Glossary of Arithmetic and Diophantine Geometry - S

S

Sato–Tate conjecture

The Sato–Tate conjecture is a conjectural result on the distribution of Frobenius elements in the Tate modules of the elliptic curves over finite fields obtained from reducing a given elliptic curve over the rationals. It is due to Mikio Sato and John Tate (independently, and around 1960, published somewhat later) and is by now supported by very substantial evidence. It is a prototype for Galois representations in general.

Skolem's method

See Chabauty's method.

Special set

The special set in an algebraic variety is the subset in which one might expect to find many rational points. The precise definition varies according to context. One definition is the Zariski closure of the union of images of algebraic groups under non-trivial rational maps; alternatively one may take images of abelian varieities; another definition is the union of all subvarieties that are not of general type. For abelian varieties the definition would be the union of all translates of proper abelian subvarieties. For a complex variety, the holomorphic special set is the Zariski closure of the images of all non-constant holomorphic maps from C. Lang conjectured that the analytic and algebraic special sets are equal.

Subspace theorem

Schmidt's subspace theorem shows that points of small height in projective space lie in a finite number of hyperplanes. A quantitative form of the theorem, in which the number of subspaces containing all solutions, was also obtained by Schmidt, and the theorem was generalised by Schlickewei (1977) to allow more general absolute values on number fields. The theorem may be used to obtain results on Diophantine equations such as Siegel's theorem on integral points and solution of the S-unit equation.