Pseudo-mesures p-adiques associées aux fonctions L de \({\mathbb{Q}}\). (p- adic pseudo-measures associated to L-functions of \({\mathbb{Q}})\) (Q1080473)

Let \(\chi =\phi \psi\) be a Dirichlet character, where \(\phi\) (resp. \(\psi)\) is of prime to p order (resp. of p-power order). We improve the result given by the author [Invent. Math. 86, 1-17 (1986; Zbl 0571.12008)] when it is not optimal (i.e. if and only if \(\omega^{-1} \phi (p)=1\) if \(p\neq 2\) (resp. \(\phi (2)=1)\); for instance this occours when \(\omega^{-1} \chi (p)=1\), the case where \(L_ p(\chi,s)\) has a trivial zero at \(s=0)\). In this case we obtain the corresponding standard lower bound C for the valuation of \(L_ p(\chi,s)\). When \(\phi =1\), we give an explicit criterium for the condition: \(v(L_ p(\chi,s))=C\), for all \(s\in {\mathbb{Z}}_ p.\) These results give a complete analytic genera theory for \(L_ p\)- functions for the base field \({\mathbb{Q}}\). They use Stickelberger's distributions. The end of the paper is concerned with an explicit general congruence for \(L_ p(\chi,t)-L_ p(\chi,s)\), \(s,t\in {\mathbb{Z}}_ p\), and we show how to find again, in a systematic way, all known congruences like recent ones of Kaplan-Williams, Desnoux, Lang, Hikita (all concerning quadratic characters and \(t=1\), \(s=0)\), and to give many other results and explicit examples.