Mishiba's Conjecture on the Coaction of $\infty$-adic Multiple Zeta Values

Abstract: We study multiple zeta values in positive characteristic. We construct, using $\infty$-adic multiple zeta values, a coaction of the $\infty$-adic multiple zeta value algebra modulo $ζ_A(q-1)$ on the $\infty$-adic multiple zeta value algebra itself, and prove that it agrees with Mishiba's coaction constructed via special values of Carlitz multiple polylogarithms. We also determine the coaction on the $\star$-inverse values and show that the antipode on the $\infty$-adic multiple zeta value algebra modulo $ζ_A(q-1)$ is given by the $\star$-inverse operation. In particular, these results prove Mishiba's conjecture concerning the coaction and the antipode for $\infty$-adic multiple zeta values.