From cryptography to the proof of Fermat's Last Theorem, elliptic curves (those curves of the form y^2 = x^3 + ax+b) are ubiquitous in modern number theory. In particular, much activity is focused on developing techniques to discover rational points on these curves. It turns out that finding a rational point on an elliptic curve is very much like finding the proverbial needle in the haystack -- in fact, there is currently no algorithm known to completely determine the group of rational points on an arbitrary elliptic curve.
I'll introduce the ''real'' picture of elliptic curves and discuss why the ambient real points of these curves seem to tell us little about finding rational points. I'll summarize some of the story of elliptic curves over finite and p-adic fields and tell you about how I study integral points on (hyper)elliptic curves via p-adic integration, which relies on doing a bit of p-adic linear algebra. Time permitting, I'll also give a short demo of some code we have to carry out these algorithms in the Sage Math Cloud.