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

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.


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


Zbl 0991.11023
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