The recent proof of Serre’s conjecture allows one to prove the finiteness of certain types of Galois representations (2-dimensional mod p, odd, semisimple respresentations of the absolute Galois group of the rationals with bounded prime to p Artin conductor). One may ask for quantitative refinements of this. It seems hard to get a grip on this, although one can formulate a natural expectation.

This is in the spirit of the theorem of Hermite-Minskowski which asserts that there are only finitely many extensions of the rationals of bounded discriminant. One can ask for quantitative versions of this: a precise count is again unknown.