We introduce intersection theory on arithmetic surfaces and use it to prove a Shioda-Tate formula and reformulate the Birch and Swinnerton-Dyer conjecture for elliptic curves in terms of elliptic surfaces.

We prove that the points dividing the lemniscate "with three leaves" in parts of equal length are algebraic, generalising results of Gauss and Abel. We also study further related questions on algebraicity of division points and transcendence of length of a class of curves containing polynomial lemniscates. A software for drawing these curves is available here.

We explain a proof, due to the Chudnovsky brothers, of the fact that two rational elliptic curves with the same number of points modulo every prime are isogenous.