Then the forward momentum under πq of the mass of f is distributed with respect to the Poincar mass overY0(1). Then the push forward under πq of the measure of f on Y0(1) distributes evenly with respect to the Poincar measure overY0(1).
Introduction
Statement of Main Result
We could normalize µin|f|2dµ as probability measures, in which case Theorem 2.1.1 asserts that |f|2dµ converges weakly to dµ. An example of F = Q theorem 2.1.1 is the famous Holowinsky-Soundararajan theorem [25], which established a quantitative degree of convergence in the limit (2.1) for the "span set".
Plan for the Chapter
The bounds we obtain are stronger than those obtained by Holowinsky and Marshall in that we have removed the factor τ(l) appearing on the RHS of (2.2) and its generalizations (see Theorem 2.4.8 and Theorem 2.6.2). Although this is not necessary for our present purposes, this refinement has applications for studying the mass distribution of large-scale holomorphic forms [47].
Acknowledgements
Preliminaries
- Number Fields
- Asymptotic Notation
- Real Embeddings
- Groups
- Measures
- Characters
- Fourier Expansions
- Automorphic Forms
- Holomorphic eigencuspforms
- Maass eigencuspforms
- Eisenstein series
- Incomplete Eisenstein series
- Masses
By the holomorphic eigencuspform f : X → C of weight k = (k1, . . . , k[F:Q]) (here and always every kj is an appositive even integer, for simplicity) we mean the arithmetically normalized base holomorphic Hilbert modular form of weight k , full level and trivial central character, which is moreover an eigenfunction of the algebra of Hecke operators. F and to the unbranched character of the idele class χ∈X(CF/ˆo∗) of a sufficiently large real part, we attach the final part of the adjoint L-function.
Brief Review of Holowinsky-Soundararajan
Holowinsky’s Independent Arguments
Recall specifically that for a completely real number field F of degree d = [F : Q], our na¨ıve generalization of (2.24) and (2.25) leaves us with the task of showing that a sum of approximately xlog(x) d −1 term is small relative to x(medxa bit larger than k1), which works beyond the limits of any method that does not exploit cancellation in the sum ofλf(m)λf(n). By trivially discarding a large number of these terms through a refinement of (2.25), we reduce to the more manageable problem of showing that a sum of roughly xlog(x)εterms is small relative tox.
Soundararajan’s Independent Arguments
The Holowinsky-Soundararajan Synthesis
The same estimate follows in the completely real case once statement 2.3.1 has been formulated.
The Key Arguments in Our Generalization
Assertion 2.4.2 shows that Theorem 2.3.1 follows from strong enough bounds for the shifted sumsSφ(Y) for φa Maass eigencuspform andSχ(Y) for χ∈X(CF/ˆo∗)(0) a character of the idel class undivided units. Holowinsky [24, Thm 2] created a slightly weaker form of the d = 1 case of Theorem 2.4.8 with the application of the large sieve; in his inequality (2.2) an additional factor of τ(l) appears on the RHS. We prove Theorem 2.4.8 by adapting his approach, with the only difficulty that the regions RT ,U are shaped quite differently when >1.
If one is willing to sacrifice uniformity in the shift, alternative proofs of the corresponding weakening of Holowinsky's [24, Thm 2] and (probably) our Theorem 2.4.8 can be obtained from the general estimates due to Nair [45 ] and Nair. -Tenenbaum [46] for amountsP. We refer to [47, Rmk 3.11] for further discussion of variations on thed= 1 case of Theorem 2.4.8 that can be derived from other works and in particular their applicability to QUE in the level aspect.
Reduction to Shifted Sums Weighted by an Integral
5This is generally known for a Maass eigencusp form [34, Prop 10.7]; an incomplete Eisenstein series vanishes from a compact subset of is an incomplete Eisenstein series according to the integral formula (2.20) because of its Fourier coefficients and the fast decay of the test function Ψ∧. Suppose φ is cuspid; the case where φ=E(Ψ,·) is an incomplete Eisenstein series proceeds in the same way after separating the constant term and invoking the formula (2.20).
