The spectral decomposition of |θ|2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\theta |^2$$\end{document}

Let θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} be an elementary theta function, such as the classical Jacobi theta function. We establish a spectral decomposition and surprisingly strong asymptotic formulas for ⟨|θ|2,φ⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle |\theta |^2, \varphi \rangle $$\end{document} as φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} traverses a sequence of Hecke-translates of a nice enough fixed function. The subtlety is that typically |θ|2∉L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\theta |^2 \notin L^2$$\end{document}. Applications to the subconvexity, quantum variance and 4-norm problems are indicated.


Introduction
Let be a congruence subgroup of the modular group SL 2 (Z), and let H denote the upper half-plane. For pair of square-integrable automorphic functions ϕ 1 , ϕ 2 ∈ L 2 ( \H), the Petersson inner product ϕ 1 , ϕ 2 := 1 vol( \H) z∈ \H ϕ 1 (z)ϕ 2 (z) dx dy y 2 (1) may be written in terms of the inner products of ϕ against the constant function 1, the elements of an orthonormal basis B cusp for the space of cusp forms, and the unitary Eisenstein series E a,1/2+it attached to the various cusps a of : with suitable normalization (see, e.g., [9], [10, §15], [6] for details), This formula may be used to establish (among other things) the equidistribution of the Hecke correspondences T n on \H through estimates such as ϕ 1 , T n ϕ 2 = ϕ 1 , 1 1, ϕ 2 + O(τ (n)n −1/2+ϑ ) for unit vectors ϕ 1 , ϕ 2 ∈ L 2 ( \H). Here the Hecke operator T n is normalized so that T n 1 = 1 (thus for = SL 2 (Z), T n f (z) is the average of f ((az + b)/d) over all factorizations n = ad and integers 0 ≤ 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-Schwarz inequality.
Variations on (3) in which ϕ 1 , ϕ 2 are not both square-integrable turn out to play a fundamental role in analytic number theory, extending far beyond the evident application to Hecke equidistribution. For instance, in the periods-based approach to the subconvexity problem on GL 2 following Venkatesh [31] and Michel-Venkatesh [18], the basic quantitative input is an analogue of (3) for • ϕ 1 = |E| 2 the squared magnitude of a unitary Eisenstein series E and • ϕ 2 = | | 2 that of a cusp form , so that ϕ 2 is rapidly-decaying but ϕ 1 fails to belong to L 2 , or even to L 1 . The inner products |E| 2 , | | 2 arise naturally after applying Cauchy-Schwarz to the integral representation L( × , 1/2) = , E for the Rankin-Selberg L-function attached to a pair cusp forms , ; the magnitudes of such L-functions are in turn related to fundamental arithmetic equidistribution problems, such as those concerning the distribution of solutions to x 2 + y 2 + z 2 = n (see for instance [17]). The standard Plancherel formula does not apply to |E| 2 , | | 2 , and indeed, its "formal application" gives the wrong answer. There arises the need for a regularized Plancherel formula which the authors of [18, §4.3.8] develop in generality sufficient for their purposes.
As we discuss below, a detailed analysis of such inner products |θ | 2 , ϕ is at the heart of each of the recent works [19][20][21][22], and also seems likely to be useful in further applications, motivating the focused discussion recorded here. Unfortunately, |θ | 2 / ∈ L 2 , so the standard Plancherel formula does not apply to the inner products |θ | 2 , ϕ . Indeed, it is immediate from (1) that a continuous function f on \H satisfying the asymptotic | f (z)| ht(z) β near the cusps is absolutely integrable if and only if β < 1, while from (4) we see that |θ | 4 (z) ht(z) near the cusp ∞. In this article, we develop a different, robust technique for decomposing and estimating such inner products. We focus for now on the following special case of the results obtained, postponing general statements to Sect. 2.
Theorem 1 Let θ denote the Jacobi theta function (4). Fix a measurable function ϕ on \H satisfying the growth condition (5).
Let n traverse a sequence of integers coprime to the level of . Then Curiously, the new bound (6) is stronger than the more straightforward estimate (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.
Before doing so, we summarize our interest in studying inner products involving |θ | 2 . Our original motivation was that asymptotic formulas such as (6) turned out to be the global quantitative inputs to the method introduced and developed in [19][20][21] for attacking the quantum variance problem. That problem concerns the sums given by for some "nice enough family of automorphic forms" F and fixed observables 1 , 2 , which we assume here for simplicity to be cuspidal. The asymptotic determination of (7) when F consists of the Maass forms of eigenvalue bounded by some parameter T → ∞ may be understood as a fundamental problem in semiclassical analysis (see for instance [35, §15.6], [23, §4.1.3], [16,36], [19, §1]). The method of [19] uses the theta correspondence to relate the sums (7) to inner products roughly of the form where h 1 , h 2 are half-integral weight Maass-Shimura-Shintani-Waldspurger lifts of 1 , 2 . The local data underlying the lifts depends rather delicately upon the family F . It turns out that when F is "nice enough," the product h 1 h 2 is essentially a translate of the product ϕ := h 0 1 h 0 2 of some fixed lifts h 0 1 , h 0 2 . If that translate is induced by (for instance) the correspondence T n , then (8) is of the form |θ | 2 , T n ϕ and so estimates like (6) become relevant for determining the asymptotics of sums like (7).
