Let $p$ be a prime number. Two integers are said to be congruent modulo $p$ if their difference is divisible by $p$. One may think of such integers as $p$-adically close to each other. Likewise, two elliptic curves are congruent modulo $p$ if they are $p$-adically close to each other from a Galois theoretic perspective. One may anticipate that congruent elliptic curves have a lot in common. The talk shall serve as a gentle introduction to investigations motivated by the Birch and Swinnerton-Dyer conjecture. It is based on joint work with R Sujatha.