Glossary of Arithmetic and Diophantine Geometry - G

G

Geometric class field theory

The extension of class field theory-style results on abelian coverings to varieties of dimension at least two is often called geometric class field theory.

Good reduction

Fundamental to local analysis in arithmetic problems is to reduce modulo all prime numbers p or, more generally, prime ideals. In the typical situation this presents little difficulty for almost all p; for example denominators of fractions are tricky, in that reduction modulo a prime in the denominator looks like division by zero, but that rules out only finitely many p per fraction. With a little extra sophistication, homogeneous coordinates allow clearing of denominators by multiplying by a common scalar. For a given, single point one can do this and not leave a common factor p. However singularity theory enters: a non-singular point may become a singular point on reduction modulo p, because the Zariski tangent space can become larger when linear terms reduce to 0 (the geometric formulation shows it is not the fault of a single set of coordinates). Good reduction refers to the reduced variety having the same properties as the original, for example, an algebraic curve having the same genus, or a smooth variety remaining smooth. In general there will be a finite set S of primes for a given variety V, assumed smooth, such that there is otherwise a smooth reduced V_p over Z/pZ. For abelian varieties, good reduction is connected with ramification in the field of division points by the Néron–Ogg–Shafarevich criterion. The theory is subtle, in the sense that the freedom to change variables to try to improve matters is rather unobvious: see Néron model, potential good reduction, Tate curve, semistable abelian variety, semistable elliptic curve, Serre–Tate theorem.

Grothendieck–Katz conjecture

The Grothendieck–Katz p-curvature conjecture applies reduction modulo primes to algebraic differential equations, to derive information on algebraic function solutions. It is an open problem as of 2005. The initial result of this type was Eisenstein's theorem.