The integral in the final expression factors in places F; if we take each hp for the characteristic function o∗p and h∞j(y) =h0(Y y) for some fixed h0∈Cc∞(R∗+), we get. The integrals here, which can be treated either by restricting κφ,∞j trivially as in (2.45) (which is essentially what Holowinsky and Marshall do) or by our sharp refinement given in Lemma 2.4.3, essentially reduce the sum overl and to a a pair of boxes instead of regions bounded by hyperbola and hyperplanes as in our approach.
Bounds for Shifted Sums Under Hyperbolas
Therefore Theorem 2.4.8 reduces to the following result, which we will establish in the remainder of this section. Let us denote the set of data allz that arise in this way andza,b,c the set of all elements∈zhaving z-datum (a,b,c), so that we have a division. Thus, the thez-datum of n∈z belongs to Z≤y if both dyz−1mandz−1 have few prime prime factors and for Z>y if z−1m or z−1n has many prime prime factors, where y defines the threshold that separates "a little" from "a lot." The latter case rarely occurs, as we now show in Lemma 2.6.3; the first case will be handled by Lemma 2.6.4.
On the other hand, if z−1m and z−1n have few small prime factors, then using the large sieve we must show that they typically have few ordinary small prime factors;. By Lemma 2.6.3 and Lemma 2.6.4 we see that Theorem 2.4.8 follows from sufficiently strong bounds on the quantity B(y, z) given by (2.100); the following lemma reduces such bounds to a classical aiming problem.
Appendix: Sieve Bounds
The duality principle for bilinear forms, which states that a form and its transpose have the same norm, implies that D(R,x,F) is the smallest non-negative real such that. Now if we assume (as we may) that the last statement holds, then each translation of the double rectangle Rb contains at most one element of the double lattice ex−1d−1, so that each sum overμin (2.117) contains at most one nonzero term is. , So.
Appendix: Bounds for Special Functions
- Statement of Result
- Plan for the Chapter
- Notation and Conventions
- Weyl’s Criterion
- Acknowledgements
Kowalski, Michel and VanderKam note that conjecture 3.1.2 in the special case of dihedral forms follows from their subconvex bounds for Rankin-SelbergL functions modulo an unestablished extension of Watson's formula [70], which is now known by theorem 3.4 . 1 of this chapter. Our extension (theorem 3.4.1) of Watson's formula [70] shows that theorem 3.1.3 would follow from subconvex boundaries L(f ×f ×φ,1/2) φ q1−δ (δ > 0) for the central L - values of the triple productL functions attached to f as above and each Maass cusp shape or unitary Eisenstein series φ on Y0(1). Rudnick [51] showed that theorem 3.1.1 implies that the zeros of new forms of level 1 and weightk→ ∞ are equally distributed onY0(1).
In §3.2 we recall some standard properties of our basic objects of study: holomorphic new forms, Maass eigencusp forms, unitary Eisenstein series and incomplete Eisenstein series. Theorem 3.1.3 thus follows if we can show that µf(φ)/µf(1) → µ(φ)/µ(1) as q → ∞for a set of bounded functions φ whose uniform closure contains Cc(Y0) ( 1)); such a set is provided [29] by the Maass eigencusp forms and the incomplete Eisenstein sets defined in §3.2.
Background on Automorphic Forms
- Holomorphic Newforms
- Maass Eigencuspforms
- Eisenstein Series
- Incomplete Eisenstein Series
A holomorphic cusp form on Γ0(q) of weight k is a holomorphic function f :H → C that satisfies f|kγ=f for allγ∈Γ0(q) and vanishes at the cusp of Γ0(q). Aholomorphic new form is a cusp form that is an eigenform of the algebra of Hecke operators and is orthogonal to the Petersson inner product with the old forms.3 We say that a holomorphic new form f is a normalized holomorphic new form if furthermore, λf(1) = 1 in the Fourier expansion. 3The terms we leave undefined are standard and their precise definitions, which can be found in the references cited above, are not necessary for our purposes.
