In mathematics, the Weil conjectures were highly influential proposals by André Weil (1949). They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. The conjectures concern the generating functions (known as local zeta functions) derived from counting points on algebraic varieties over finite fields. A variety V over a finite field with q elements has a finite number of rational points (with coordinates in the original field), as well as points with coordinates in any finite extension of the original field. The generating function has coefficients derived from the numbers Nk of points over the extension field with qk elements. Weil conjectured that such zeta functions for smooth varieties are rational functions, satisfy a certain functional equation, and have their zeros in restricted places. The last two parts were consciously modelled on the Riemann zeta function, a kind of generating function for prime integers, which obeys a functional equation and (conjecturally) has its zeros restricted by the Riemann hypothesis. The rationality was proved by Bernard Dwork (1960), the functional equation by Alexander Grothendieck (1965), and the analogue of the Riemann hypothesis by Pierre Deligne (1974).
Background and history The earliest antecedent of the Weil conjectures is by Carl Friedrich Gauss and appears in section VII of his Disquisitiones Arithmeticae (Mazur 1974), concerned with roots of unity and Gaussian periods. In article 358, he moves on from the periods that build up towers of quadratic extensions, for the construction of regular polygons; and assumes that p is a prime number congruent to 1 modulo 3. Then there is a cyclic cubic field inside the cyclotomic field of pth roots of unity, and a normal integral basis of periods for the integers of this field (an instance of the Hilbert–Speiser theorem). Gauss constructs the order-3 periods, corresponding to the cyclic group (Z/pZ)× of non-zero residues modulo p under multiplication and its unique subgroup of index three. Gauss lets R {\displaystyle {\mathfrak {R}}} , R ′ {\displaystyle {\mathfrak {R}}'} , and R ″ {\displaystyle {\mathfrak {R}}''} be its cosets. Taking the periods (sums of roots of unity) corresponding to these cosets applied to exp(2πi/p), he notes that these periods have a multiplication table that is accessible to calculation. Products are linear combinations of the periods, and he determines the coefficients. He sets, for example, ( R R ) {\displaystyle ({\mathfrak {R}}{\mathfrak {R}})} equal to the number of elements of Z/pZ which are in R {\displaystyle {\mathfrak {R}}} and which, after being increased by one, are also in R {\displaystyle {\mathfrak {R}}} . He proves that this number and related ones are the coefficients of the products of the periods. To see the relation of these sets to the Weil conjectures, notice that if α and α + 1 are both in R {\displaystyle {\mathfrak {R}}} , then there exist x and y in Z/pZ such that x3 = α and y3 = α + 1; consequently, x3 + 1 = y3. Therefore ( R R ) {\displaystyle ({\mathfrak {R}}{\mathfrak {R}})} is related to the number of solutions to x3 + 1 = y3 in the finite field Z/pZ. The other coefficients have similar interpretations. Gauss's determination of the coefficients of the products of the periods therefore counts the number of points on these elliptic curves, and as a byproduct he proves the analog of the Riemann hypothesis. The Weil conjectures in the special case of algebraic curves were conjectured by Emil Artin (1924). The case of curves over finite fields was proved by Weil, finishing the project started by Hasse's theorem on elliptic curves over finite fields. Their interest was obvious enough from within number theory: they implied upper bounds for exponential sums, a basic concern in analytic number theory (Moreno 2001). 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. 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. The rationality part of the conjectures was proved first by Bernard Dwork (1960), using p-adic methods. Grothendieck (1965) and his collaborators established the rationality conjecture, the functional equation and the link to Betti numbers by using the properties of étale cohomology, a new cohomology theory developed by Grothendieck and Michael Artin for attacking the Weil conjectures, as outlined in Grothendieck (1960). Of the four conjectures, the analogue of the Riemann hypothesis was the hardest to prove. Motivated by the proof of Serre (1960) of an analogue of the Weil conjectures for Kähler manifolds, Grothendieck envisioned a proof based on his standard conjectures on algebraic cycles (Kleiman 1968). However, Grothendieck's standard conjectures remain open (except for the hard Lefschetz theorem, which was proved by Deligne by extending his work on the Weil conjectures), and the analogue of the Riemann hypothesis was proved by Deligne (1974), using the étale cohomology theory but circumventing the use of standard conjectures by an ingenious argument. Deligne (1980) found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf.
