Weil conjectures

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In mathematics, the Weil conjectures were highly influential proposals by Template:Harvs. 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 Template:Mvar over a finite field with Template:Mvar 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 Template:Math of points over the extension field with Template:Math 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 Template:Harvs, the functional equation by Template:Harvs, and the analogue of the Riemann hypothesis by Template:Harvs.

Background and historyEdit

The earliest antecedent of the Weil conjectures is by Carl Friedrich Gauss and appears in section VII of his Disquisitiones Arithmeticae Template:Harv, 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 Template:Mvar is a prime number congruent to 1 modulo 3. Then there is a cyclic cubic field inside the cyclotomic field of Template:Mvarth 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 Template:Math of non-zero residues modulo Template:Mvar under multiplication and its unique subgroup of index three. Gauss lets <math>\mathfrak{R}</math>, <math>\mathfrak{R}'</math>, and <math>\mathfrak{R}</math> be its cosets. Taking the periods (sums of roots of unity) corresponding to these cosets applied to Template:Math, 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, <math>(\mathfrak{R}\mathfrak{R})</math> equal to the number of elements of Template:Math which are in <math>\mathfrak{R}</math> and which, after being increased by one, are also in <math>\mathfrak{R}</math>. 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 Template:Mvar and Template:Math are both in <math>\mathfrak{R}</math>, then there exist Template:Mvar and Template:Mvar in Template:Math such that Template:Math and Template:Math; consequently, Template:Math. Therefore <math>(\mathfrak{R}\mathfrak{R})</math> is related to the number of solutions to Template:Math in the finite field Template:Math. 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 Template:Harvs. 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 Template:Harv.

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 Template:Harvs, using [[p-adic number|Template:Mvar-adic]] methods. Template:Harvtxt 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 Template:Harvtxt. Of the four conjectures, the analogue of the Riemann hypothesis was the hardest to prove. Motivated by the proof of Template:Harvtxt of an analogue of the Weil conjectures for Kähler manifolds, Grothendieck envisioned a proof based on his standard conjectures on algebraic cycles Template:Harv. 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 Template:Harvs, using the étale cohomology theory but circumventing the use of standard conjectures by an ingenious argument.

Template:Harvtxt found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf.

Statement of the Weil conjecturesEdit

Suppose that Template:Mvar is a non-singular Template:Mvar-dimensional projective algebraic variety over the field Template:Math with Template:Mvar elements. The zeta function Template:Math of Template:Mvar is by definition

<math>\zeta(X, s) = \exp\left(\sum_{m = 1}^\infty \frac{N_m}{m} q^{-ms}\right)</math>

where Template:Math is the number of points of Template:Mvar defined over the degree Template:Mvar extension Template:Math of Template:Math.

The Weil conjectures state:

1. (Rationality) Template:Math is a rational function of Template:Math. More precisely, Template:Math can be written as a finite alternating product

<math>\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)},</math>

where each Template:Math is an integral polynomial. Furthermore, Template:Math, Template:Math, and for Template:Math, Template:Math factors over Template:Math as <math>\textstyle\prod_j (1 - \alpha_{ij}T)</math> for some numbers Template:Math.

2. (Functional equation and Poincaré duality) The zeta function satisfies

or equivalently

where Template:Mvar is the Euler characteristic of Template:Mvar. In particular, for each Template:Mvar, the numbers Template:Math, Template:Math, ... equal the numbers Template:Math, Template:Math, ... in some order.

3. (Riemann hypothesis) Template:Math for all Template:Math and all Template:Mvar. This implies that all zeros of Template:Math lie on the "critical line" of complex numbers Template:Mvar with real part Template:Math.

4. (Betti numbers) If Template:Mvar is a (good) "[[reduction mod p|reduction mod Template:Mvar]]" of a non-singular projective variety Template:Mvar defined over a number field embedded in the field of complex numbers, then the degree of Template:Math is the Template:Mvar^th Betti number of the space of complex points of Template:Mvar.

