(I'm a number theorist, and this is very close to my research area.) The Gross-Zagier formula (along with work of Kolyvagin) from the 1980s proved the Birch an

(I'm a number theorist, and this is very close to my research area.) The Gross-Zagier formula (along with work of Kolyvagin) from the 1980s proved the Birch and Swinnerton-Dyer conjecture when "r_an <= 1"; this conjecture is an amazing link between analysis and arithmetic, and also one of the Clay Problems. This number "r_an" is the order of vanishing of a certain "generating function" that counts the number of solution to y^2 = x^3 + Ax + B modulo all prime numbers. For about 30 years now, people have been completely stumped at coming up with even a wishy-washy conjectural half guess* as to what to do when r_an is 2 or larger. In number theory a standard approach is to replace the integers Z = {... -2, -1, 0, 1, 2, ...} with a polynomial ring F_p[X] over finite field, and try to solve analogous problems (this is called working "over a functional field"). The paper that this article is about has come up with exactly what to do in case r_an is 2 or larger in the function field setting. So far they haven't figured out how to do anything over the rational numbers yet. Sometimes solving a problem over function fields provides incredible and valuable insight into how to solve the analogous problem over the rational numbers, and sometimes it doesn't (e.g., the function field analogue of Fermant's Last Theorem and the ABC conjecture are both basically trivial to deal with, whereas the same problem over the rational numbers is ridiculously hard).