Universal families of Eulerian multiple zeta values in positive characteristic

We study positive characteristic multiple zeta values associa- ted to general curves over Fq together with an Fq-rational point ∞ as introduced by Thakur. For the case of the projective line these values were defined as analogues of classical multiple zeta values. In the present paper we first establish a general non-commutative factorization of exponential series associated to certain lattices of rank one. Next we introduce universal families of multiple zeta values of Thakur and show that they are Eulerian in full generality. In particular, we prove a conjecture of Lara Rodríguez and Thakur in [28]. One of the main ingredients of the proofs is the notion of L-series in Tate algebras introduced by the third author [31] in 2012.