Statement of the Weil conjectures Suppose that X is a non-singular n-dimensional projective algebraic variety over the field Fq with q elements. The zeta function ζ(X, s) of X is by definition
ζ ( X , s ) = exp ( ∑ m = 1 ∞ N m m q − m s ) {\displaystyle \zeta (X,s)=\exp \left(\sum _{m=1}^{\infty }{\frac {N_{m}}{m}}q^{-ms}\right)}
where Nm is the number of points of X defined over the degree m extension Fqm of Fq. The Weil conjectures state:
1. (Rationality) 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 = 0 2 n P i ( q − s ) ( − 1 ) i + 1 = P 1 ( T ) ⋯ P 2 n − 1 ( T ) P 0 ( T ) ⋯ P 2 n ( T ) , {\displaystyle \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)}},}
where each Pi(T) is an integral polynomial. Furthermore, P0(T) = 1 − T, P2n(T) = 1 − qnT, and for 1 ≤ i ≤ 2n − 1, Pi(T) factors over C as ∏ j ( 1 − α i j T ) {\displaystyle \textstyle \prod _{j}(1-\alpha _{ij}T)} for some numbers αij. 2. (Functional equation and Poincaré duality) The zeta function satisfies
ζ ( X , n − s ) = ± q n E / 2 − E s ζ ( X , s ) {\displaystyle \zeta (X,n-s)=\pm q^{nE/2-Es}\zeta (X,s)}
or equivalently
ζ ( X , q − n T − 1 ) = ± q n E / 2 T E ζ ( X , T ) {\displaystyle \zeta (X,q^{-n}T^{-1})=\pm q^{nE/2}T^{E}\zeta (X,T)}
where E is the Euler characteristic of X. In particular, for each i, the numbers α2n−i,1, α2n−i,2, ... equal the numbers qn/αi,1, qn/αi,2, ... in some order. 3. (Riemann hypothesis) |αi,j| = qi/2 for all 1 ≤ i ≤ 2n − 1 and all j. This implies that all zeros of Pk(T) lie on the "critical line" of complex numbers s with real part k/2. 4. (Betti numbers) 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 Pi is the ith 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 qm elements is just Nm = qm + 1 (where the "+ 1" comes from the "point at infinity"). The zeta function is just
1 ( 1 − q − s ) ( 1 − q 1 − s ) . {\displaystyle {\frac {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 qm elements is just Nm = 1 + qm + q2m + ⋯ + qnm. The zeta function is just
1 ( 1 − q − s ) ( 1 − q 1 − s ) … ( 1 − q n − s ) . {\displaystyle {\frac {1}{(1-q^{-s})(1-q^{1-s})\dots (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 number of points on the projective line and projective space are so easy to calculate because they can be written as disjoint unions of a finite number of copies of affine spaces. It is also easy to prove the Weil conjectures for other spaces, such as Grassmannians and flag varieties, which have the same "paving" 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 qm elements is 1 − αm − βm + qm, where α and β are complex conjugates with absolute value √q. The zeta function is
( 1 − α q − s ) ( 1 − β q − s ) ( 1 − q − s ) ( 1 − q 1 − s ) . {\displaystyle {\frac {(1-\alpha q^{-s})(1-\beta q^{-s})}{(1-q^{-s})(1-q^{1-s})}}.}
The Betti numbers are given by the torus, 1,2,1, and the numerator is a quadratic.
