Rational points on smooth cubic hypersurfaces

• Main Author: Papanikolopoulos, Stafanos
• Published: University of Warwick 2014
• Subjects: 510; QA Mathematics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.647958

Let S be a smooth n-dimensional cubic variety over a field K and suppose that K is finitely generated over its prime subfield. It is a well-known fact that whenever we have a set of K-points on S, we may obtain new ones, using secant and tangent constructions. A Mordell-Weil generating set B ⊆ S(K) is a subset of minimal cardinality that generates S(K) via these operations; we define the Mordell-Weil rank as r(S,K) = #B. The Mordell-Weil theorem asserts that in the case of an elliptic curve E defined over a number field K, we have that r(E,K) < 1. Manin [11] asked whether this is true or not for surfaces. Our goal is to settle this question for higher dimensions, and for as many fields as possible. We prove that when the dimension of the cubic hypersurface is big enough, if a point can generate another point, then it can generate all the points in the hypersurface that lie in its tangent plane. This gives us a powerful tool, yet a simple one, for generating sets of points starting with a single one. Furthermore, we use this result to prove that if K is a finite field and the dimension of the hypersurface is at least 5, then r(S,K) = 1. On the other hand, it is natural to ask whether r(S,K) can be bounded by a constant, depending only on the dimension of S. It is conjectured that such a constant does not exist for the elliptic curves (the unboundedness of ranks conjecture for elliptic curves). In the case of cubic surfaces, Siksek [16] has proven that such a constant does not exist when K = Q. Our goal is to generalise this for cubic threefolds. This is achieved via an abelian group HS(K), which holds enough information about the Mordell-Weil rank r(S,K) in the following manner; if HS(K) becomes large, so does r(S,K). Then, by using a family of cubic surfaces that is known to have an unbounded number of Mordell-Weil generators over Q, we prove that the number of Mordell-Weil generators is unbounded in the case of threefolds too.