Glossary of Arithmetic and Diophantine Geometry - D

D

Diagonal forms

Diagonal forms are some of the simplest projective varieties to study from an arithmetic point of view (including the Fermat varieties). Their local zeta-functions are computed in terms of Jacobi sums. Waring's problem is the most classical case.

Diophantine dimension

The Diophantine dimension of a field is the smallest natural number k, if it exists, such that the field of is class C_k: that is, such that any homogeneous polynomial of degree d in N variables has a non-trivial zero whenever N > dk. Algebraically closed fields are of Diophantine dimension 0; quasi-algebraically closed fields of dimension 1.

Discriminant of a point

The discriminant of a point refers to two related concepts relative to a point P on an algebraic variety V defined over a number field K: the geometric (logarithmic) discriminant d(P) and the arithmetic discriminant, defined by Vojta. The difference between the two may be compared to the difference between the arithmetic genus of a singular curve and the geometric genus of the desingularisation. The arithmetic genus is larger than the geometric genus, and the height of a point may be bounded in terms of the arithmetic genus. Obtaining similar bounds involving the geometric genus would have significant consequences.

Dwork's method

Bernard Dwork used distinctive methods of p-adic analysis, p-adic algebraic differential equations, Koszul complexes and other techniques that have not all been absorbed into general theories such as crystalline cohomology. He first proved the rationality of local zeta-functions, the initial advance in the direction of the Weil conjectures.