Absolute zeta functions and the automorphy.

*(English)*Zbl 1390.14067The paper investigates the zeta function \(\zeta_f(s)\) associated to a particular type of “absolute automorphic form” \(f: \mathbb R_+\to \mathbb C\) that satisfies a functional equation
\[
f(x^{-1}) = Cx^{-D}f(x)\qquad D\in\mathbb Z,\quad C=\pm 1.
\]
This study takes the inspiration from the results and the techniques developed in [A. Connes and C. Consani, Compos. Math. 146, No. 6, 1383–1415 (2010; Zbl 1201.14001)] where the equation
\[
-\frac{\partial_s\zeta_N(s)}{\zeta_N(s)}=\int_1^\infty N(x)\, x^{-s}\frac{dx}x\tag{1}
\]
defines, up-to a multiplication by a non-zero scalar, an integral formula for the Hasse-Weil zeta function \(\zeta_N(s)\) associated, in the limiting case \(q\to 1\), to the counting function \(f(x)=N(x)\) defined for \(x\in [1,\infty)\).

The integration in \(s\) of both sides of (1), produces a divergence, at \(x=1\), of the term \(\frac{x^{-s-1}}{\log x}\) on the right hand-side, thus one has to choose a principal value in \[ \log(\zeta_N(s)) = \int_1^\infty \frac{N(x)x^{-s-1}}{\log x}\, dx+c \] near \(x=1\) (the term \(\frac{N(x)}{\log(x)}\) is singular). Here, the authors use the technique that implements a new variable \(w\) to define the principal value as \[ \int_1^\infty \frac{N(x)x^{-s-1}}{\log x}dx :=\frac{\partial}{\partial w}~\frac{1}{\Gamma(w)}\int_1^\infty \frac{N(x)x^{-s-1}}{(\log x)^{1-w}}\, dx\bigg|_{w=0}. \] Thus they obtain, after exponentiation, the formula \[ \zeta_N(s) = \exp\left(\frac{\partial}{\partial w}~\frac{1}{\Gamma(w)}\int_1^\infty \frac{N(x)x^{-s-1}}{(\log x)^{1-w}}\, dx\bigg|_{w=0}\right)\tag{2} \] which fully determines the Hasse-Weil zeta function \(\zeta_N(s)\) associated to \(N\). Notice that the equation (1) of [loc. cit.] defines \(\zeta_N(s)\) only up to a multiplication by a non-zero scalar.

The point of view of the authors of the paper under review is that equation (2) can be applied to define the so called “absolute zeta function” \[ \zeta_f(s):=\exp\left(\frac{\partial}{\partial w}~Z_f(w,s)\bigg|_{w=0}\right) \] associated to an “absolute automorphic form” \(f(x)\), where \[ Z_f(w,s) := \frac{1}{\Gamma(w)}\int_1^\infty \frac{f(x)x^{-s-1}}{(\log x)^{1-w}}\, ~dx. \] In Theorem 1, the authors show that when \(f(x)\) is cyclotomic i.e. \(f(x)\) is rationally described in terms of cyclotomic polynomials, then \(\zeta_f(s)\) is a meromorphic function that can be written in terms of multiple gamma functions and has a functional equation. In particular, \(\zeta_f(s)\) is a rational function if and only if \(f(x)\in\mathbb Z[x]\) and in this case the functional equation is of the form \[ \zeta_f(D-s)^C = (-1)^{f(1)}\zeta_f(s). \] Examples of “cyclotomic absolute automorphic forms” are the rational functions which provide the counting of points defined over finite fields extensions, for affine and projective spaces, Grassmannians and for general, special and symplectic groups.

