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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ... In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ... In number theory, a local zeta-function is a generating function Z(t) for the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F. The analogy with the Riemann zeta-function comes via consideration of the logarithmic derivative . Given... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ...

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. In number theory, a K-rational point is a point on an algebraic variety where each coordinate of the point belongs to the field K. Rational points of varieties constitute a major area of current research. ...

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. In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ... In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always ½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ...

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. In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... In mathematics, Hasses theorem on elliptic curves bounds the number of points on an elliptic curve over a finite field. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ... In mathematics, an exponential sum may be a finite Fourier series (i. ... Analytic number theory is the branch of number theory that uses methods from mathematical analysis. ...

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. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In algebraic topology, the Betti number of a topological space is, in intuitive terms, a way of counting the maximum number of cuts that can be made without dividing the space into two pieces. ... In mathematics, the Lefschetz fixed-point theorem counts the number of fixed points of a mapping from a topological space X to itself (subject to some mild conditions on X), by means of traces of the induced mappings on the homology groups of X. The counting is subject to some...

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 1960, using p-adic methods. The rest awaited the construction of étale cohomology, a theory whose very definition lies quite deep. The proofs were completed by Pierre Deligne in 1974, roughly speaking using a painstaking induction argument on dimension. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Alexander Grothendieck (Berlin, March 28, 1928) is one of the most important mathematicians of the 20th century. ... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... Bernard Morris Dwork (May 27, 1923 – May 9, 1998) was an American mathematician, known for his application of p-adic analysis to local zeta functions, and in particular for the first general results on the Weil conjectures. ... The p-adic number systems were first described by Kurt Hensel in 1897. ... In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ... Pierre Deligne, March 2005 Pierre Deligne (born 3 October 1944) is a Belgian mathematician. ...

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. The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary number theory. ...

Statement 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 In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ... This article is about algebraic varieties. ...

where N_m is the number of points of X defined over the field of order q^m.

The Weil conjectures state:

1. ζ(X, s) is a rational function of T=q^−s. More precisely, ζ(X, s) can be written as a finite product ∏ (−1)ⁱP_i(q^−s) where each P_i(T) is some integral polynomial, of the form ∏(1-α_i,jT). (Rationality).
2. ζ(X, s)=ζ(X, n−s), or equivalently, by arranging the notation, the map taking α to qⁿ/α takes the numbers α_i,j to the numbers α_2n-i,j. (Functional equation or Poincaré duality.)
3. |α_i,j| = q^i/2, by arranging notation. This is the analogue of the classical Riemann hypothesis and proved to be the hardest part of the conjectures. It can be rephrased as saying that all zeros of P_i(q^−s) lie on the "critical line" of complex numbers s with real part i/2.
4. If X is the "reduction mod p" of a nonsingular complex projective variety Y, then the degree of P_i is the ith Betti number of Y.

In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... This is a glossary of arithmetic and Diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. ... In algebraic topology, the Betti number of a topological space is, in intuitive terms, a way of counting the maximum number of cuts that can be made without dividing the space into two pieces. ...

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 is just N_m = q^m + 1 (where the "+ 1" comes from the "point at infinity"). The zeta function is just The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line, . Nota Bene: The real projective line is not equivalent to the extended real number line. ...

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. A rendering of the Riemann Sphere. ...

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 is just N_m = 1 + q^m + q^2m + ... + 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.) In mathematics, complex projective space, or CPn, is the projective space of (complex) lines in Cn+1. ...

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 −αq^−s)(1 −βq^−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. In algebraic geometry, a motive (or sometimes motif) refers to some essential part of an algebraic variety. Mathematically, the theory of motives is then the conjectural universal cohomology theory for such objects. ... In mathematics, the Frobenius automorphism is an automorphism induced by a prime power mapping defined for various extensions of fields. ... In mathematics, the Lefschetz fixed-point theorem counts the number of fixed points of a mapping from a topological space X to itself (subject to some mild conditions on X), by means of traces of the induced mappings on the homology groups of X. The counting is subject to some... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ...

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. In mathematics, the Hasse-Witt matrix H of a non-singular algebraic curve C over a finite field F is the matrix of the Frobenius mapping (p-th power mapping where F has q elements, q a power of the prime number p) with respect to a basis for the... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... In mathematics, a quaternion algebra over a field L is a particular kind of central simple algebra A over L, namely such an algebra that has dimension 4, and therefore becomes the 2×2 matrix algebra over some field extension of L, by extending scalars. ... Michael Artin Michael Artin (born 1934) is an American mathematician and a professor at MIT, known for his contributions to algebraic geometry. ... In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ...

References

• Weil, André Numbers of solutions of equations in finite fields. Bull. Amer. Math. Soc. 55, (1949). 497--508. Reprinted in Oeuvres Scientifiques/Collected Papers by Andre Weil ISBN 0-387-90330-5
• Deligne, Pierre La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math. No. 43 (1974), 273--307. 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) [Results in Mathematics and Related Areas (3)], 13. Springer-Verlag, Berlin, 1988. ISBN 0-387-12175-7
• 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.