In this lecture we lay the groundwork needed to prove the Hasse-Minkowski theorem for Q, which states that a quadratic form over Q represents 0 if and only if it represents 0 over every completion of Q (as proved by Minkowski). The statement still holds if Q is replaced by any number field (as proved by Hasse), but we will restrict our attention to Q. Unless otherwise indicated, we use p throug...

A partially ordered set (poset) is planar if it has a planar Hasse diagram. The dimension of a bounded planar poset is at most two. We show that the dimension of a planar poset having a greatest lower bound is at most three. We also construct four-dimensional planar posets, but no planar poset with dimension larger than four is known. A poset is called a tree if its Hasse diagram is a tree in t...

Let Eq : Y 2 + XY = X3 + h4X + h6 be the Tate curve with canonical differential, ω = dX/(2Y + X). If the characteristic is p > 0, then the Hasse invariant, H, of the pair (Eq, ω) should equal one. If p > 3, then calculation of H leads to a nontrivial separable relation between the coefficients h4 and h6. If p = 2 or p = 3, Thakur related h4 and h6 via elementary methods and an identity of Raman...

We study Brauer–Manin obstructions to the Hasse principle and to weak approximation, with special regard to effectivity questions.

We prove that the types in Separably Closed Hasse Fields which have transcendence degree 1 are separably thin

We investigate the Hasse principle for complete intersections cut out by a quadric and cubic hypersurface defined over the rational numbers.