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

$\begin{pmatrix} \chi_1 &0 \\ 0& \chi_2 \end{pmatrix}$

fits into a family of representations

$\begin{pmatrix} a(g) &b(g) \\ c(g)& d(g) \end{pmatrix}$

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:

$\begin{pmatrix} \chi_1 &\ast \\ 0& \chi_2 \end{pmatrix}$.

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.