no rational point. The failure of the Hasse principle for curves of genus 1 is measured by the Tate–Shafarevich group. If is a curve of genus 1 with a -rational point , then is called an elliptic curve over . In this case, has the structure of a commutative algebraic group (with as the zero element), and so the set of -rational points is an abelian group. The Mordell–Weil theorem says that for an elliptic curve (or, more generally, an abelian variety) over a number field , the abelian group is finitely generated. Computer algebra programs can determine the Mordell–Weil group in many examples, but it is not known whether there is an algorithm that always succeeds in computing this group. That would follow from the conjecture that the Tate–Shafarevich group is finite, or from the related Birch–Swinnerton-Dyer conjecture. Genus at least 2 Faltings's theorem (formerly the Mordell conjecture) says that for any curve of genus at least 2 over a number field , the set is finite. Some of the great achievements of number theory amount to determining the rational points on particular curves. For example, Fermat's Last Theorem (proved by Richard Taylor and Andrew Wiles) is equivalent to the statement that for an integer at least 3, the only rational points of the curve in over are the obvious ones: and ; and for even; and for odd. The curve (like any smooth curve of degree in ) has genus It is not known whether there is an algorithm to find all the rational points on an arbitrary curve of genus at least 2 over a number field. There is an algorithm that works in some cases. Its termination in general would follow from the conjectures that the Tate–Shafarevich group of an abelian variety over a number field is finite and that the Brauer–Manin obstruction is the only obstruction to the Hasse principle, in the case of curves. Higher dimensions Varieties with few rational points In higher dimensions, one unifying goal is the Bombieri–Lang conjecture that, for any variety of general type over a number field , the set of -rational points of is not Zariski dense in . (That is, the -rational points are contained in a finite union of lower-dimensional subvarieties of .) In dimension 1, this is exactly Faltings's theorem, since a curve is of general type if and only if it has genus at