Algebraic varieties, arithmetic of

arithmetical algebraic geometry

The branch of algebraic geometry in which one studies properties of algebraic varieties defined over fields of so-called arithmetic type, i.e. finite, local and global fields of algebraic numbers or algebraic functions. In the case of a finite field, its main subject is the study of the number of rational points of the algebraic variety in this field and in finite extensions of it. The zeta-function of the variety which is used in these studies strongly influenced the development of the methods of algebraic geometry. Estimates from below of the number of (rational) points [1], [4] are also important.

If $X$ is an algebraic variety (or scheme) over a local field $K$ with field of residues $k$, then the study of the set $X(K)$ of rational points with values in $K$ combines two different problems: To find solutions of congruences (or points of the variety over a finite field) and to find integral or rational solutions of Diophantine equations (cf. Hasse principle). If the variety $X$ is defined by a set of equations with coefficients from the ring $A$ of integers of the field $K$, then it is possible to define the reduction of this variety by the same set of equations, but with coefficients taken modulo the maximal ideal of $A$. A "variety" $X_0$ over the field of residues $k$ and a canonical mapping, or reduction,

$$\mathrm{Red}:X(K)\to X_0(k)$$

are obtained. This description of the reduction is difficult to explain in terms of classical algebraic geometry. This was one of the reasons for the introduction of the concept of a scheme — a language suitable for a rigorous description of the process. The main problem is to determine the image of the mapping Red, i.e. to find the points $x\in X_0(k)$ that come from the rational $K$- points of the variety. The Hensel lemma states that $x$ is such a point if $x$ is a non-singular point. For more general results on this subject see [4].

Another type of problems concerned with the local arithmetic of algebraic varieties is the study of forms over such fields. Let $F$ be a form in $n$ variables of degree $d$ over a local field; Artin's conjecture is that if $n>d^2$, then the equation $F=0$ has a non-trivial solution. It is known that this statement is true in the case of function fields. It was proved for $p$- adic fields that for each $d$ there exists a finite number $A(d)$ of primes such that Artin's conjecture holds for forms of degree $d$, if $p\notin A(d)$. It was shown in 1966 that the set $A(4)$ is non-empty, which proved Artin's conjecture to be false [4]. It is not yet (1977) known whether or not it is valid for forms of odd degree.

The arithmetic of algebraic varieties over global fields is the largest and most diverse part of algebraic geometry. It includes Diophantine geometry, class field theory, the theory of zeta-functions of varieties, and complex multiplication of Abelian functions (or varieties). All these theories were developed in parallel for number and function fields. Such a possibility was first demonstrated by the development of class field theory in the 1930s; it is based on the rough analogy between these fields which is most clearly manifested in the construction of the theory of schemes.

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