Another motivation for the present study comes from the appearance of elementary theta functions in the Shimura integral representation [5,28] L(sym 2 ϕ, 1/2) ≈ ϕθẼ (9) for symmetric square L-functions on GL 2 ; here ϕ is a cusp form andẼ is a suitable halfintegral weight Eisenstein series, and ≈ denotes equality up to local factors. One also knows the cuspidal analogue of (9), namely for cusp forms with half-integral lift h (see Qiu [24,Thm 4.5]). An application of the Cauchy-Schwartz inequality to such integrals yields inner products involving |θ | 2 . By study-ing those inner products using the results of this paper, a surprising implication concerning the subconvexity problem for (twisted) symmetric square L-functions was obtained in [22].
A third source of motivating applications may be obtained by summing (precise forms of) the identity (10) over either ϕ or h in an orthonormal basis. The LHS of the resulting identity is a moment of L-functions, while the RHS, by Parseval, is an inner product involving |θ | 2 . By applying the results of this paper to such inner products, one may hope to obtain summation formulas for certain moments of L-functions. Summing over ϕ yields the moments relevant for the quantum variance problem discussed above, while taking a suitably normalized sum over h yields a summation formula for twisted symmetric square L-functions to which one might hope to apply the techniques of [20]. We plan to pursue this idea separately.
For the applications motivating this work, it is indispensible to have more flexible forms of Theorem 1 in which: • The squared magnitude |θ | 2 is generalized to any product θ 1 θ 2 of unary theta series θ 1 , θ 2 , such as those obtained by imposing congruence conditions in the summation defining θ . • One allows greater variation than ϕ → T n ϕ for n coprime to the level of . For instance, one would like to consider variation under the diagonal flow on \SL 2 (R), or with respect to "Hecke operators" at primes dividing the level. • The dependence of the error term upon ϕ and θ 1 , θ 2 is quantified.
The main purpose of this article is to supply such flexible forms. Our results, to be formulated precisely in Sect. 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].
The regularized Plancherel formula of Michel-Venkatesh, which builds on a method of Zagier [34], does not seem to apply to the inner products |θ | 2 , ϕ considered here: that method would involve finding an Eisenstein series E with parameter to the right of the unitary axis for which |θ | 2 − E ∈ L 2 , but it is easy to see from the expansion |θ | 2 (z) ∼ y 1/2 + · · · near the cusp ∞ that such an E does not exist. The singular nature of the parameter 1/2 presents further difficulties if one tries to adapt that method. We are not aware of an adequate formal reduction (e.g., via a simple approximation argument or truncation) by which one may deduce flexible forms of estimates such as (6) directly from (2). The technique developed here is more direct, and specific to |θ | 2 . 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 Sect. 3 give a pointwise expansion where a traverses the cusps of 0 (4), E * a,1/2+it := 2ξ(1 + 2it)E a,1/2+it , ξ(s) := π −s/2 (s/2)ζ (s), and the complex coefficients c(a, t) are explicit and uniformly bounded (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 a,1/2+it , as given in the simplest case a = ∞, = SL 2 (Z) by for Re(s) > 1, vanishes like O(t) as t → 0, while ξ(1 + 2it) has a simple pole at t = 0, so the normalized variant E * a,1/2+it is well-defined and E * a,1/2 is not identically zero. (2) From the expansion (11) we deduce first that |θ | 2 , 1 = 1 and then that (3) We deduce Theorem 1 in the expected way by replacing ϕ with T n ϕ and appealing to the consequence E * a,1/2+it , T n ϕ (1 +|t|) −100 τ (n)n −1/2 of standard bounds for unitary Eisenstein series and the rapid decay of ξ(1 + 2it).
The main analytic difference between the integrands on the RHS of (2) and (13) is in their t → 0 behavior: for nice enough ϕ, ϕ 1 , ϕ 2 , the typical magnitude of the integrand in (2) is |t| while that in (13) is 1. The more glaring difference between the two expansions is that (13) 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. [18, §4.3.8] or [34], one can define a regularized inner product |θ | 2 , ϕ reg whenever ϕ admits an asymptotic expansion near each cusp in terms of finite functions y β log(y) m with exponent of real part Re(β) = 1/2. The regularization takes the form |θ | 2 , ϕ reg := |θ | 2 , ϕ−E for an auxiliary Eisenstein series E. The difference ϕ−E satisfies the growth condition (5), so the results of this paper apply directly to such regularized inner products.

