2. 6 THE SPECTRAL DECOMPOSITION OF |θ|2 9 4.3. Conventions on multiplicative characters. We represent the group X(k×):=Hom(k×,C×) of continuous homomorphisms additively. Denote by yχ the value taken by the character χ X(k×) at the element y k×, by 0 X(k×) the trivial character ∈ ∈ ∈ y y0 :=1,byαtheabsolutevaluecharactery yα := y ,and,foranycomplex 7→ 7→ | | number s, by sα the character y ysα := y s; thus 7→ | | yχ1+χ2 =yχ1yχ2, (y y )χ =yχyχ, y−χ =(1/y)χ, y0 =1. 1 2 1 2 Every χ may be written uniquely as cα + χ for some c R and χ unitary; 0 0 ∈ Re(χ):=c is called the real part of χ. 4.4. The metaplectic group. Denoteby Mp (k)the metaplectic doublecoverof 2 SL (k), defined using Kubota cocycles [14] as the set of all pairs (σ,ζ) SL (k) 2 2 ∈ × 1 with the multiplication law (σ ,ζ )(σ ,ζ )=(σ σ ,ζ ζ c(σ ,σ )) where 1 1 2 2 1 2 1 2 1 2 {± } x(σ σ ) x(σ σ ) d if c=0, 1 2 1 2 c(σ1,σ2):=(cid:18) x(σ1) , x(σ2) (cid:19), x(cid:18)(cid:18)∗c ∗d(cid:19)(cid:19):=(c if c6=0 (22) with(,):k×/k×2 k×/k×2 1 the Hilbertsymbol. As generatorsforMp (k) × →{± } 2 we take for a k×,b k and ζ 1 the elements ∈ ∈ ∈{± } 1 b a n(b)= ,1 , t(a)= ,1 , 1 a−1 (cid:18)(cid:18) (cid:19) (cid:19) (cid:18)(cid:18) (cid:19) (cid:19) 1 w = ,1 , ε(ζ)=(1,ζ) 1 (cid:18)(cid:18)− (cid:19) (cid:19) which satisfy the relations n(b )n(b )=n(b +b ), t(a )t(a )=t(a a )ε((a ,a )), 1 2 1 2 1 2 1 2 1 2 t(a)n(b)=n(a2b)t(a),wt(a)=t(a−1)w,w2 =t( 1)and(whenb=0)wn( b−1)= − 6 − n(b)t(b)wn(b)w−1. Identify functions on SL (k) with their pullbacks to Mp (k). 2 2 4.5. Principal congruence subgroups. Suppose for this subsection that k is non-archimedean. Set K := SL (o). For m Z , denote by K [m] := SL2 2 ∈ >0 SL2 K (1+pmM (o)) the mth principal congruence subgroup. Define a map 2 ∩ σ :K Mp (k) SL2 → 2 (c,d)ν(c) if c=0, σ ∗ ∗ := 6 (cid:18)(cid:18)c d(cid:19)(cid:19) (1 if c=0 whereν denotesthenormalizedvaluationonk. Thereexistsanm ,dependingonly 0 upon ν(2), so that the restriction of σ to K [m ] is a homomorphism ([8, Prop SL2 0 2.8], [15]); in fact, one may take m :=0 in the unramified case ( 4.2). In general, 0 § denote for m > m by K[m] := σ(K [m]) < Mp (k) the image. It defines a 0 SL2 2 filtration K[m ] K[m +1] of congruence subgroups of Mp (k). One has 0 ⊃ 0 ⊃ ··· 2 w K[0] whenever K[0] is defined and n(b),t(a) K[m] for all b,a pm,1+pm ∈ ∈ ∈ whenever K[m] is defined. ForasmoothrepresentationV ofMp (k)andm>m ,denotebyV[m]:=VK[m] 2 0 the subspace of K[m]-invariant vectors. For m < m , write V[m] := 0 . Thus 0 { } 0 =V[ 1] V[0] V[1] and V = V[m]. { } − ⊆ ⊆ ⊆··· ∪ THE SPECTRAL DECOMPOSITION OF |θ|2 10 4.6. Sobolev norms. For eachinteger d and unitary admissible representationV of Mp (k), denote by V the Sobolev norm on V defined by the recipe of [18, 2]. 