Because conjecture

The Weil conjectures , which have been theorems since their final proof in 1974, have been a driving force for a long time in the border area between number theory and algebraic geometry since they were formulated 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 History

The case of algebraic curves over finite fields was proved by Weil himself. 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 of interest e.g. B. for the estimation of exponential sums of the analytic number theory . The only surprising thing was the emergence of topological concepts ( Betti numbers of the underlying spaces, fixed point theorem of Lefschetz and others) that were supposed to determine the geometry over finite fields (i.e. in number theory). Weil himself is never said 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 (the étale cohomology was necessary for the Riemann assumption ). First in 1960 the rationality of the zeta function was proven by Bernard Dwork using p-adic methods. In 1964 Grothendieck gave a more general L-adic proof of this, and he also proved the second and fourth Weil conjectures in the 1960s (with Michael Artin and Jean-Louis Verdier ). The most difficult and last part of the Weil conjectures, the analogues to the Riemann hypothesis, proved the Grothendieck student Pierre Deligne in 1974. Deligne proved in 1980 in a second proof (La conjecture de Weil II) a generalization of the Weil conjectures with which he was able to prove the hard Lefschetz theorem, part of Grothendieck's standard conjectures. His second proof used an analogue of the proof of the prime number theorem by Jacques Hadamard and Charles-Jean de La Vallée Poussin , which was led about the non-existence of a zero of the Riemann zeta function with real part 1 (transferred by Deligne to L-functions). Gérard Laumon simplified the proof in 1987 by using the L-adic Fourier transform introduced by Deligne and an analogue to the classical estimation of Gaussian sums.

Grothendieck was dissatisfied with Deligne's proof because, in his opinion, he used a “trickery” with modular shapes in the Riemann assumption (a classic result by Robert Alexander Rankin ). In his opinion, the proof of the theory of motives and its fundamental assumptions (Standard conjectures) should take place on algebraic cycles (still largely open even as difficult to attack claimed and) and outlined a derivative thereof, as well as independent at the same time Enrico Bombieri on these guesses came up. Grothendieck attended the seminar at IHES in 1973, in which Deligne presented his proof and discussed with Deligne, but was not interested in the proof of the Riemann conjecture for the reasons mentioned.

Formulation of the Weil conjectures

${\displaystyle X}$be a non-singular -dimensional projective algebraic variety over the finite field with elements. Then the zeta function of is defined as a function of a complex number by: ${\displaystyle n}$ ${\displaystyle F_{q}}$ ${\displaystyle q}$ ${\displaystyle \zeta (X,\ s)}$ ${\displaystyle X}$ ${\displaystyle s}$

${\displaystyle \zeta (X,s)=\exp \left(\sum _{m=1}^{\infty }{\frac {N_{m}}{m}}(q^{-s})^{m}\right)}$

with the number of points from above the body of order . ${\displaystyle N_{m}}$ ${\displaystyle X}$ ${\displaystyle q^{m}}$

The Weil conjectures are:

1. (Rationality) is a rational function of . More specifically, where each is a polynomial with integer coefficients that factors over the complex numbers in the form . Furthermore, , . ${\displaystyle \zeta (X,\ s)}$ ${\displaystyle T=q^{-s}}$ ${\displaystyle \zeta (X,\ s)=\prod _{i=0}^{2n}\,P_{i}(q^{-s})^{(-1)^{i+1}}={\frac {P_{1}(T)\dotsb P_{2n-1}(T)}{P_{0}(T)\dotsb P_{2n}(T)}}}$ ${\displaystyle P_{i}(T)}$ ${\displaystyle \prod _{j}(1-\alpha _{i,j}T)}$ ${\displaystyle P_{0}(T)=1-T}$ ${\displaystyle P_{2n}(T)=1-q^{n}T}$
2. (Functional equation and Poincaré duality) , where the Euler characteristic of is. The numbers are the numbers displayed. ${\displaystyle \zeta (X,\ n-s)=\pm q^{E({\frac {n}{2}}-s)}\zeta (X,\ s)}$ ${\displaystyle E}$ ${\displaystyle X}$ ${\displaystyle {\frac {q^{n}}{\alpha _{i,j}}}}$ ${\displaystyle \alpha _{2n-i,j}}$
3. (Riemann assumption) for everyone and everyone . 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 on the critical straight line lie in the numerical plane of the real part . ${\displaystyle |\alpha _{i,j}|=q^{\tfrac {i}{2}}}$ ${\displaystyle 1\leq i\leq n-1}$ ${\displaystyle j}$ ${\displaystyle P_{k}(q^{-s})}$ ${\displaystyle s}$ ${\displaystyle {\tfrac {k}{2}}}$
4. (Betti numbers) If a good reduction mod is a non-singular complex projective variety , the degree of is the -th Betti number of . ${\displaystyle X}$ ${\displaystyle p}$ ${\displaystyle Y}$ ${\displaystyle P_{i}}$ ${\displaystyle i}$ ${\displaystyle Y}$