Remark 2 It follows in particular from (11) that
|θ | 2 is orthogonal to every cusp form, as does not appear to be widely known; this feature is crucial for the application to subconvexity pursued in [22]. By taking a residue in Shimura's integral [28], (14) 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 arguments amount to viewing |θ | 2 as the restriction to the first factor of a theta kernel for (SL 2 , O 2 ), where O 2 denotes the orthogonal group of the split binary quadratic form (x, y) → x 2 − y 2 ; the expansion (11) then amounts to the (regularized) decomposition of that kernel with respect to the O 2 -action. We may then understand (11) as asserting that cusp forms do not participate in the global theta correspondence with the split O 2 , as should be well-known to experts.

Statements of main results
We refer to Sect. 5 for detailed definitions in what follows and to [7,8,29,33] for general background.
Let k be a global field of characteristic = 2 with adele ring A. Let ψ : A/k → C (1) be a non-trivial additive character. Denote by Mp 2 (A) the metaplectic double cover of SL 2 (A), A(SL 2 ) (resp. A(Mp 2 )) the space of automorphic forms on SL 2 (k)\SL 2 (A) (resp. SL 2 (k)\Mp 2 (A)), ω the Weil representation of Mp 2 (A) attached to ψ and the dual pair the standard theta intertwiner parametrizing the elementary theta functions, I(χ ) the unitary induction to SL 2 (A) of a unitary character χ : A × /k × → C (1) , and Eis : I(χ ) → A(SL 2 ) the standard Eisenstein intertwiner defined by averaging over * * * \SL 2 (k) and analytic continuation along flat sections. Choose a Haar measure d × y on A × , hence on A × /k × .
Using d × y, we define in Sect. 5.11 for all nontrivial unitary characters χ of A × /k × an Mp 2 (A)-equivariant map I χ : ω ψ ⊗ ω ψ → I(χ ); it may be regarded as a restricted tensor product of local maps, regularized by the local factors of L(χ, 1). As we explain in Sect. 5.11, the composition Eis • I χ makes sense for all unitary χ.
Denote by (0) the integral over unitary characters χ of A × /k × with respect to the measure dual to d × y. Write simply for an integral over SL 2 (k)\SL 2 (A) with respect to Tamagawa measure, or equivalently, the probability Haar.
The map ω ψ φ → θ ψ,φ has kernel given by the subspace of odd functions, so we restrict attention to the even subspace ω (That subspace is reducible, but its further reduction is unimportant for us.) ψ , then we have the inner product formula Finally, for ϕ ∈ A(SL 2 ) satisfying the growth condition (30) analogous to (5), we have the inner product expansion Theorem 2 specializes to (11) upon taking k = Q and φ 1 , φ 2 as in the example of Sect. 5.4. A special case of Theorem 2 was proved by us in [19, §10]; the proofs are presented differently, and their comparison may be instructive. The contribution from the trivial character χ to the RHS of (17) should be compared with the case of Siegel-Weil discussed in [4, §7.2]. More generally, suppose φ 1 ∈ ω ψ , φ 2 ∈ ω ψ for some nontrivial characters ψ, ψ of A/k. One can write ψ (x) = ψ(ax) for some a ∈ k × . If a ∈ k ×2 , then ω ψ ∼ = ω ψ , and so one can study θ ψ,φ 1 θ ψ ,φ 2 using Theorem 2. If a / ∈ k ×2 , one can prove (more easily) an analogue of Theorem 2 involving dihedral forms for the quadratic space k 2 (x, y) → x 2 − ay 2 ; see Sect. 5.12. One finds in particular that