Hyperelliptic curves As an example, consider the hyperelliptic curve
C : y 2 + y = x 5 , {\displaystyle C:y^{2}+y=x^{5},}
which is of genus g = 2 {\displaystyle g=2} and dimension n = 1 {\displaystyle n=1} . At first viewed as a curve C / Q {\displaystyle C/\mathbb {Q} } defined over the rational numbers Q {\displaystyle \mathbb {Q} } , this curve has good reduction at all primes 5 ≠ q ∈ P {\displaystyle 5\neq q\in \mathbb {P} } . So, after reduction modulo q ≠ 5 {\displaystyle q\neq 5} , one obtains a hyperelliptic curve C / F q : y 2 + h ( x ) y = f ( x ) {\displaystyle C/{\bf {F}}_{q}:y^{2}+h(x)y=f(x)} of genus 2, with h ( x ) = 1 , f ( x ) = x 5 ∈ F q [ x ] {\displaystyle h(x)=1,f(x)=x^{5}\in {\bf {F}}_{q}[x]} . Taking q = 41 {\displaystyle q=41} as an example, the Weil polynomials P i ( T ) {\displaystyle P_{i}(T)} , i = 0 , 1 , 2 , {\displaystyle i=0,1,2,} and the zeta function of C / F 41 {\displaystyle C/{\bf {F}}_{41}} assume the form
ζ ( C / F 41 , s ) = P 1 ( T ) P 0 ( T ) ⋅ P 2 ( T ) = 1 − 9 ⋅ T + 71 ⋅ T 2 − 9 ⋅ 41 ⋅ T 3 + 41 2 ⋅ T 4 ( 1 − T ) ( 1 − 41 ⋅ T ) . {\displaystyle \zeta (C/{\bf {F}}_{41},s)={\frac {P_{1}(T)}{P_{0}(T)\cdot P_{2}(T)}}={\frac {1-9\cdot T+71\cdot T^{2}-9\cdot 41\cdot T^{3}+41^{2}\cdot T^{4}}{(1-T)(1-41\cdot T)}}.}
The values c 1 = − 9 {\displaystyle c_{1}=-9} and c 2 = 71 {\displaystyle c_{2}=71} can be determined by counting the numbers of solutions ( x , y ) {\displaystyle (x,y)} of y 2 + y = x 5 {\displaystyle y^{2}+y=x^{5}} over F 41 {\displaystyle {\bf {F}}_{41}} and F 41 2 {\displaystyle {\bf {F}}_{41^{2}}} , respectively, and adding 1 to each of these two numbers to allow for the point at infinity ∞ {\displaystyle \infty } . This counting yields N 1 = 33 {\displaystyle N_{1}=33} and N 2 = 1743 {\displaystyle N_{2}=1743} . It follows:
c 1 = N 1 − 1 − q = 33 − 1 − 41 = − 9 {\displaystyle c_{1}=N_{1}-1-q=33-1-41=-9} and
c 2 = ( N 2 − 1 − q 2 + c 1 2 ) / 2 = ( 1743 − 1 − 41 2 + ( − 9 ) 2 ) / 2 = 71. {\displaystyle c_{2}=(N_{2}-1-q^{2}+c_{1}^{2})/2=(1743-1-41^{2}+(-9)^{2})/2=71.}
The zeros of P 1 ( T ) {\displaystyle P_{1}(T)} are z 1 := 0.12305 + − 1 ⋅ 0.09617 {\displaystyle z_{1}:=0.12305+{\sqrt {-1}}\cdot 0.09617} and z 2 := − 0.01329 + − 1 ⋅ 0.15560 {\displaystyle z_{2}:=-0.01329+{\sqrt {-1}}\cdot 0.15560} (the decimal expansions of these real and imaginary parts are cut off after the fifth decimal place) together with their complex conjugates z 3 := z ¯ 1 {\displaystyle z_{3}:={\bar {z}}_{1}} and z 4 := z ¯ 2 {\displaystyle z_{4}:={\bar {z}}_{2}} . So, in the factorisation P 1 ( T ) = ∏ j = 1 4 ( 1 − α 1 , j T ) {\displaystyle P_{1}(T)=\prod _{j=1}^{4}(1-\alpha _{1,j}T)} , we have α 1 , j = 1 / z j {\displaystyle \alpha _{1,j}=1/z_{j}} . As stated in the third part (Riemann hypothesis) of the Weil conjectures, | α 1 , j | = 41 {\displaystyle |\alpha _{1,j}|={\sqrt {41}}} for j = 1 , 2 , 3 , 4 {\displaystyle j=1,2,3,4} . The non-singular, projective, complex manifold that belongs to C / Q {\displaystyle C/\mathbb {Q} } has the Betti numbers B 0 = 1 , B 1 = 2 g = 4 , B 2 = 1 {\displaystyle B_{0}=1,B_{1}=2g=4,B_{2}=1} . As described in part four of the Weil conjectures, the (topologically defined!) Betti numbers coincide with the degrees of the Weil polynomials P i ( T ) {\displaystyle P_{i}(T)} , for all primes q ≠ 5 {\displaystyle q\neq 5} : d e g ( P i ) = B i , i = 0 , 1 , 2 {\displaystyle {\rm {deg}}(P_{i})=B_{i},\,i=0,1,2} .
