Weil conjectures

In mathematics, the Weil conjectures, which had become theorems by 1974, were some highly-influential proposals from the late 1940s by André Weil on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields.

A variety "V" over a finite field with "q" elements has a finite number of rational points, and over every finite field with "q"^"k" elements containing that field. The generating function has coefficients derived from the numbers "N"_"k" of points over the (essentially unique) field with "q"^"k" elements.

The main burden was that such "zeta-functions" should be rational functions, should satisfy a form of functional equation, and should have their zeroes in restricted places. The last two parts were quite consciously modelled on the Riemann zeta function and Riemann hypothesis.

Background and history

In fact the case of curves over finite fields had been proved by Weil himself, finishing the project started by Hasse's theorem on elliptic curves over finite fields. The conjectures were natural enough in one direction, simply proposing that known good properties would extend. Their interest was obvious enough from within number theory: they implied the existence of machinery that would provide upper bounds for exponential sums, a basic concern in analytic number theory.

What was really eye-catching, from the point of view of other mathematical areas, was the proposed connection with algebraic topology. Given that finite fields are "discrete" in nature, and topology speaks only about the "continuous", the detailed formulation of Weil (based on working out some examples) was striking and novel. It suggested that geometry over finite fields should fit into well-known patterns relating to Betti numbers, the Lefschetz fixed-point theorem and so on.

Weil himself, it is said, never seriously tried to prove the conjectures. The analogy with topology suggested that a new homological theory be set up applying within algebraic geometry. This took two decades (it was a central aim of the work and school of Alexander Grothendieck) building up on initial suggestions from Serre and others. The rationality part of the conjectures was proved first by Bernard Dwork in 1959, using "p"-adic methods. In 1964, Grothendieck and his collaborators were able to establish the rationality conjecture (see (Grothendieck 1965)), the functional equation and the link to Betti numbers by using the properties of étale cohomology, a subtle new cohomology theory developed by Grothendieck with the specific aim of attacking the Weil conjectures, as outlined in his address (Grothendieck 1958) to the 1958 ICM congress. Of the four conjectures it was the analogue of the Riemann hypothesis that turned out to be the hardest to prove. Grothendieck envisioned a proof based on his standard conjectures on algebraic cycles (see (Kleiman 1968) for details). However, the standard conjectures remain open, and the analogue of the Riemann hypothesis was proved in 1973 by Pierre Deligne and published in the landmark article (Deligne 1974), using the full force of the étale cohomology theory but circumventing the use of standard conjectures by a very ingenious argument.

The conjectures of Weil have therefore taken their place within the general theory (of L-functions, in the broad sense). Since étale cohomology has had many other applications, this development exemplifies the relationship between conjectures (based on examples, guesswork and intuition), theory-building, problem-solving, and spin-offs, even in the most abstract parts of pure mathematics.

tatement of the Weil conjectures

Suppose that "X" is a non-singular "n"-dimensional projective algebraic variety over the field F_"q" with "q" elements. The zeta function ζ("X", "s") of "X" is by definition:$zeta(X,\; s)\; =\; expleft(sum\_\{m\; =\; 1\}^infty\; frac\{N\_m\}\{m\}\; (q^\{-s\})^m\; ight)$

where "N"_"m" is the number of points of "X" defined over the degree "m" extension F_"q"^"m" of F_"q".

The Weil conjectures state:
# ζ("X", "s") is a "rational function" of "T"="q^−s". More precisely, ζ("X", "s") can be written as a finite alternating product ∏_"i" "P"_"i"("q^−s")^(−1)"i+1" where each "P"_"i"("T") is an integral polynomial of the form ∏_"j"(1-α_"i","j""T") for some complex numbers α_"i","j". (Rationality)
#ζ("X", "n−s")=±"q"^"nE/2-Es"ζ("X", "s") where "E" is the self-intersection number of the diagonal in "X" × "X". In particular, for each "i", the numbers α_2"n"-"i",1, α_2"n"-"i",2, … equal the numbers "q"^"n"/α_"i",1, "q"^"n"/α_"i",2, … in some order. (Functional equation or Poincaré duality)
#|α_"i,j"| = "q"^"i"/2 for all "i" and "j". When "n"=1, this implies that all zeros of ζ("X", "s") lie on the "critical line" of complex numbers "s" with real part 1/2. ("Riemann hypothesis")
#If "X" is a (good) "reduction mod "p"" of a non-singular projective variety "Y" defined over a number field embedded in the field of complex numbers, then the degree of "P"_"i" is the "i"^th Betti number of the space of complex points of "Y".

Examples

The projective line

The simplest example (other than a point) is to take "X" to be the projective line. The number of points of "X" over a field with "q"^"m" elements isjust "N"_"m" = "q"^"m" + 1 (where the "+ 1" comes from the "point at infinity"). The zeta function is just

:1/(1−"q"^−"s")(1−"q"^1−"s").