Theorem 3 There exists an integer d depending only upon the degree of k so that for nontrivial characters
The implied constant depends at most upon (k, ψ, ψ ).
One should understand the conclusion of Theorem 3 as follows: if θ 1 , θ 2 are a pair of essentially fixed elementary theta functions, ϕ is an essentially fixed automorphic form on SL 2 of sufficient decay, and σ ∈ SL 2 (A) traverses a sequence that eventually escapes any fixed compact, then θ 1 θ 2 , σ ϕ tends to θ 1 θ 2 , 1 1, ϕ as rapidly as the Ramanujan conjecture would predict if θ 1 θ 2 were square-integrable, and with polynomial dependence on the parameters of the "essentially fixed" quantities. An inspection of the proof also reveals polynomial dependence upon the heights of the characters ψ, ψ . The estimate (19) can be sharpened a bit at the cost of lengthening the argument (see e.g. Remark 6), but already suffices for our intended applications; the groundwork has been laid here for the pursuit of more specialized refinements should motivation arise.

Remark 4
We have already indicated that Theorem 1 follows by specializing Theorem 2 to (13). Alternatively, the T n 2 case of Theorem 1 may be recovered from Theorem 3 by taking k = Q and for σ the finite-adelic matrix diag(n, 1/n). One can deduce from (17) a more general form of (19) involving an extension of θ φ 1 θ φ 2 to the similitude group PGL 2 (A) which then specializes to the general case of Theorem 1; 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.
Detect the condition 1 μ≡ν(2) as 1 2 ξ =0,1 (−1) ξμ e πiξν and apply Poisson summation to (say) the ν sum: For simplicity, we now consider instead of |θ | 2 (z) the closely related sum obtained by stripping (21) 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 (12). It is known that E s vanishes to order one as s → 1/2, while E * s is holomorphic for Re(s) ≥ 1/2 except for a simple pole at s = 1 of constant residue 1. Shifting contours, we obtain By standard bounds on E * s (z) that take into account the rapid decay of the factor (s), Therefore (23) holds not only pointwise but also weakly when tested against functions ϕ satisfying (5). In particular, E, 1 = 2, 1 = 2. The expansion (11) 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 Sect. 2, we must keep track of how more complicated variants of the 2-adic factors in (21) 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) One can realize θ as the residue as ε → 0 of a 1/2-integral weight Eisenstein series E 3/4+ε and then subtract off a weight zero Eisenstein series E 1+ε to regularize the inner product θẼ 3/4+ε , ϕ reg := θẼ 3/4+ε − E 1+ε , ϕ following the scheme of [18, §4.3.5] with some necessary modifications; the hypotheses do not literally apply, but the method can be adapted with some work. 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 . (2) As in the proof of the standard Plancherel formula (see e.g. [6,9]), one can reduce first to understanding |θ | 2 , ϕ when ϕ is an incomplete Eisenstein series, expand ϕ =  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 approach is more direct.

