A local Riemann hypothesis. I. (English) Zbl 0991.11022

Bump, Daniel; Choi, Kwok-Kwong; Kurlberg, Pär; Vaaler, Jeffrey

Math. Z. 233, No. 1, 1-19 (2000).

This is a joint review for Parts I and II (see below).
Consider the integral \(Z(f,\nu,s) = \int_{F^\times}f(x)\nu(x)|x|^{s} d^{\times}x\), where \(F\) is a local field, \(\nu\) a character of \(F^\times\) and \(f\) a Schwartz function on \(F\). It is a meromorphic function of the complex variable \(s\) and for suitable \(f\) it is a local \(L\)-function. The authors discovered a set of Schwartz functions for which all zeros of \(Z(f,\nu,s)\) lie on the line \(\text{Re}(s) = {1\over 2}\), if \(Z(f,\nu,s)\) is not identically zero. These Schwartz functions are the eigenfunctions of the Weil representation of the unitary group \(\text{U}(1) = \{z\in E^{\times}:\text{N}_{E/F} (z) = 1\}\), where \(E = F(\sqrt{-1})\), assuming that \(-1\) is not a square in \(F\). The case \(F = {\mathbb R}\) is treated in Part I, the case of a non-archimedean \(F\) with residual characteristic \(\neq 2\) in Part II. In the latter case there are more quadratic extensions of \(F\), for the other three the assertion about the zeros of \(Z(f,\nu,s)\) does not hold.
The proof in Part II relies on two facts: all eigencharacters of \(\text{U}(1)\) have multiplicity 1 and Fourier transform is in the image of the representation. The latter is true for \(E = F(\sqrt{-1})\) only. The corresponding split case is also treated.
In Part I an extension to the \(n\)-dimensional harmonic oscillator is also proved.

Reviewer: J.G.M.Mars (Utrecht)

MSC:

11F70	Representation-theoretic methods; automorphic representations over local and global fields
11S40	Zeta functions and \(L\)-functions
11M38	Zeta and \(L\)-functions in characteristic \(p\)
22E50	Representations of Lie and linear algebraic groups over local fields
11F27	Theta series; Weil representation; theta correspondences

Keywords:

zeta function; local field; Schwartz function; local \(L\)-function; \(n\)-dimensional harmonic oscillator

Citations:

Zbl 0991.11023