On a Noncommutative Iwasawa Main Conjecture for Function Fields

We formulate and prove an analogue of the non-commutative Iwasawa Main Conjecture for $\mathrm{\ell }$-adic representations of the Galois group of a function field of characteristic $p$. We also prove a functional equation for the resulting non-commutative $L$-functions. As corollaries, we obtain non-commutative generalisations of the main conjecture for Picard-$1$-motives of Greither and Popescu and a main conjecture for abelian varieties over function fields in precise analogy to the ${\mathrm{GL}}_2$ main conjecture of Coates, Fukaya, Kato, Sujatha and Venjakob.