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 self-dual for the character ψ 2 defined by ψ 2 (x) := ψ(2x).
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.

"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.
Every χ may be written uniquely as cα + χ 0 for some c ∈ R and χ 0 unitary; Re(χ) := c is called the real part of χ.

The metaplectic group
Denote by Mp 2 (k) the metaplectic double cover of SL 2 (k), defined using Kubota cocycles [15] as the set of all pairs (σ, ζ ) ∈ SL 2 (k) × {±1} with the multiplication law the Hilbert symbol. As generators for Mp 2 (k) we take for a ∈ k × , b ∈ k and ζ ∈ {±1} the elements Identify functions on SL 2 (k) with their pullbacks to Mp 2 (k).

Principal congruence subgroups
Suppose for this subsection that k is non-archimedean.

Sobolev norms
For each integer d and unitary admissible representation V of Mp 2 (k), denote by S V d the Sobolev norm on V defined by the recipe of [18, §2]. 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 have the shape S V d (v) := d v for a positive self-adjoint operator on V , whose definition is given the archimedean case by (essentially) := 1 − X ∈B(LieMp 2 (k)) X 2 and in the non-archimedean case by multiplication by q m on the orthogonal complement in V [m] of V [m −1]. A number of useful properties of such norms ("axioms (S1a) through (S4d)") are established in [18, §2]; for the purposes of Sect. 4, we shall need only the following: (S1b) The distortion property. There is a constant κ, depending only upon deg(k), so that for all g, v ∈ Mp 2 (k), V , one has S V d (gv) Reduction to the case of -eigenfunctions. If W is a normed vector space and : V → W a linear functional with the property that When the representation V is clear from context, we abbreviate S d := S V d .

Conventions on implied constants
Implied constants in this section are allowed to depend upon (k, ψ) except in the unramified case (Sect. 4.2), in which implied constants are required to be depend at most upon deg(k). Similarly, we abbreviate S := S d when d (the implied index) may be chosen with the above dependencies. The purpose of this convention is to ensure that implied constants are uniform when (k, ψ) traverses the local components of analogous global data.

The Weil representation
For (V , q) a quadratic space over k, the Weil representation ω ψ,V of Mp 2 (k) on the Schwartz-Bruhat space S(V ) is defined on the generators as follows: there is a quartic character χ ψ,V : k × → μ 4 < C (1) and an eighth root of unity γ ψ,V ∈ μ 8 < C (1) , whose precise definitions are unimportant for our purposes (see [8,11,28] for details), so that The complex conjugate representation is given by ω ψ,V ∼ = ω ψ,V − where V − := (V , −q) denotes the quadratic space "opposite" to V = (V , q) obtained by negating the quadratic form.
We are concerned here primarily, although not exclusively, with the case that V is the one-dimensional quadratic space V 1 ∼ = k with the quadratic form x → x 2 . In that case, we write simply ω ψ := ω V 1 ,ψ . Since ψ is fixed throughout Sect. 4, we accordingly abbreviate ω := ω ψ . It is realized on the space S(k). Note that the Fourier transform φ → φ ∧ attached above to this space differs from the "usual one" by a factor of 2 in the argument of the phase, i.e., φ ∧ (x) = y∈V 1 φ(y)ψ(2x y) dy. We normalized the Haar measure on k as we did in Sect. 4.1 so that the isomorphism V 1 ∼ = k is measure-preserving and ω is unitary. In the unramified case, the space ω K [0] of K [0]-invariant vectors in ω is one-dimensional and spanned by the characteristic function 1 o of the maximal order o in k.

