Pseudodifferential Operators with Automorphic Symbols

Abstract

11Pseudo-Differential Operators: Theory and Applications is a series of moderately priced graduate-level textbooks and monographs appealing to students and experts alike. Pseudo-differential operators are understood in a very broad sense and include such topics as harmonic analysis, PDE, geometry, mathematical physics, microlocal analysis, time-frequency analysis, imaging and computations. Modern trends and novel applications in mathematics, natural sciences, medicine, scientific computing, and engineering are highlighted.

The Ramanujan-Petersson conjecture for Maass forms

“…The present section is a summary, brought down to a minimum, of developments made over a number of years [12,13,14,15,16]. Chapter 1 in [15] gives complete proofs of the facts recalled in this section and a summary, somewhat more detailed than the present one, is given in [16,Section 6.3…”

“…When χ = χ 0 , we set T χ 0 ,iλ = 1 2 E iλ : as a distribution-valued function of iλ, it extends as an analytic function E ν for ν = ±1, admitting when Re ν < 0 and ν = −1 the Fourier expansion [15,Theor.1.1.7] such that…”

A unified scheme of approach to the Ramanujan conjecture

“…This shows that L-functions can be defined in a spectral-theoretic role. If one develops in the first case the symbolic calculus of modular distributions, the same is true [12] or [13, section 6.4] for a variety of composite objets generalizing L-functions; we have not, or not yet, pushed anaplectic pseudodifferential analysis in the modular case to a comparable extent.…”

“…There are sharp composition formulas (the composition of symbols corresponding to the composition of operators), by which we mean the following: given two symbols h 1 and h 2 , give the explicit decomposition into homogeneous components (metaplectic case) of into isotypic components (anaplectic case) of the symbol h 1 ♯ h 2 : this is to be found in [10,Section 1.2] in the first case, and [9, Section 3.4] in the second. In the automorphic situation, only the metaplectic case has been completed ( [12] or [13,Section 6.4] for a summary). This might be of interest to arithmeticians, as it gives a new way to approach multilinear operations on L-functions of non-holomorphic modular forms, such as convolution L-functions or triple products: these enter the sharp composition formula of automorphic symbols as coefficients.…”

“…The L-function L(s, f ) = n≥1 b n n −s can be defined in spectraltheoretic terms: the decomposition of S m χ into isotypic components is given as an integral superposition of the distributions iso m, ν+m 2 with Re ν = 1, the coefficient being L( m−ν+2 2 , f ) up to a simple non-arithmetic factor. Note that (this was the policy chosen [12,Chap.1] in the non-holomorphic case) we might have started, in place of a cusp-form of Hecke type, from an arbitrary character χ of Q × : the function θ m S m χ would then be automorphic (and then, automatically, of Hecke type), if and only if the L-function so defined in spectral-theoretic terms did satisfy the required functional equation.…”