Abelian surfaces An Abelian surface is a two-dimensional Abelian variety. This is, they are projective varieties that also have the structure of a group, in a way that is compatible with the group composition and taking inverses. Elliptic curves represent one-dimensional Abelian varieties. As an example of an Abelian surface defined over a finite field, consider the Jacobian variety X := Jac ( C / F 41 ) {\displaystyle X:={\text{Jac}}(C/{\bf {F}}_{41})} of the genus 2 curve
C / F 41 : y 2 + y = x 5 , {\displaystyle C/{\bf {F}}_{41}:y^{2}+y=x^{5},}
which was introduced in the section on hyperelliptic curves. The dimension of X {\displaystyle X} equals the genus of C {\displaystyle C} , so n = 2 {\displaystyle n=2} . There are algebraic integers α 1 , … , α 4 {\displaystyle \alpha _{1},\ldots ,\alpha _{4}} such that
the polynomial P ( x ) = ∏ j = 1 4 ( x − α j ) {\displaystyle P(x)=\prod _{j=1}^{4}(x-\alpha _{j})} has coefficients in Z {\displaystyle \mathbb {Z} } ;
M m := | Jac ( C / F 41 m ) | = ∏ j = 1 4 ( 1 − α j m ) {\displaystyle M_{m}:=|{\text{Jac}}(C/{\bf {F}}_{41^{m}})|=\prod _{j=1}^{4}(1-\alpha _{j}^{m})} for all m ∈ N {\displaystyle m\in \mathbb {N} } ; and
| α j | = 41 {\displaystyle |\alpha _{j}|={\sqrt {41}}} for j = 1 , … , 4 {\displaystyle j=1,\ldots ,4} . The zeta-function of X {\displaystyle X} is given by
ζ ( X , s ) = ∏ i = 0 4 P i ( q − s ) ( − 1 ) i + 1 = P 1 ( T ) ⋅ P 3 ( T ) P 0 ( T ) ⋅ P 2 ( T ) ⋅ P 4 ( T ) , {\displaystyle \zeta (X,s)=\prod _{i=0}^{4}P_{i}(q^{-s})^{(-1)^{i+1}}={\frac {P_{1}(T)\cdot P_{3}(T)}{P_{0}(T)\cdot P_{2}(T)\cdot P_{4}(T)}},}
where q = 41 {\displaystyle q=41} , T = q − s = d e f exp ( − s ⋅ log ( 41 ) ) {\displaystyle T=q^{-s}\,{\stackrel {\rm {def}}{=}}\,{\text{exp}}(-s\cdot {\text{log}}(41))} , and s {\displaystyle s} represents the complex variable of the zeta-function. The Weil polynomials P i ( T ) {\displaystyle P_{i}(T)} have the following specific form (Kahn 2020):
P i ( T ) = ∏ 1 ≤ j 1 < j 2 < … < j i − 1 < j i ≤ 4 ( 1 − α j 1 ⋅ … ⋅ α j i T ) {\displaystyle P_{i}(T)=\prod _{1\leq j_{1}<j_{2}<\ldots <j_{i-1}<j_{i}\leq 4}(1-\alpha _{j_{1}}\cdot \ldots \cdot \alpha _{j_{i}}T)}
for i = 0 , 1 , … , 4 {\displaystyle i=0,1,\ldots ,4} , and
P 1 ( T ) = ∏ j = 1 4 ( 1 − α j T ) = 1 − 9 ⋅ T + 71 ⋅ T 2 − 9 ⋅ 41 ⋅ T 3 + 41 2 ⋅ T 4 {\displaystyle P_{1}(T)=\prod _{j=1}^{4}(1-\alpha _{j}T)=1-9\cdot T+71\cdot T^{2}-9\cdot 41\cdot T^{3}+41^{2}\cdot T^{4}}
is the same for the curve C {\displaystyle C} (see section above) and its Jacobian variety X {\displaystyle X} . This is, the inverse roots of P i ( T ) {\displaystyle P_{i}(T)} are the products α j 1 ⋅ … ⋅ α j i {\displaystyle \alpha _{j_{1}}\cdot \ldots \cdot \alpha _{j_{i}}} that consist of i {\displaystyle i} many, different inverse roots of P 1 ( T ) {\displaystyle P_{1}(T)} . Hence, all coefficients of the polynomials P i ( T ) {\displaystyle P_{i}(T)} can be expressed as polynomial functions of the parameters c 1 = − 9 {\displaystyle c_{1}=-9} , c 2 = 71 {\displaystyle c_{2}=71} and q = 41 {\displaystyle q=41} appearing in P 1 ( T ) = 1 + c 1 T + c 2 T 2 + q c 1 T 3 + q 2 T 4 . {\displaystyle P_{1}(T)=1+c_{1}T+c_{2}T^{2}+qc_{1}T^{3}+q^{2}T^{4}.} Calculating these polynomial functions for the coefficients of the P i ( T ) {\displaystyle P_{i}(T)} shows that
P 0 ( T ) = 1 − T P 1 ( T ) = 1 − 3 2 ⋅ T + 71 ⋅