Basic estimates in the Weil representation
The following estimate is cheap, but adequate for us.
Proof The L ∞ bound follows from the L 1 -bound, Fourier inversion and the distortion property applied to the Weyl element w. We turn to the L 1 -bound. In the real case k = R, there is an element Z in the complexified Lie algebra of Mp 2 (k) so that ω(Z )φ(x) = x 2 φ(x). By Cauchy-Schwarz, the contribution to φ L 1 from the range |x| ≤ 1 is bounded by O( φ ) and that from the remaining range by x∈k:|x|>1 |φ(x)| = x∈k: The complex case is similar. In the non-archimedean case, it suffices by reduction to the case of -eigenfunctions (S4d) to show for each m ≥ 0 and φ ∈ ω[m] that φ L 1 q Am φ for some absolute A. The condition φ ∈ ω[m] implies that for all b ∈ p m , one has ω(n(b))φ = φ, that is to say, (ψ(bx 2 ) − 1)φ(x) = 0 for all x ∈ k. Therefore φ is supported on elements x ∈ k satisfying the constraint ψ(bx 2 ) = 1 for all b ∈ p m . That constraint implies |x| q m/2 , and the set of elements satisfying it has volume O(q m/2 ), so Cauchy-Schwarz gives as required that

Induced representations
Denote by I(χ ) the unitarily normalized induction to SL 2 (k) of a character χ of k × , realized in its induced model as a space of functions f :

Change of polarization
Recall from Sect. 4.8 that V 1 ∼ = k is the one-dimensional quadratic space with the form x → x 2 underlying ω and V − 1 ∼ = k that with x → −x 2 underlying ω. We abbreviate the tensor product of ω and ω as ω 2 := ω ⊗ ω; it is not in any literal sense the square of the representation ω, but we shall have no occasion to refer to the latter. Then ω 2 identifies with the Weil representation ω ψ,V 2 attached to the quadratic space The latter quadratic space is split, and so by a well-known procedure (see e.g. [25, §0, (VII)]) involving a change of polarization in the symplectic space W ⊗ V 2 underlying the construction of ω ψ,V 2 (here W is the symplectic space for which SL 2 (k) = Sp(W )), there exists an intertwiner F : under which the representation ω 2 on the source corresponds to a natural geometric action of Mp 2 (k) = Mp 2 (W ) on the target: Denote by V s ∼ = k 2 the standard split quadratic space with the form (y 1 , y 2 ) → y 1 y 2 . The map ρ : V s ∼ = k 2 → V 2 ∼ = k 2 given by ρ(y 1 , y 2 ) := y 1 +y 2 2 , y 1 −y 2 2 is an isometry: if (x 1 , x 2 ) = ρ(y 1 , y 2 ), then x 2 1 − x 2 2 = y 1 y 2 . Define for φ ∈ ω ⊗ ω the partial Fourier transform

Lemma 5
In the unramified case, Proof By direct calculation.

The local intertwiner
Let χ be a character of k × with Re(χ) > −1. By Lemma 6, the map I χ : ω ⊗ ω → I(χ ) defined by the convergent integral is equivariant. The normalized local Tate integral χ → I χ (φ)/L(χ, 1) extends to an entire function of χ. We will ultimately only need to consider the range Re(χ) ≥ 0. Proof Our assumptions imply by Lemma 5 that F φ = 1 o 2 , so we conclude by the standard evaluation of unramified local Tate integrals.

Proposition 8 Suppose χ is unitary, so that I(χ ) is unitary. For each d there exists d so that for all
Proof By the equivariance of I χ , we reduce to showing that I χ (φ) S(φ 1 )S(φ 2 ). The norm on I χ (φ) is given by integration over the maximal compact, so by the distortion property (S1b) and -once again -the equivariance of I χ , we reduce to establishing the pointwise bound I χ (φ) (1) S(φ). But so we conclude by Lemma 4.

