Final Report Summary - FANTAST (Frontiers of Analytic Number Theory And Selected Topics)
Diophantine equations are systems of polynomial equations with rational coefficients. Even very simple equations can mask a rich arithmetic structure that can be very difficult to penetrate. Nonetheless a suitable blend of analytic and algebro-geometric tools can sometimes be used to resolve deep questions about whether or not the particular system of Diophantine equations is soluble in rational numbers and, if it is, how the rational solutions are distributed. Passing to the language of Diophantine geometry, my project has established decidability criteria (the Hasse principle and weak approximation) for several significant families of algebraic varieties. These include:
(i) a family of conic bundle surfaces with arbitrarily many singular fibres (using additive combinatorics),
(ii) the family of all smooth and projective varieties whose dimension is sufficiently large in terms of the degree (using the circle method),
(iii) the family of all smooth cubic hypersurfaces of dimension at least 8 over any global field (using a variant of the circle method especially developed for global fields).
(i) a family of conic bundle surfaces with arbitrarily many singular fibres (using additive combinatorics),
(ii) the family of all smooth and projective varieties whose dimension is sufficiently large in terms of the degree (using the circle method),
(iii) the family of all smooth cubic hypersurfaces of dimension at least 8 over any global field (using a variant of the circle method especially developed for global fields).