A motivic pairing and the Mellin transform in function fields (Q6634755)

For a real-valued function \(f(x)\), the Mellin transform of \(f\) is defined by \({\mathscr M}(f)(s)=\int_0^{\infty} f(x) x^{s-1}dx\), \(s\in{\mathbb C}\). A classical formula relates the Mellin transform of the exponential function to the Riemann zeta function \(\zeta(s)\) and the gamma function \(\Gamma(s)\): \({\mathscr M}\big(\frac 1{e^x-1}\big)(s)= \Gamma(s)\zeta(s)\).\N\NThe main result of this paper establishes an analogue of the above formula for global function fields. A special case is the following. Let \(A={\mathbb F}_q[\theta]\), \(K={\mathbb F}_q(\theta)\), \(K_{ \infty}={\mathbb F}_q((1/\theta))\), the completion of \(K\) at \(\infty\). Let \(C\) be the Carlitz module. Let \({\mathbb C}_{\infty}\) be the completion of an algebraic closure of \(K_{\infty}\). Set \(u=\frac {\tilde \pi}{\theta-t}\in {\mathbb C}_{\infty}(t)\), where \(\tilde \pi\) is the Carlitz period, and \(t\) is an independent variable.\N\NFor a specified element \({\mathbf z}\in{\mathbb C}_{\infty}\), the author defines a map \(\delta_{1,{\mathbf z}}^M\) from a \(t\)-motive \(M\) of dimension \(n\) to \({\mathbb C}_{\infty}\). Then, for \(n=1\), \({\mathbf z}=1\), we have \(\delta_{1,{\mathbf z}}^M\big(\frac u{ \exp_C(u)}\big)=\Gamma_A(n)\zeta_A(n)\in K_{\infty}\), where \(\exp_C\) is the Carlitz exponential, and \(\Gamma_A\) and \(\zeta_A\) are function field versions of the gamma and the Riemann zeta functions, respectively. The general results are Theorem 5.10 and Corollary 5.11.\N\NThe author defines two pairings: the exponential and the logarithm motivic pairings. He also gives an example showing how the results apply to Carlitz multiple zeta values, Example 5.15. In the last section, the pairings are applied to give a new log-algebraicity criterion.