A current trend in number theory is understanding how modular forms (or more generally automorphic representations) can be put into $p$-adic families, as well as their associated galois representations. One application of this is as follows:
Suppose you know that a certain semisimple reducible representation
\chi_1 &0 \\
fits into a family of representations
a(g) &b(g) \\
which is generically irreducible (so the reducible representation is somehow a degeneration of the family). Then one can sometimes use the existence of the function $b$ (or $c$) to "deform" the reducible representation into a nontrivial extension:
\chi_1 &\ast \\
While this may or may not impress you, this is exactly the technique used by Wiles in his proof of the Iwasawa Main Conjecture for totally real fields.