Values at $T$-tuples of negative integers of twisted multivariable zeta series associated to polynomials of several variables

Abstract

We consider twisted multivariable zeta series associated to polynomials of several variables. We introduce a new class of polynomials, namely HDF, that contains strictly non-degenerate and hypoelliptic polynomials. For polynomials belonging to the HDF class, we show that we can extend holomorphically our series to $\mathbb{C}^T$. Then, thanks to a new principle called ‘the Exchange Lemma’, we give very simple formulae for the values of our series at $T$-tuples of negative integers. Finally, we make the $p$-adic interpolation of those values. Thus, we have generalized the results of Cassou-Noguès (that she used to construct the $p$-adic $L$-functions of totally real fields) in two ways: we consider multivariable series and our series are associated to more general polynomials. In addition, our proof is completely different.