Since q is square-free, the numbers dσ and q/dσ are coprime, so that wσ is the Atkin-Lehner operator “WQ” in the sense of [1, p.138]. AMaass eigencuspform is a Maass cusp form that is an eigenfunction of (Nearhimedean) Hecke operators and involutions T−1:φ7→[z7→φ(−¯z)] commuting between and with them.
Main Estimates
Reduction to Shifted Sums
Our aim in this section is to reduce Theorem 3.3.1 to the limit problem of such shifted sums. Next we will refer to the statement below of Proposition 3.3.3, but not to the details of its proof. It follows by integrating the Fourier expansion (3.2) of a new form, the Fourier expansion (3.6) of a peak Maass form and formula (3.11) for the non-constant Fourier coefficients of an Eisenstein series.
When φ is an incomplete Eisenstein series, the integral formula (3.11) and standard limits for the K-Bessel function show that for every positive integer A φl(y)φ,ε,A τ(l)YA−1/2 has |l|−A(1 +Y /|l|)ε; the claim then follows by summarizing.
Bounds for Individual Shifted Sums
Define the z part of a positive integer to be the greatest divisor of the integer supported by primes p≤z. 4This limit is slightly inferior to the one obtained by Holowinsky, because we have been more precise in our calculation of the residual classes sifted out by prime divisors of c−1l; the discrepancy here doesn't matter in the end. A bound of the form (3.16) but with an unspecified dependence on the parameter l can be derived from the work of Nair [45].
The condition 4l2≤x makes their result useless in our consideration of the level aspect of QUE, where l can be almost as large as x. Here we used the inequality of the arithmetic mean and the geometric mean, a well-known limit [72, Ch 7, Misc.
Bounds for Sums of Shifted Sums
It follows from the Cauchy-Schwarz inequality, partial summation, the Rankin-Selberg bound (3.7) forλφ and the uniform bound|λit(l)| ≤τ(l) before.
An Extension of Watson’s Formula
Let dgv denote the Haar measure on the group Gv with respect to which vol(Kv) = 1. We have taken into account the relation between classical modular forms and automorphic forms on the adele group GA (see Gelbart [12]) and the comparison (see for example Vign´eras [69, §III.2]) between the Poincare measure on the upper half-plane and the Tamagawa measure on the GA. Before proceeding with the proof, let us introduce some notation and recall formulas for the matrix coefficients Φφ,p and Φf,p.
The matrix coefficient Φf,pis is bi-Kp-invariant, so with the Cartan decomposition we only need to specify Φφ,p(a(pm)) for m ≥ 0, which is given by the Macdonald formula [5, Theorem 4.6 .6]. The vectorifp lies on the unique Ip-fixed line inπf,p, where Ip is the Iwahori subgroup of Kp consisting of matrices that are mod p in upper triangle.
Proof of Theorem 3.1.3
Ifφ is a Maass eigencusp form, then the analytic conductor of φ×f ×f (qk)4, thus theorem 3.4.1 and the arguments of Soundararajan [66, Example 2] with “k” replaced by “qk”. Ifφ=E(Ψ,·) is an incomplete Eisenstein series, then the unfolding method as in Lemma 3.3.5 and the bound forRf(q) given by Lemma 3.5.1 show this. 2] Don Blasius, Hilbert modular forms and the Ramanujan conjecture, Noncommutative Geometry and Number Theory, Aspects Math., E37, Vieweg, Wiesbaden, 2006, pp.
Schulze-Pillot, On the central critical value of the triple product L-function, Number theory (Paris London Math. 27] Atsushi Ichino and Tamutsu Ikeda, On the periods of automorphic forms of special orthogonal groups and the Gross-Prasad conjecture, Geom.