ExamplesEdit

The projective lineEdit

The simplest example (other than a point) is to take Template:Mvar to be the projective line. The number of points of Template:Mvar over a field with Template:Math elements is just Template:Math (where the "Template:Math" comes from the "point at infinity"). The zeta function is just

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 spaceEdit

It is not much harder to do Template:Mvar-dimensional projective space. The number of points of Template:Mvar over a field with Template:Math elements is just Template:Math. The zeta function is just

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 curvesEdit

These give the first non-trivial cases of the Weil conjectures (proved by Hasse). If Template:Mvar is an elliptic curve over a finite field with Template:Mvar elements, then the number of points of Template:Mvar defined over the field with Template:Math elements is Template:Math, where Template:Mvar and Template:Mvar are complex conjugates with absolute value Template:Math. The zeta function is

<math>\frac{(1-\alpha q^{-s})(1-\beta q^{-s})}{(1-q^{-s})(1-q^{1-s})}. </math>

The Betti numbers are given by the torus, 1,2,1, and the numerator is a quadratic.

Hyperelliptic curvesEdit

As an example, consider the hyperelliptic curve<ref>LMFDB: Genus 2 curve 3125.a.3125.1</ref>

which is of genus <math>g=2</math> and dimension <math>n=1</math>. At first viewed as a curve <math>C/\mathbb{Q}</math> defined over the rational numbers <math>\mathbb{Q}</math>, this curve has good reduction at all primes <math>5\ne q\in\mathbb{P}</math>. So, after reduction modulo <math>q\ne 5</math>, one obtains a hyperelliptic curve <math>C/{\bf F}_q: y^2 + h(x)y=f(x)</math> of genus 2, with <math>h(x)=1, f(x)=x^5\in {\bf F}_q[x]</math>. Taking <math>q=41</math> as an example, the Weil polynomials <math>P_i(T)</math>, <math>i=0,1,2,</math> and the zeta function of <math>C/{\bf F}_{41}</math> assume the form

The values <math>c_1=-9</math> and <math>c_2=71</math> can be determined by counting the numbers of solutions <math>(x,y)</math> of <math>y^2 + y=x^5</math> over <math>{\bf F}_{41}</math> and <math>{\bf F}_{41^2}</math>, respectively, and adding 1 to each of these two numbers to allow for the point at infinity <math>\infty</math>. This counting yields <math>N_1=33</math> and <math>N_2=1743</math>. It follows:<ref>Chapter 6, Theorem 5.1 in Template:Cite book</ref>

<math>c_1=N_1-1-q=33-1-41=-9</math> and

The zeros of <math>P_1(T)</math> are <math>z_1:=0.12305+\sqrt{-1}\cdot 0.09617</math> and <math>z_2:=-0.01329+\sqrt{-1}\cdot 0.15560</math> (the decimal expansions of these real and imaginary parts are cut off after the fifth decimal place) together with their complex conjugates <math>z_3:=\bar z_1</math> and <math>z_4:=\bar z_2</math>. So, in the factorisation <math>P_1(T)=\prod_{j=1}^4 (1-\alpha_{1,j} T)</math>, we have <math>\alpha_{1,j}=1/z_j</math> . As stated in the third part (Riemann hypothesis) of the Weil conjectures, <math>|\alpha_{1,j}|=\sqrt{41}</math> for <math>j=1,2,3,4</math>.

The non-singular, projective, complex manifold that belongs to <math>C/\mathbb{Q}</math> has the Betti numbers <math>B_0=1, B_1=2g=4, B_2=1</math>.<ref>Chapter 7, Paragraph §7B in Template:Cite book</ref> As described in part four of the Weil conjectures, the (topologically defined!) Betti numbers coincide with the degrees of the Weil polynomials <math>P_i(T)</math>, for all primes <math>q\ne 5</math>: <math>{\rm deg}(P_i)=B_i,\,i=0,1,2</math>.

