| We demonstrate a sufficient condition, in terms of the values taken by certain Hilbert modular forms, for the mod p local Galois representation associated to a Hilbert-Blumenthal abelian variety to be finite. More precisely, we define a discriminantal set of Hilbert modular forms to be a set $/[f/sb1,/...,f/sb[r]/]$ of forms whose q-expansions satisfy a certain linear independence condition. We then show that if $/[f/sb1,/...,f/sb[r]/]$ is a discriminantal set of modular forms, X is an HBAV, and the value of $f/sb[i]$ at X has order divisible by p for every i, then the local mod p Galois representation associated to X is finite. We also prove that discriminantal sets exist for every choice of real multiplication ring. As an application, we show that a solution in non-zero coprime integers to the equation $A/sp4 + B/sp2 = C/sp[p]$, with A even, B odd, and $13 < p /equiv 1(4)$, would give rise to a non-modular Hilbert-Blumenthal abelian variety. |