Arithmetic of Elliptic Curves and Ogg's Conjecture

Which rational numbers can be expressed as the area of a right-angle triangle whose sides all have rational length? This question sounds simple enough, but the answer has eluded number theorists for ages. In the 17th century, Pierre de Fermat made remarkable progress on this question by connecting it to what is now known as the arithmetic of elliptic curves. I'll discuss Fermat's ideas, elliptic curves, and a theorem of Barry Mazur from nearly fifty years ago on a conjecture of Andrew Ogg. I'll end by discussing recent progress on generalizations of Ogg's conjecture, including a result of mine with Ken Ribet. I won't assume any familiarity with number theory or algebraic geometry, and intend the talk to be accessible to graduate students in all areas of math.