An interesting theme developed in this paper deals with the absolute zeta function \(\zeta_\rho(s)= \frac{\det(s-D_{\rho_-})}{\det(s-D_{\rho_+})}\) associated to a virtual complex representation \(\rho=(\rho_+,\rho_-)\) of \(\mathbb R_+\), where \[ D_{\rho_\pm} = \lim_{x\to 1}\frac{\rho_\pm(x)-\rho_\pm(1)}{x-1}. \] Let \(f(x) = \text{tr}(\rho_+(x))-\text{tr}(\rho_-(x))\). Then in Theorem 2 the authors show that \(\zeta_\rho(s) = \zeta_f(s)\). If \(\rho\) is self-dual unitary, a suitable analogue of the Riemann Hypothesis is also formulated.

The integration in \(s\) of both sides of (1), produces a divergence, at \(x=1\), of the term \(\frac{x^{-s-1}}{\log x}\) on the right hand-side, thus one has to choose a principal value in \[ \log(\zeta_N(s)) = \int_1^\infty \frac{N(x)x^{-s-1}}{\log x}\, dx+c \] near \(x=1\) (the term \(\frac{N(x)}{\log(x)}\) is singular). Here, the authors use the technique that implements a new variable \(w\) to define the principal value as \[ \int_1^\infty \frac{N(x)x^{-s-1}}{\log x}dx :=\frac{\partial}{\partial w}~\frac{1}{\Gamma(w)}\int_1^\infty \frac{N(x)x^{-s-1}}{(\log x)^{1-w}}\, dx\bigg|_{w=0}. \] Thus they obtain, after exponentiation, the formula \[ \zeta_N(s) = \exp\left(\frac{\partial}{\partial w}~\frac{1}{\Gamma(w)}\int_1^\infty \frac{N(x)x^{-s-1}}{(\log x)^{1-w}}\, dx\bigg|_{w=0}\right)\tag{2} \] which fully determines the Hasse-Weil zeta function \(\zeta_N(s)\) associated to \(N\). Notice that the equation (1) of [loc. cit.] defines \(\zeta_N(s)\) only up to a multiplication by a non-zero scalar.

The point of view of the authors of the paper under review is that equation (2) can be applied to define the so called “absolute zeta function” \[ \zeta_f(s):=\exp\left(\frac{\partial}{\partial w}~Z_f(w,s)\bigg|_{w=0}\right) \] associated to an “absolute automorphic form” \(f(x)\), where \[ Z_f(w,s) := \frac{1}{\Gamma(w)}\int_1^\infty \frac{f(x)x^{-s-1}}{(\log x)^{1-w}}\, ~dx. \] In Theorem 1, the authors show that when \(f(x)\) is cyclotomic i.e. \(f(x)\) is rationally described in terms of cyclotomic polynomials, then \(\zeta_f(s)\) is a meromorphic function that can be written in terms of multiple gamma functions and has a functional equation. In particular, \(\zeta_f(s)\) is a rational function if and only if \(f(x)\in\mathbb Z[x]\) and in this case the functional equation is of the form \[ \zeta_f(D-s)^C = (-1)^{f(1)}\zeta_f(s). \] Examples of “cyclotomic absolute automorphic forms” are the rational functions which provide the counting of points defined over finite fields extensions, for affine and projective spaces, Grassmannians and for general, special and symplectic groups.

An interesting theme developed in this paper deals with the absolute zeta function \(\zeta_\rho(s)= \frac{\det(s-D_{\rho_-})}{\det(s-D_{\rho_+})}\) associated to a virtual complex representation \(\rho=(\rho_+,\rho_-)\) of \(\mathbb R_+\), where \[ D_{\rho_\pm} = \lim_{x\to 1}\frac{\rho_\pm(x)-\rho_\pm(1)}{x-1}. \] Let \(f(x) = \text{tr}(\rho_+(x))-\text{tr}(\rho_-(x))\). Then in Theorem 2 the authors show that \(\zeta_\rho(s) = \zeta_f(s)\). If \(\rho\) is self-dual unitary, a suitable analogue of the Riemann Hypothesis is also formulated.

Reviewer: Caterina Consani (Baltimore)