There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Special Functions and Their Applications. By N. N. Lebedev, translated by R. A. Silverman. Pp. xii, 308. 96s. 1965. (Prentice-Hall)

An old question of Linnik asks about the equidistribution of integral points on a large sphere. This question proved to be very rich: it is intimately linked to modular forms, to subconvex estimates for L-functions, and to dynamics of torus actions on homogeneous spaces. Indeed, Linnik gave a partial answer using ergodic methods, and his question was completely answered by Duke using harmonic analysis and modular forms. We survey the context of these ideas and their developments over the last decades. Mathematics Subject Classification (2000).11F66; Secondary: 11F67, 11M41

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Equidistribution, L-functions and Ergodic Theory: on some problems of Y.V Linnik

We prove a formula relating the Fourier coefficients of a modular form of half-integral weight to the special values of L-functions. The form in question is an explicit theta lift from the multiplicative group of an indefinite quaternion algebra over Q. This formula has applications to proving the nonvanishingof this lift and to an explicit version of the Rallis inner product formula.

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On the Fourier coefficients of modular forms of half-integral weight

A conjecture on Whittaker-Fourier coefficients of cusp forms

Let G = GLn(K) where K is either R or C and let P = Pn(K) be the subgroup of matrices in GLn(K) consisting of matrices whose last row is (0, 0, . . . , 0, 1). Let π be an irreducible unitary representation of G. Gelfand and Neumark [Gel-Neu] proved that if K = C and π is in the Gelfand-Neumark series of irreducible unitary representations of G then the restriction of π to P remains irreducible. Kirillov [Kir] conjectured that this should be true for all irreducible unitary representations π of GLn(K), where K is R or C:

https://annals.math.princeton.edu/wp-content/uploads/annals-v158-n1-p04.pdf

A proof of Kirillov's conjecture

We prove a generalization to the totally real field case of the Waldspurger’s formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger’s formula as a combination of two ingredients – an equality between global distributions, and a dichotomy result for theta correspondence. As applications we generalize the Kohnen–Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindelof hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting.

Central Value Of Automorphic L-Functions

We prove a generalization to the totally real field case of the Waldspurger's formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger's formula in terms of equality between global distributions. As applications we generalize the Kohnen-Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindelof hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting.

Central value of automorphic $L-$functions

We prove certain identities between Bessel functions attached to irreducible unitary representations of PGL2(R) and Bessel functions attached to irreducible unitary representations of the double cover of SL2(R). These identities give a correspondence between such representations which turns out to be the Waldspurger correspondence. In the process we prove several regularity theorems for Bessel distributions which appear in the relative trace formula. In the heart of the proof lies a classical result of Weber and Hardy on a Fourier transform of classical Bessel functions. This paper constitutes the local (real) spectral theory of the relative trace formula for the Waldspurger correspondence for which the global part was developed by Jacquet. CONTENTS

Bessel identities in the waldspurger correspondence over the real numbers

We prove an explicit identity relating the central values of the twist L-functions of weight 0
Maass forms to the Fourier coefficients of weight ±1
2 Maass forms. We also give a bound on
Fourier coefficients of weight ±1
2 Maass forms.

Ehud Moshe Baruch

Papers

A proof of Kirillov's conjecture

Central Value Of Automorphic L-Functions

Central value of automorphic $L-$functions

Bessel identities in the waldspurger correspondence over the real numbers

A generalized Kohnen–Zagier formula for Maass forms