Desingularization of complex multiple zeta-functions and fundamentals of p-adic multiple L-functions

The talk is based on my current joint work with Y.Komori, K.Matsumoto and H.Tsumura.

In the first half of my talk, I will introduce our method of desingularization of multiple zeta-functions. I will explain that multiple zeta-functions (which are known to be meromorphic with infinitely many singular loci in the whole space) turn to be entire by our integral method. I will show various properties of our methods, particularly I will reveal that our method is essentially taking a finite linear combination of multiple zeta-functions.

Based on them, in the second half, I will explain our construction of (several variable) p-adic multiple L-functions which generalizes that of Kubota-Leopoldt's p-adic L-functions. Then I will show their various fundamental properties including multiple Kummer congruences.