Linking Numbers and the Tame Fontaine-Mazur Conjecture

TL;DR

This paper investigates the conditions under which certain Galois representations are trivial, linking number properties to the tame Fontaine-Mazur conjecture, and providing criteria for triviality based on reduction mod p.

Contribution

It establishes new conditions involving linking numbers that guarantee the triviality of Galois representations in the context of the tame Fontaine-Mazur conjecture.

Findings

Galois representation r is trivial if its reduction mod p is 1 under certain linking number conditions.

Reduction of r mod p is 1 if r can be expressed in triangular form mod p^3.

Provides criteria connecting linking numbers to the triviality of Galois representations.

Abstract

Let p be an odd prime, let S be a finite set of primes q congruent to 1 mod p but not mod p^2 and let G_S be the Galois group of the maximal p-extension of Q un-ramified outside of S. If r is a continuous homomorphism of G_S into GL_2(Z_p) then under certain conditions on the linking numbers of S we show that r=1 if its reduction mod p is 1. We also show that the reduction of r mod p is 1 if r can be put in triangular form mod p^3.