# What are the because conjectures

## Because guesswork

The **Because guesswork**Theorems, which have been theorems since their final proof in 1974, have long been a driving force in the border area between number theory and algebraic geometry since their formulation by André Weil in 1949.

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

### Motivation and story

The case of algebraic curves over finite fields was proved by Weil himself^{[1]}. Before that, Helmut Hasse had already proven the Riemann hypothesis for the case of elliptical curves (gender 1). In this regard, many of the Weil conjectures were naturally embedded in the main developments in this area and were of interest, for example, for the estimation of exponential sums in analytical number theory. The only surprising thing was the emergence of topological concepts (Betti numbers of the underlying spaces, Lefschetz's fixed point theorem) that were supposed to determine the geometry over finite bodies (i.e. in number theory). Weil himself is said never to have bothered seriously with the evidence in the general case, since his conjectures suggested the need for the development of new topological concepts in algebraic geometry. The development of these concepts by the Grothendieck School took 20 years (étale cohomology). First, in 1960, the rationality of the zeta function was proven by Bernard Dwork using p-adic methods. The most difficult and final part of the Weil conjectures, the analogues to the Riemann hypothesis, was proven by Grothendieck's student Pierre Deligne in 1974.

### Formulation of the Weil conjectures

*X* be a non-singular one *n*-dimensional projective algebraic variety over the body **F.**_{q} With *q* Elements. Then it is *Zeta function* ζ (*X*, *s*) of *X* defined as a function of a complex number s by:

With *N*_{m} the number of points from *X* above the body of order *q*^{m}.

The Weil conjectures are:

- ζ (
*X*,*s*) is a rational function of*T*=*q*. More precisely, ζ (^{−s}*X*,*s*) = ∏ (−1)^{i}*P.*_{i}(*q*) where each^{−s}*P.*_{i}(*T*) is a polynomial of the form ∏ (1-α_{i, j}*T*) is (rationality). - ζ (
*X*,*s*) = ζ (*X*,*n − s*). In other words, the mapping that α is based on*q*^{n}/ α, forms the numbers α_{i, j}on the numbers α_{2n-i, j}from. (Functional equation or Poincare duality) - | α
_{i, j}| =*q*^{i / 2}This is the analogue of the Riemann Hypothesis and the hardest part of the guesswork. It can also be formulated in such a way that all zeros of*P.*_{i}(*q*) on the^{−s}*critical straight line*in the numerical level of the*s*lie with real part*i*/2. - If
*X*the reduction mod*p*a non-singular complex projective variety*Y*is is the degree of*P.*_{i}the*i*te Betti number of*Y*.

### Examples

### The projective straight line

The simplest example apart from the point is the case of the projective straight line *X*. The number of points from *X* over a body with *q*^{m} Elements is *N*_{m} = *q*^{m} +1 (where the “+1” comes from the “point at infinity”). The zeta function is 1 / (1−*q*^{−s})(1−*q*^{1−s}). Further verification of the Weil conjectures is straightforward.

### Projective space

The case of the *n* dimensional projective space is not much more difficult. The number of points from *X* over a body with *q*^{m} Elements is *N*_{m} = 1 + *q*^{m} + *q*^{2m} + ... + *q*^{nm}. The zeta function is

- 1/(1−
*q*^{−s})(1−*q*^{1−s})(1−*q*^{2−s})...(1−*q*^{n−s}).

Again, the Weil conjectures are easy to test.

The reason why projective lines and spaces are so simple is that they can be written as disjoint copies of a finite number of affine spaces. The proof is just as easy for rooms with a similar structure such as Grassmann varieties.

### Elliptic curves

They are the first non-trivial case of the Weil presumption (it was treated by Helmut Hasse in the 1930s). *E.* be an elliptic curve over a finite field of *q* Elements, then the number of points is from *E.* over bodies with *q ^{m}* Elements 1 − α

^{m}−β

^{m}+

*q*

^{m}, where α and β are complex conjugate to each other with absolute value √

*q*. The zeta function is

- ζ (
*E.*,*s*) = (1 −α*q*^{−s}) (1 −β*q*^{−s}) / (1 −*q*^{−s})(1−*q*^{1−s})

### Because cohomology

Weil suggested that the conjectures would follow from the existence of a suitable "Weil cohomology theory" for varieties over finite fields, similar to the usual cohomology with rational coefficients for complex varieties. According to his plan of evidence, the points are variety *X* over a body of order *q*^{m} Fixed points of the Frobenius automorphism*F.* this body. In algebraic topology, the number of fixed points of an automorphism is expressed using Lefschetz's fixed point theorem as the alternating sum of the traces of the effect of this automorphism in the cohomology groups. If similar cohomology groups were defined for varieties over finite fields, the zeta function could be expressed by them.

The only problem was that the coefficient field of the Weil cohomologies could not be that of the rational numbers. For example, consider a supersingular elliptic curve over a body of the characteristic *p*. The endomorphism ring of this curve is a quaternion algebra over the rational numbers. It should act accordingly on the first cohomology group, a 2-dimensional vector space. But this is impossible for a quaternionalgebra over the rational numbers if the vector space over the rational numbers is explained. The real and p-adic numbers are also ruled out. That would be an option, however *l*-adic numbers for a prime number *l* ≠ *p*, since the division algebra of the quaternions then splits up and becomes a matrix algebra which can then operate on 2-dimensional vector spaces. This construction was carried out by Grothendieck and Michael Artin (l-adic cohomology).

### literature

- André Weil
*Numbers of solutions of equations in finite fields.*Bull. Amer. Math. Soc. Vol. 55, 1949, pp. 497-508. - Deligne, Pierre
*La conjecture de Weil I.*, Publications Math. IHES, No. 43, 1974, pp.273-307,*La conjecture de Weil II*, ibid., No.52, 1980, pp.137-252, Online: * Part 1, Part 2 - Eberhard Friday, Reinhardt Kiehl
*Etale cohomology and the Weil conjecture*, Springer 1988, ISBN 0-387-12175-7 - Nicholas Katz
*An overview of Deligne's work on Hilbert's twenty-first problem*, in Browder (Ed.) Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Vol. 28), American Mathematical Society 1976, pp. 537-557 - Robin Hartshorne
*Algebraic Geometry*, Appendix, Springer 1997, ISBN 0387902449 - Ireland, roses
*A classical introduction to Modern Number Theory*, Springer, 2nd ed. 2006, ISBN 038797329X

### References

- ↑ one
*elementary*Proof of algebraic curves over finite fields was given in 1969 by Sergei Alexandrowitsch Stepanow, represented in Enrico Bombieri*Counting points on curves over finite fields (d´apres Stepanov)*, Seminaire Bourbaki No. 431, 1972/73, Stepanow:*On the number of points of a hyperelliptic curve over a prime field*, Izvestija Akad. Nauka vol. 33, 1969, p.1103, Stepanow*Arithmetic of Algebraic Curves*1994

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