Fields, groups, spaces, measures
Let k, A, ψ be as in Sect. 2. In what follows, equip all discrete spaces with discrete measures and quotient spaces with quotient measures. Equip A with Tamagawa measure, so that vol(A/k) = 1. Fix an arbitrary Haar measure d × y on A × . Denote by p a typical place of k.
Denote by P < SL 2 the upper-triangular subgroup and U < P the strictly upper-triangular subgroup. Write e 1 := (1, 0), e 2 := (0, 1). Equip U (A), SL 2 (A) with Tamagawa measures, so that the map U (A)\SL 2 (A) σ → e 2 σ ∈ A 2 is measure-preserving. Equip P(A) with the left Haar measure compatible with the natural isomorphism P(A)/U (A) ∼ = A × and the chosen Haar measure on A × . Set X := SL 2 (k)\SL 2 (A) equipped with the Tamagawa measure (i.e., the probability Haar). We retain and adapt to the adelic setting the conventions of Sect. 4.3 concerning multiplicative characters. For instance, we denote now by y α := |y| the adelic absolute value of y ∈ A × .

Siegel domains
For convergence issues, we assume basic familiarity with Siegel domains (see e.g. [27, §4] or [2,6] or [2, §12]); the reader may alternatively trust that the general analytic issues concerning convergence are not qualitatively different from those in the model example of Sect. 3. In particular, denote by ht : X → R >0 the function ht(g) := max γ ∈SL 2 (k) ht A (γ g) where ht A is defined with respect to the Iwasawa decomposition by ht A (n(x)t(y)k) := |y| 1/2 . Then ht(x) ≥ c > 0 for some c > 0 depending only upon k; moreover, ht is proper.

Sobolev norms
We briefly recall the adelic Sobolev norms introduced in [18, §2] which were inspired in turn by [1,31]. For an integer d and a unitary admissible representation V of Mp 2 (A), define the Sobolev where denotes the restricted tensor product of the operators defined in Sect. 4.6. This definition applies also to SL 2 (A)modules, which we regard as non-genuine Mp 2 (A)-modules. These norms take finite values on smooth vectors and apply in particular when V = L 2 (X), but for that space, a finer Sobolev norm S X d is also useful: for f ∈ C ∞ (X), set S X d ( f ) := ht d d f L 2 (X) . We omit the superscript, writing S V d := S d , when V is clear from context. The indices d, d appearing here and below are implicitlyw restricted to depend only upon deg(k). We employ as in Sect. 4.6 and [18, §2] the convention of omitting the index when it is implied.
As in [18, §2.6.5], we set We now record some specialized and annotated forms of the axioms from [20, §2] relevant for Sect. 5: (S1c) Sobolev embedding. Let V be a unitary irreducible admissible representation of Mp 2 (A). Then for each d there exists a d so that the inclusion of Hilbert spaces is trace class; moreover (see [18, §2.6.3]), there exists d 0 so that the trace of −d 0 is bounded uniformly in V (with the global field k held fixed).

Mellin transforms and Tate integrals
Recall that we have fixed a Haar measure d × y on A × . It defines a quotient measure, which we also denote by d × y, on A × /k × , hence a dual measure dχ on the space of characters χ : A × /k × → C × of given real part Re(χ) = c (defined as in Sect. 4.3), so that the Mellin We summarize here some standard consequences of the theory of Tate integrals (see [26,30]). For a Schwartz-Bruhat function φ ∈ S(A) and a character χ of A × /k × with Re(χ) > 1, the integral y∈A × y χ φ(y) d × y converges absolutely for Re(χ) > 1 and extends meromorphically to all χ. We denote by reg y∈A × y χ φ(y) d × y that meromorphic extension. The possible poles are at χ = α and χ = 0. One has the global Tate functional equation where φ ∧ denotes the Fourier transform with respect to any non-trivial additive character of A/k, such as ψ or ψ 2 ; it does not matter which.

Induced representations and Eisenstein series
For each character χ : A × → C (1) whose local components have real part ≥ 0, denote by I(χ ) its unitary induction to SL 2 (A), which consists of smooth functions f : SL 2 (A) → C satisfying f (n(x)t(y)σ ) = y χ +α f (σ ) for x, y, σ ∈ A, A × , SL 2 (A); it is the restricted tensor product of the representations defined in Sect. 4.10. When χ is trivial on k × , so that it defines an automorphic unitary character χ : A × /k × → C (1) , denote by Eis χ : I(χ ) → A(SL 2 ), or simply Eis := Eis χ when χ is clear from context, the standard Eisenstein intertwiner obtained by averaging over P(k)\SL 2 (k) and analytic continuation along holomorphic sections (see e.g. [6]). When χ is unitary, so is I(χ ), and we equip it with the product of the invariant norms defined in Sect. 4.10. As in Sect. 4.10, the representation I(χ ) is reducible when χ is a nontrivial quadratic character; the results of Sect. 5.3 nevertheless apply, either by continuity from the irreducible case or by inspection of the proofs.
The Eisenstein intertwiner Eis χ has a simple zero for χ the trivial character, so the normalized variant L(χ, 1)Eis χ makes sense for any unitary χ.

