Linking numbers and the tame Fontaine–Mazur conjecture

Abstract

Let \(p\) be an odd prime, let \(S\) be a finite set of primes \(q\equiv 1\ \mathrm{mod}\ p\) but \(q\not \equiv 1\ \mathrm{mod}\ p^2\) and let \(G_S\) be the Galois group of the maximal \(p\)-extension of \({\mathbb {Q}}\) unramified outside of \(S\). If \(\rho \) is a continuous homomorphism of \(G_S\) into \(\mathrm{GL}_2({\mathbb {Z}}_p)\), then under certain conditions on the linking numbers of \(S\) we show that \(\rho =1\) if \(\overline{\rho }=1\). We also show that \(\overline{\rho }=1\) if \(\rho \) can be put in triangular form mod \(p^3\).