Weil conjectures

The Weil conjectures, which are theorems in 1974 since their final proof had, since its formulation by André Weil in 1949 for a long time been a driving force in the border area between number theory and algebraic geometry.

You make statements about the algebraic solutions of the number of varieties over finite fields formed generating functions, the local zeta functions called. Because suspected that these rational functions, they obey a functional equation, and that the zeros on certain geometric Örtern are ( analogue of the Riemann Hypothesis ), similar to the Riemann zeta function as a carrier of information about the distribution of prime numbers. He also suspected that her behavior of certain topological invariants of the underlying manifolds is determined.

Motivation and History

The case of algebraic curves over finite fields was proved by Weil himself. Prior to that already Helmut Hasse, the Riemann hypothesis for the case of elliptic curves ( genus 1 ) had been proved. In this respect many of the Weil conjectures were embedded in a natural way into the main developments in this area and of interest eg exponential for estimating sums of analytic number theory. What was surprising was only the emergence of topological concepts ( Betti numbers of the underlying spaces of Lefschetz fixed point theorem, etc.) that should the geometry over finite fields (ie in number theory ) determine. Because even should never have taken care seriously about the evidence in the general case, as his suspicions indicated a need to need to develop new topological concepts in algebraic geometry. The development of these concepts by the Grothendieck school took 20 years ( étale cohomology ). First, the rationality of the zeta function by Bernard Dwork in 1960 proved with p- adic methods. The hardest and last part of the Weil conjectures, the analogues in the Riemann hypothesis, the Grothendieck -student Pierre Deligne proved 1974.

Formulation of the Weil conjectures

Let X be a nonsingular projective n-dimensional algebraic variety over the field Fq with q elements. Then the zeta function ζ ( x, s ) is defined by X as a function of a complex number s by:

With Nm the number of points of X over the field of order qm.

The Weil conjectures are:

Examples

The projective line

The point other than the simplest example is the case of the projective line X. The number of points of X over a body with square elements is Nm = qm 1 (where the " 1 " from " point at infinity " is derived ). Zeta function is 1 / (1 -q -s ) (1- q1 -s). The further review of the Weil conjectures is simple.

Projective space

The case of the n -dimensional projective space is not much more difficult. The number of points of X over a body with square elements is Nm = 1 qm ... QNM Q2M. The zeta function is

Again, let the Weil conjectures easily verify.

The reason why projective line and space are so simple is that they can be written as a disjoint copies of a finite number of affine spaces. For similarly structured spaces such as Grassmann varieties the evidence is just as easy.

Elliptic Curves

They are the first non- trivial case of the Weil conjectures ( he was treated in the 1930s by Helmut Hasse ). Let E be an elliptic curve over a finite field of q elements, then the number of points of E over fields with square elements 1 -? M - βm sq. ft., where α and β are complex conjugate to each other with absolute value √ q. The zeta function is

Weil cohomology

Because suggested that the presumptions of the existence of a suitable " Weil cohomology theory " would follow for varieties over finite fields, similar to the usual cohomology with rational coefficients for complex varieties. After his proof plan, the points of the variety X over a field of order qm fixed points of the Frobenius automorphism F of this body. In algebraic topology, the number of fixed points of an automorphism on the Lefschetz fixed point theorem is expressed as an alternating sum of the traces of the action of this automorphism in cohomology. Would like cohomology groups defined for varieties over finite fields, the zeta function can be expressed by this.

The first problem was that the coefficient field of the Weil - Kohomologien could not be the rational numbers. For example, consider a supersingular elliptic curve over a field of characteristic p. The endomorphism ring of the curve is a quaternion algebra over the rational numbers. You should act according to the first cohomology group, a 2-dimensional vector space. But this is impossible for a Quaternionalgebra over the rational numbers, if the vector space is declared over the rational numbers. The real and p- adic numbers are eliminated. In question, however, would l - adic integers for a prime l ≠ p, since the division algebra of quaternions then splits and a matrix algebra that can operate on two -dimensional vector spaces. This construction was by Grothendieck and Michael Artin executed ( l -adic cohomology ).