This is the last of three talks on the non-existence of certain Galois extensions of the rational number field Q with prescribed ramification. In this talk, I will discuss work of Hyunsuk Moon and Yuichiro Taguchi refining Tate's discriminant bound, and giving non-existence theorems for mod p Galois representations. In particular, I will discuss Moon and Taguchi's refinement of Tate's discriminant bound to get the case when p<=31 with small Serre weight k.