We give necessary and sufficient conditions for an integral polynomial without linear factors to be the characteristic of isometry some even, unimodular lattice given signature. This gives rise Hasse principle questions, which we answer in a more general setting. As application, prove signatures knots.

We introduce an analog of the L-function for noncommutative tori. It is proved that such a function coincides with Hasse–Weil elliptic curve complex multiplication. As corollary, one gets localization formula tori real

This article explains the Hasse principle and gives a self-contained development of certain counterexamples to this principle. The counterexamples considered are similar to the earliest counterexample discovered by Lind and Reichardt. This type of counterexample is important in the theory of elliptic curves: today they are interpreted as nontrivial elements in Tate– Shafarevich groups.

In both of these examples, we have proved that X(Q) = ∅ by showing that X(Qv) = ∅ for some place v. In the first case it was v = ∞, the real place. In the second case we showed that X(Q2) was empty: the argument applies equally well to a supposed solution over Q2. Given a variety X over a number field k and a place v of k, it is a finite procedure to decide whether X(kv) is empty. Moreover, X(k...

Around 1940, Lind [Lin] and (independently, but shortly later) Reichardt [Re] discovered that some genus 1 curves over Q, such as 2y = 1− 17x, violate the Hasse principle; i.e., there exist x, y ∈ R satisfying the equation, and for each prime p there exist x, y ∈ Qp satisfying the equation, but there do not exist x, y ∈ Q satisfying the equation. In fact, even the projective nonsingular model h...

We construct families of log K3 surfaces and study the arithmetic their members. use this to produce explicit with an order 5 Brauer–Manin obstruction integral Hasse principle.