Obstructions to rational and integral points

Abstract

In this thesis, I study two examples of obstructions to rational and integral points on varieties. The first concerns the S-unit equation, which asks for solutions to x + y = 1 with x and y both S-units, or units in Z = Z[1/S]. This is equivalent to finding the set of Z-points of P1 \ {0, 1, [infinity]}. We follow work of Dan-Cohen-Wewers and Brown in applying a motivic version of the non-abelian Chabauty's method of Minhyong Kim to find polynomials in p-adic polylogarithms that vanish on this set of integral points. More specifically, we extend the computations already done by Dan-Cohen-Wewers to the integer ring Z[1/3], and we provide some significant simplifications to a previous algorithm of Dan-Cohen, especially in the case of Z[1/S]. One of the reasons for doing this is to verify cases of Kim's conjecture, which states that these p-adic functions precisely cut out the set of integral points. This is joint work with Ishai Dan-Cohen. The second is about obstructions to the local-global principle. The étale Brauer-Manin obstruction of Skorobogatov can be used to explain the failure of the local-global principle for many algebraic varieties. In 2010, Poonen gave the first example of failure of the local-global principle that cannot be explained by the étale Brauer-Manin obstruction. Further obstructions such as the étale homotopy obstruction and the descent obstruction are unfortunately equivalent to the étale Brauer-Manin obstruction. However, Poonen's construction was not accompanied by a definition of a new, finer obstruction. Here, we present a possible definition for such an obstruction by applying the Brauer-Manin obstruction to each piece of every stratification of the variety. We prove that this obstruction is necessary and sufficient, over imaginary quadratic fields and totally real fields unconditionally, and over all number fields conditionally on the section conjecture. This is part of a joint project with Tomer Schlank.