Let k be the rational function field over a finite field and fix a finite place v and an infinite place of k. In this talk, I give a simultaneous vanishing principle for the v-adic Carlitz multiple polylogarithms (abbreviated as CMPLs) at algebraic points. This principle establishes the fact that the v-adic vanishing of CMPLs at algebraic points is equivalent to its infinite-adic counterpart being Eulerian. Here an infinite-adic convergence of a CMPL is called Eulerian if the ratio of this value over the w-th power of the fundamental period of the Carlitz module lies in k,where w is the weight of the CMPL. This reveals a nontrivial connection between the v-adic and infinite-adic worlds in positive characteristic.

I also try to explain a relation to v-adic multizeta values. This is a joint work with Chieh-Yu Chang.