hasse

Hasse is a German surname and, less commonly, a given name. The most prominent bearer in mathematics is Helmut Hasse (1898–1979), a German mathematician who made foundational contributions to number theory and algebra. His work helped formalize several ideas that carry his name, including the Hasse diagram, the Hasse principle, and the Hasse–Minkowski theorem.

Hasse diagram: A graphical representation of a finite partially ordered set. In a Hasse diagram, elements are placed so that if x < y, then x appears below y, and cover relations are connected by edges while transitive relations are not drawn. The diagram provides an intuitive view of the order structure and is widely used in lattice theory and related fields.

Hasse principle: A local-global principle in number theory. Roughly, if a diophantine equation has a solution in every completion of a number field (for example, over the real numbers and over all p-adic fields), then it has a solution over the field itself. The principle is exact for quadratic forms as described by the Hasse–Minkowski theorem, but not in full generality for all equations.

Hasse–Minkowski theorem: A foundational result stating that a quadratic form over a number field represents zero nontrivially if and only if it does so over every completion of the field. In particular, a rational quadratic form is isotropic over Q precisely when it is isotropic over R and over Q_p for every prime p.

Hasse invariant: An arithmetic-geometric invariant named for Hasse, used in the study of abelian varieties and elliptic curves in characteristic p and in reductions modulo p. It captures certain Frobenius-related properties of the variety.