p-adic multiple zeta values at roots of unity and p-adic pro-unipotent harmonic actions - IV-1 : p-adic multiple zeta values at roots of unity extended to sequences of integers of any sign · arXivDesk
1712.09976Dec 28, 201718 pages
p-adic multiple zeta values at roots of unity and p-adic pro-unipotent harmonic actions - IV-1 : p-adic multiple zeta values at roots of unity extended to sequences of integers of any sign
This work is a study of p-adic multiple zeta values at roots of unity (pMZVμN's), the p-adic periods of the crystalline pro-unipotent fundamental groupoid of (P1−{0,μN,∞})/Fq
Nearby in the stack
. The main tool is new objects which we call
p
-adic pro-unipotent harmonic actions. In this part IV we define and study
p
-adic analogues of some elementary complex analytic functions which interpolate multiple zeta values at roots of unity such as the multiple zeta functions. The indices of
p
MZV
μN
's involve sequences of positive integers ; in this IV-1, by considering an operation which we call localization (inverting certain integration operators) in the pro-unipotent fundamental groupoid of
P1−{0,μN,∞}
, and by using
p
-adic pro-unipotent harmonic actions, we extend the definition of
p
MZV
μN
's to indices for which these integers can be negative, and we study these generalized