2 Sd § Strictly speaking, that article considers the case of reductive groups and not their finite covers, but the definitions and results apply verbatim in our context (using the principal subgroups defined above in the non-archimedean case). These norms havethe shape V(v):= ∆dv for a positive self-adjointoperator Sd k k ∆ on V, whose definition is given the archimedean case by (essentially) ∆ := 1 X2 and in the non-archimedeancase by multiplication by qm − X∈B(LieMp2(k)) onthe orthogonalcomplement in V[m]of V[m 1]. A number ofuseful properties P − of such norms (“axioms (S1a) through (S4d)”) are established in [18, 2]; for the § purposes of 4, we shall need only the following: § (S1b) Thedistortionproperty. Thereisaconstantκ,dependingonlyupondeg(k), so that for all g,v Mp (k),V, one has V(gv) Ad(g) κd V(v) ∈ 2 Sd ≪k k Sd (S4d) Reduction to the case of ∆-eigenfunctions. If W is a normed vector space andℓ:V W alinearfunctionalwiththepropertythat ℓ(v) 6A ∆dv V → | | k k for each ∆-eigenfunction v, then ℓ(v) 6 A′ V(v) for all v V, where | | Sd′ ∈ (A′,d′) depends only upon (A,d) and deg(k). When the representation V is clear from context, we abbreviate := V. Sd Sd 4.7. Conventions on implied constants. Implied constants in this section are allowedtodependupon(k,ψ)exceptintheunramifiedcase( 4.2),inwhichimplied § constantsarerequiredto be depend atmostupon deg(k). Similarly,we abbreviate := when d (the implied index) may be chosen with the above dependencies. d S S The purpose of this convention is to ensure that implied constants are uniform when (k,ψ) traverses the local components of analogous global data. 4.8. The Weil representation. For (V,q) a quadratic space over k, the Weil representation ω of Mp (k) on the Schwartz–Bruhat space (V) is defined on ψ,V 2 S the generators as follows: there is a quartic character χ : k× µ < C(1) and ψ,V 4 → aneighthrootofunityγ µ <C(1),whoseprecisedefinitionsareunimportant ψ,V 8 ∈ for our purposes (see [29, 11, 8] for details), so that ω (n(b))φ(x)=ψ(bq(x))φ(x), ω (t(a))φ(x)=aχψ,V+(d/2)αφ(ax), ψ,V ψ,V ω (w)φ=γ φ∧, ω (ε(ζ))φ=ζdimVφ ψ,V ψ,V ψ,V whereφ∧(y):= φ(x)ψ( x,y )dxwith x,y :=q(x+y) q(x) q(y)andtheHaar V h i h i − − measure dx normalized so that ((φ)∧)∧(x) = φ( x). The assignment V ω ψ,V R − 7→ is compatible with direct sums in the sense that if V = V′ V′′, then ω = ψ,V ⊕ ωψ,V′ ωψ,V′′ with respect to the dense inclusion (V′) (V′′) ֒ (V). The ⊗ S ⊗S → S complex conjugate representation is given by ωψ,V ∼= ωψ,V− where V− := (V,−q) denotes the quadratic space “opposite” to V = (V,q) obtained by negating the quadratic form. We are concernedhere primarily, althoughnot exclusively,with the case that V is the one-dimensional quadratic space V = k with the quadratic form x x2. 1 ∼ 7→ In that case, we write simply ω := ω . Since ψ is fixed throughout 4, we ψ V1,ψ § accordingly abbreviate ω := ω . It is realized on the space (k). Note that the ψ S Fouriertransformφ φ∧ attachedabovetothisspacediffersfromthe“usualone” 7→ byafactorof2intheargumentofthephase,i.e.,φ∧(x)= φ(y)ψ(2xy)dy. We y∈V1 normalizedtheHaarmeasureonkaswedidin 4.1sothattheisomorphismV =k § R 1 ∼ is measure-preserving and ω is unitary. In the unramified case, the space ωK[0] of

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