It is easy to check all parts of the Weil conjectures directly. For example, the corresponding complex variety is the Riemann sphere and its initial Betti numbers are 1, 0, 1.

Projective space

It is not much harder to do "n" dimensional projective space.The number of points of "X" over a field with "q"^"m" elements isjust "N"_"m" = 1 + "q"^"m" + "q"^2"m" + ... + "q"^"nm". The zeta function is just :1/(1−"q"^−"s")(1−"q"^1−"s")(1−"q"^2−"s")...(1−"q"^"n"−"s"). It is again easy to check all parts of the Weil conjectures directly. (Complex projective space gives the relevant Betti numbers, which nearly determine the answer.)

The reason why the projective line and projective space were so easy is that they can be written as disjoint unions of a finite number of copies of affine spaces, which makes the number of points on them particularly easy to calculate. It is also easy to prove the Weil conjectures for other spaces, such as Grassmannians, that have the same property.

Elliptic curves

These give the first non-trivial cases of the Weil conjectures (proved by Hasse).If "E" is an elliptic curve over a finite field with "q" elements, then the number of points of "E" defined over the field with "q^m" elements is 1−α^"m"− β^"m"+"q"^"m", where α and β are complex conjugates 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").

Weil 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.His idea was that if "F" is the Frobenius automorphism over the finite field, then the number of points of the variety "X" over the field of order "q"^"m" is the number of fixed points of "F"^"m" (acting on all points of the variety "X" defined over the algebraic closure). In algebraic topology the number of fixed points of an automorphism can be worked out using the Lefschetz fixed point theorem, given as an alternating sum of traces on the cohomology groups. So if there were similar cohomology groups for varieties over finite fields, then the zeta function could be expressed in terms of them.

The first problem with this is that the coefficient field for a Weil cohomology theory cannot be the rational numbers. To see this consider the case of a supersingular elliptic curve over a finite field of characteristic "p". The endomorphism ring of this is a quaternion algebra over the rationals, and should act on the first cohomology group, which should a 2 dimensional vector space over the coefficient field by analogy with the case of a complex elliptic curve. However a quaternion algebra over the rationals cannot act on a 2 dimensional vector space over the rationals. The same argument eliminates the possibility of the coefficient field being the reals or the "p"-adic numbers, because the quaternion algebra is still a division algebra over these fields. However it does not eliminate the possibility that the coefficient field is the field of "l"-adic numbers for some prime "l"≠"p", because over these fields the division algebra splits and becomes a matrix algebra, which can act on a 2-dimensional vector space. Grothendieck and Michael Artin managed to construct suitable cohomology theories over the field of "l"-adic numbers for each prime "l"≠"p", called l-adic cohomology.

References

*Deligne, Pierre [http://www.numdam.org/item?id=PMIHES_1974__43__273_0 "La conjecture de Weil. I."] Inst. Hautes Études Sci. Publ. Math. No. 43 (1974), 273--307. [http://www.numdam.org/item?id=PMIHES_1980__52__137_0 "La conjecture de Weil : II."] Publications Mathématiques de l'IHÉS, 52 (1980), p. 137-252
*Freitag, Eberhard; Kiehl, Reinhardt "Étale cohomology and the Weil conjecture." Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 13. Springer-Verlag, Berlin, 1988. ISBN 0-387-12175-7
*Alexander Grothendieck, "The cohomology theory of abstract algebraic varieties". Proc. Int. Congr. Math. Edinburgh (1958), 103-118.
*Alexander Grothendieck, "Formule de Lefschetz et rationalité des fonctions L". Séminaire Bourbaki 279, Secréteriat Mathématique, Paris (1965). Reprinted recently by Société Mathematique de France.
*Katz, Nicholas M. "An overview of Deligne's work on Hilbert's twenty-first problem." Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974), pp. 537--557. Amer. Math. Soc., Providence, R. I., 1976.
*S. L. Kleiman, "Algebraic cycles and the Weil conjectures". Dix Exposes Cohomologie Schemas, Advanced Studies Pure Math. 3, 359-386 (1968).
*Citation | last1=Weil | first1=André | author1-link=André Weil | title=Numbers of solutions of equations in finite fields | url=http://www.ams.org/bull/1949-55-05/S0002-9904-1949-09219-4/home.html | doi=10.1090/S0002-9904-1949-09219-4 | id=MathSciNet | id = 0029393 | year=1949 | journal=Bulletin of the American Mathematical Society | issn=0002-9904 | volume=55 | pages=497–508 Reprinted in Oeuvres Scientifiques/Collected Papers by Andre Weil ISBN 0-387-90330-5