On the Fontaine-Mazur Conjecture for number fields and an analogue for function fields

A conjecture of Jean-Marc Fontaine and Barry Mazur says that given any number field k, there are no infinite unramified {dollar}ell{dollar}-adic analytic {dollar}ell{dollar}-extensions of k. (This conjecture comes from two sources; one is a broad conjecture of Fontaine and Mazur about irreducible {dollar}ell{dollar}-adic representations of absolute Galois groups. The other source is the Golod-Shafarevich proof that infinite class field towers exist.) It seems reasonable and straightforward to pose the same conjecture in the function field case. However, we immediately see that the naive version must be false, because the extension of a function field k over a finite field F given by constant field {dollar}ell{dollar}-extensions of F is already infinite and {dollar}ell{dollar}-adic analytic. The statement of the conjecture that I suspect to be correct is as follows:; Conjecture 1. Let k be a function field over a finite field F of characteristic p and order q, and {dollar}ell{dollar} a prime not equal to p. Let K be obtained from k by taking the maximal {dollar}ell{dollar}-extension of the constant field. If M is an unramified {dollar}ell{dollar}-adic analytic {dollar}ell{dollar}-extension of k, and M does not contain K, then M is a finite extension of k.; I believe this conjecture is of interest not only independently, but because the methods I use to address it, which involve basic facts about abelian varieties over finite fields, can be adapted to number fields through the use of Iwasawa {dollar}{lcub}bf Z{rcub}sb{lcub}ell{rcub}{dollar}-extensions and {dollar}ell{dollar}-adic L-functions. The methods I use, which stem from work of Nigel Boston's using group automorphisms, also introduce {dollar}ell{dollar}-adic Lie algebras in a crucial way, relating them to standard constructions of arithmetic algebraic geometry, such as the Tate module. As progress towards the conjecture I offer a number of original theorems proving it in special cases, as well as a theorem on some special cases of the original conjecture for number fields, and some related results.