(I&#x27;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 &quot;r_an &lt;= 1&quot;; this conjecture is an amazing link between analysis and arithmetic, and also one of  the Clay Problems.  This number &quot;r_an&quot; is the order of vanishing of a certain &quot;generating function&quot; that counts the number of solution to y^2 = x^3 + A<i>x + 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 </i>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 &quot;over a functional field&quot;).  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&#x27;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&#x27;t (e.g., the function field analogue of Fermant&#x27;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).