Zeta function and distributions. (Fonction zéta et distributions.) (French) Zbl 0226.12008

Weil, André

Sém. Bourbaki 1965/1966, Exp. No. 312, 9 p. (1966).

The author gives a new interpretation of the Tate-Iwasawa proof for the functional equation of the zeta functions of Riemann, Dedekind and Hecke. Let \(X\) be a vector space (of finite dimension) over a (commutative) local field \(X\) or the ring of adeles of an \(A\)-field \(K\) and \(G\) a group acting on \(X\); if \(\mathcal S\) is the space of Schwartz-Bruhat functions on \(X\), then \(G\) acts on \(\mathcal S\) through the maps \(f\mapsto f^\gamma\) where \(f\in\mathcal S\), \(\gamma\in G\), \(f^\gamma(x) = f(\gamma x)\) for \(x\in X\) and further \(G\) acts by duality on the space of tempered distributions \(\Delta\) on \(X\) through the maps \(\Delta \rightarrow \gamma\Delta\) where \((\gamma\Delta(f) = \Delta(f^\gamma)\). Let \(\omega\) be a 1-dimensional representation of \(G\).
The author poses the problem \(P(X,G,\omega)\) of determining all tempered distributions \(\Delta\) such that \((\gamma\Delta = \omega(\gamma)^{-1}\Delta\). If \(X = k\), \(G= k^\times = k - \{0\}\), the author shows that for \(\omega \ne \omega_0\) (the trivial character), the problem has up to a constant scalar factor, a unique solution for \(k\ne \mathbb R, \mathbb C\) and he also determines the solutions for \(\omega = \omega_0\); for \(k = \mathbb R, \mathbb C\), such “uniqueness” holds but for countably many \(\omega\).
Let, for complex \(s\), \(\omega = \omega_s\), be defined by \(\omega_s(x) = \vert x\vert_k^s\), for \(x\in k\) (with \(\vert x\vert_k = \) normalised value of \(x\) in \(k)\); then \(\Delta_{\omega_s}\) defined (for \(f\in\mathcal S\)) by
\[ \Delta_{\omega_s}(f) = \int_{k^\times} f(x) \omega_s(x) \,d^\times x \]
gives a solution for the problem for \(\operatorname{Re} s>0\) (where \(d^\times x\) is a normalised measure on \(k^\times\)).
To obtain the solutions for \(\operatorname{Re} s \le 0\), the author sketches three methods of which the first is that of Fourier transforms: indeed, if \(\Delta_\omega\) is a solution for \(P(k,k^\times,\omega)\), the Fourier transform \(\Delta_\omega^*\) of \(\Delta_\omega\) is a solution for \(P(k,k^\times,\omega_1\omega^{-1})\). By the “uniqueness” mentioned above, \(\Delta_{\omega_1\omega^{-1}}\) and \((\Delta_\omega)^*\) differ by a scalar factor, say \(\gamma_k(\omega)\) which is determined later (for every local field \(k)\).
The author then takes up the corresponding “global” problem, namely \(P(K_{\mathbb A},K_{\mathbb A}^\times,\omega)\) where \(K_{\mathbb A}\) is the ring of \(K\)-adeles, \(K_{\mathbb A}^\times\) the group of \(K\)-ideles, \(K^\times = K - \{0\}\) and \(\omega\) is a character of \(K_{\mathbb A}^\times/K^\times\).
The problem now is to define \(\Delta_\omega\) analogously and get its analytic continuation (as a function of \(\omega)\); this is done by the Tate-Iwasawa method (which involves, inter alia, the use of Poisson summation formula for Schwartz-Bruhat functions on \(K_{\mathbb A}\)); the factor corresponding to \(\gamma_k(\omega)\) above turns out to be 1 here. Now \(\Delta_\omega\) can be expressed as a product of corresponding \(\Delta_{\omega_s}\) for the completions \(K_v\) of \(K\). Taking \(\omega(x)=\omega_s(x) = \vert x\vert_A^s\) (where \(\vert x\vert_A\) is the “module” of \(x\) in \(K_{\mathbb A}/K\) and comparing \(\Delta_\omega^*\), \(\Delta_{\omega_1\omega^{-1}}\) give the functional equation of the Dedekind zeta function of \(K\).
The author then discusses generalisations of the problem above to simple algebras over \(K\); some of these problems have been solved since by Stein and Cartier. As far as \(P(k,k^\times,\omega)\) with \(k = \mathbb R\) or \(\mathbb C\) is concerned, the results above are referred to as “Tate’s lemma” by the author on p. 91 of his recent book [Dirichlet series and automorphic forms. Berlin etc.: Springer Verlag (1971; Zbl 0218.10046)] and used in connection with certain “local functional equations”.
[For the entire collection see Zbl 0192.31105.]

Reviewer: S. Raghavan

MSC:

11R42

Zeta functions and \(L\)-functions of number fields

Keywords:

Tate-Iwasawa proof; functional equation; zeta function

Citations:

Zbl 0192.31105; Zbl 0218.10046