## 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.

### 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

Zbl 0991.11023
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