(*) Provided a "point at infinity" is added, any straight line drawn through two points on one of these objects must pass through a third point on it.
A paper subtitled "Wild 3-adic exercises" by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor proved a landmark theorem involving a feature of these objects. The problem of determining whether a square-free positive integer is a congruent number is equivalent to determining whether one of these mathematical objects has a positive rank. If one of these objects is defined over the rationals, it is necessarily a finitely generated Abelian group according to Mordell's theorem. They can be used to perform prime factorization in the Lenstra method, and to test whether a number is prime using the Atkin-Morain algorithm. (*) Provided a "point at infinity" is added, any straight line drawn through two points on one of these objects must pass through a third point on it. The modularity theorem that relates these objects over the field of rationals to modular forms used to be known as the Taniyama-Shimura conjecture. These objects are described by the equation y squared equals x cubed plus A x plus B. For 10 points, name these curves that were famously used by Andrew Wiles to prove Fermat's Last Theorem.
A paper subtitled "Wild 3-adic exercises" by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor proved a landmark theorem involving a feature of these objects.
elliptic curves ["curve" unnecessary after spoken, prompt on "modular forms" in the first sentence]