The goal is legitimization of term-wise differentiation of L spectral expansions, so that computations producing a classical outcome are correct. We are fond of L expansions because they are what Plancherel gives. Typically, L expansions are not continuous, much less differentiable, so the issue cannot be proving classical differentiability, which does not hold. To say that L spectral expansion...

For recent developments of this work in the classical direction, especially to generalizing to modular groups acting on higher dimensional spaces, see papers of Min Ho Lee: http://www.math.uni.edu/ lee/pub.html. He has, for example, developed the Hilbert modular case. Also, Olav Richter’s work on Rankin-Cohen brackets: http://www.math.unt.edu/ richter/. Work of Conley on 1/2-integral weight: ht...

Understanding the implications of SL(2,Z) S-duality on the hypermultiplet moduli space of type II string theories has led to much progress recently in uncovering D-instanton contributions. In this note, we suggest that the extended duality group SL(3,Z), which includes both S-duality and Ehlers symmetry, potentially determines the contributions of NS5-branes. We support this claim by automorphi...

Many computations in the harmonic analysis of automorphic forms, especially concerning Eisenstein series, or, even worse, trace formulas, expose one to the danger that a naive formal approach leads to incorrect manipulation of expressions whose convergence is fragile or even volatile. At the same time it is often clear that some improved formalism can be correct, and may be more intelligible th...

We present a 1-loop toroidal membrane winding sum reproducing the conjectured M-theory, four-graviton, eight derivative, R amplitude. The U -duality and toroidal membrane world-volume modular groups appear as a Howe dual pair in a larger, exceptional, group. A detailed analysis is carried out for M-theory compactified on a 3-torus, where the target-space Sl(3,Z)× Sl(2,Z) U -duality and Sl(3,Z) ...

It is proved that every endomorphism preserving the automorphic orbit of a nontrivial element of the rank two polynomial algebra over the complex number field is an automorphism.

We state sufficient conditions for the existence of positive bounded, almost automorphic or almost periodic solutions of the following nonlinear infinite delay integral equation: