  In 1922, Mordell conjectured the striking statement that for a polynomial
equation $f(x,y)=0$, if the topology of the set of complex number solutions is
complicated enough, then the set of rational number solutions is finite. This
was proved by Faltings in 1983, and again by a different method by Vojta in
1991, but neither proof provided a way to provably find all the rational
solutions, so the search for other proofs has continued. Recently, Lawrence and
Venkatesh found a third proof, relying on variation in families of $p$-adic
Galois representations; this is the subject of the present exposition.