Examples

The projective straight line

The simplest example apart from the point is the case of the projective straight line . Is the number of points from above a body with elements (where the " " comes from the "point at infinity"). The zeta function is . Further verification of the Weil conjectures is straightforward. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle q^{m}}$ ${\displaystyle N^{m}=q^{m}+1}$ ${\displaystyle +1}$ ${\displaystyle {\tfrac {1}{({1-q}^{-s})({1-q}^{1-s})}}}$

Projective space

The case of -dimensional projective space is not much more difficult. The number of points from above a body with elements is . The zeta function is ${\displaystyle n}$ ${\displaystyle X}$ ${\displaystyle q^{m}}$ ${\displaystyle N^{m}=1+q^{m}+q^{2m}+\cdots +q^{nm}}$

${\displaystyle {\frac {1}{({1-q}^{-s})({1-q}^{1-s})({1-q}^{2-s})\cdots ({1-q}^{n-s})}}}$.

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

The first non-trivial case of the because conjectures are elliptic curves . These were already dealt with by Helmut Hasse in the 1930s . Let be an elliptic curve over a finite field with elements. Then the formula applies to the number of points above a body extension with elements ${\displaystyle E}$ ${\displaystyle q}$ ${\displaystyle N_{m}}$ ${\displaystyle E}$ ${\displaystyle q^{m}}$

${\displaystyle N_{m}=q^{m}+1-\alpha ^{m}-\beta ^{m}}$,

where and are complex conjugate to each other and each have an absolute value (Riemann Hypothesis). The zeta function of the elliptic curve is ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle {\sqrt {q}}}$

${\displaystyle \zeta (E,s)={\frac {(1-\alpha q^{-s})(1-\beta q^{-s})}{(1-q^{-s})(1-q^{1-s})}}}$.

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 proof, the points of the variety over a field of order are fixed points of the Frobenius automorphism of this field. 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. ${\displaystyle X}$ ${\displaystyle q^{m}}$ ${\displaystyle F}$

The first problem was only 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 . 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 -adic numbers are also ruled out. -Adic numbers for a prime number would come into question, however , since the division algebra of the quaternions then splits up and becomes a matrix algebra that can operate on 2-dimensional vector spaces. This construction was carried out by Grothendieck and Michael Artin ( l-adic cohomology ). ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle l}$ ${\displaystyle l\not =p}$

The étale cohomology, which was introduced by Grothendieck and Michael Artin and was developed in the IHES seminar (SGA), was necessary to prove the Riemann hypothesis .

literature

• André Weil Numbers of solutions of equations in finite fields. Bull. Amer. Math. Soc. Vol. 55, 1949, pp. 497-508 .
• Pierre Deligne 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 Freitag , Reinhardt Kiehl Étale 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
• Kenneth Ireland, Michael Rosen A classical introduction to Modern Number Theory , Springer, 2nd ed. 2006, ISBN 038797329X

Web links

• Raskin: The Because Conjectures for Curves
• nLab, because conjectures

References

1. ↑ An elementary proof for algebraic curves over finite fields was given in 1969 by Sergei Alexandrowitsch Stepanov , presented in Enrico Bombieri Counting points on curves over finite fields (d'apres Stepanov) . In: Seminaire Bourbaki . No. 431, 1972/73 ( numdam.org [PDF]). , 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
2. ↑ Laumon, Transformation de Fourier, constantes d'équations fonctionnelles et conjecture de Weil , Publications Mathématiques de l'IHÉS, Volume 65, 1987, pp. 131-210
3. ↑ Allyn Jackson, Comme appelé du Néant, Notices AMS, Oct. 2004, p 1203
4. ↑ Chapter V, Theorem 2.3.1 in Joseph H. Silverman: The Arithmetic of Elliptic Curves . 2nd Edition. Springer, 2009, ISBN 978-0-387-09493-9 .