The residue of the Eisenstein intertwiner
The residue of the association χ → Eis χ : I(χ ) → A(SL 2 ) as χ approaches the character α = |.| 1 is given by "integration over P(A)\SL 2 (A)" in the following sense (see e.g. [6]): 2 Let f χ ∈ I(χ ) be a holomorphic family defined in a vertical strip containing the character χ = α. Suppose also that f χ has sufficient decay as C(χ) → ∞. Then for σ ∈ SL 2 (A), where P(A)\SL 2 (A) denotes the equivariant functional I χ (α) → C compatible with the chosen Haar measures on P(A) and SL 2 (A). As a "dimensionality test," note that both dχ and P(A)\SL 2 (A) scale inversely with respect to the measure d × y on A × .

Bounds for the Eisenstein projector
For ϕ ∈ L ∞ (X) there exists, by duality, an element χ (ϕ) ∈ I(χ ) so that for all f ∈ I(χ ), The map χ is linear and equivariant. By continuity and the discussion of Sect. 5.6, we may consider L(χ, 1) χ even when χ = 0.

Lemma 11
Let χ be a unitary character of A × /k × . For any d there exists d so that Proof By (34), Lemma 10 and the automorphic Sobolev inequality (S2a), we have f , χ (ϕ) S( f )S X (ϕ). The conclusion follows from Sobolev embedding (S1c) in the form (29).

The regularized global intertwiner
Let χ : A × /k × → C × be a nontrivial Hecke character with Re(χ) > −1. We now define an Mp 2 (A)-equivariant map I χ : ω ⊗ ω → I(χ ). It may be characterized most simply as the unique equivariant map for which We estimate the size and support of the non-archimedean components and decay of the archimedean components as in the proof of Lemma 4. In the number field case, we then modify y by a suitable element of k × , using the compactness of A (1) /k × , to reduce to showing: for L a fixed lattice in V , the quantity #{δ ∈ hL : q(δ) ≤ Q} − 1 vanishes for Q small enough and is otherwise O(Q O (1) ) for all h in a fixed compact subset of SO(V )(A), where q(δ) denotes the maximum of the norms coming from the archimedean completions. The function field case is similar.

The 4-function
The Harish-Chandra function : SL 2 (A) → R >0 is the matrix coefficient of the normalized spherical vector in I(0). It controls the matrix coefficients of tempered representations in the following sense: Lemma 14 applies in particular to π = I(χ ) for any unitary character χ of A × /k × or to any of the non-split dihedral theta lifts π = θ(τ ) as in Sect. 5.12. For orientation, we record that factors as (σ ) = p p (σ p ), is left and right invariant under the standard maximal compact, and is given locally at a finite place p with uniformizer and | | −1 = q for m ≥ 0 by p (t( m )) = (2m + 1)/q m − 1 m>0 (2m − 1)/q m+1 (2m + 1)/q m .

Proof of Theorem 3 By
We deduce the ψ = ψ case by integrating over χ and applying Lemma 9 and Theorem 2.
The ψ = ψ case is proved similarly using Lemma 13.

Remark 6
A slightly lengthier argument gives a stronger (but more complicated, and not obviously more useful) estimate with S X d (ϕ 0 ) replaced by ht 1/2+ε d S ϕ 0 for S the finite set of places p for which σ p is not in the maximal compact and S the product of local Laplacians (Sect. 4.6) at those places. One can also specify more precisely the dependence upon φ 1 , φ 2 .