Abelian surfacesEdit

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 <math>X:=\text{Jac}(C/{\bf F}_{41})</math> of the genus 2 curve <ref>LMFDB: Abelian variety isogeny class 2.41.aj_ct over F(41)</ref>

which was introduced in the section on hyperelliptic curves. The dimension of <math>X</math> equals the genus of <math>C</math>, so <math>n=2</math>. There are algebraic integers <math>\alpha_1,\ldots,\alpha_4</math> such that<ref>Chapter V, Theorem 19.1 in Template:Cite book</ref>

the polynomial <math>P(x)=\prod_{j=1}^4 (x-\alpha_j)</math> has coefficients in <math>\mathbb{Z}</math>;
<math>M_m:=|\text{Jac}(C/{\bf F}_{41^m})|=\prod_{j=1}^4 (1-\alpha_j^m)</math> for all <math>m\in\mathbb{N}</math>; and
<math>|\alpha_j|=\sqrt{41}</math> for <math>j=1,\ldots,4</math>.

The zeta-function of <math>X</math> is given by

<math>\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)},

</math> where <math>q=41</math>, <math>T=q^{-s}\,\stackrel{\rm def}{=}\,\text{exp}(-s\cdot\text{log}(41))</math>, and <math>s</math> represents the complex variable of the zeta-function. The Weil polynomials <math>P_i(T)</math> have the following specific form Template:Harv:

for <math>i=0,1,\ldots,4</math>, and

<math>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</math>

is the same for the curve <math>C</math> (see section above) and its Jacobian variety <math>X</math>. This is, the inverse roots of <math>P_i(T)</math> are the products <math>\alpha_{j_1}\cdot\ldots\cdot\alpha_{j_i} </math> that consist of <math>i</math> many, different inverse roots of <math>P_1(T)</math>. Hence, all coefficients of the polynomials <math>P_i(T)</math> can be expressed as polynomial functions of the parameters <math>c_1=-9</math>, <math>c_2=71</math> and <math>q=41</math> appearing in <math>P_1(T)=1+c_1 T + c_2 T^2 + q c_1 T^3 + q^2 T^4.</math> Calculating these polynomial functions for the coefficients of the <math>P_i(T)</math> shows that

<math>\begin{alignat}{2}

P_0(T) &= 1 - T\\ P_1(T) &= 1 - 3^2\cdot T + 71\cdot T^2 - 3^2\cdot 41\cdot T^3 + 41^2\cdot T^4\\ P_2(T) &=(1 - 41\cdot T)^2 \cdot (1 + 11\cdot T + 3\cdot 7\cdot 41\cdot T^2 + 11\cdot 41^2\cdot T^3 + 41^4\cdot T^4)\\ P_3(T) &=1 - 3^2\cdot 41\cdot T + 71\cdot 41^2\cdot T^2 - 3^2\cdot 41^4\cdot T^3 + 41^6\cdot T^4\\ P_4(T) &=1 - 41^2\cdot T \end{alignat}</math> Polynomial <math>P_1</math> allows for calculating the numbers of elements of the Jacobian variety <math>\text{Jac}(C)</math> over the finite field <math>{\bf F}_{41}</math> and its field extension <math>{\bf F}_{41^2}</math>:<ref>Chapter 6, Theorem 5.1 in Template:Cite book</ref><ref>LMFDB: Abelian variety isogeny class 2.41.aj_ct over F(41)</ref>

<math>\begin{alignat}{2}

M_1 &\;\overset{\underset{\mathrm{def}}{}}{=}\; |\text{Jac}(C/{\bf F}_{41})|=P_1(1)=\prod_{j=1}^4 [1-\alpha_j T]_{T=1}\\ &= [1 - 9\cdot T + 71\cdot T^2 - 9\cdot 41\cdot T^3 + 41^2\cdot T^4]_{T=1} = 1 - 9 + 71 - 9\cdot 41 + 41^2=1375=5^3\cdot 11\text{, and}\\ M_2 &\;\overset{\underset{\mathrm{def}}{}}{=}\; |\text{Jac}(C/{\bf F}_{41^2})|=\prod_{j=1}^4 [1-\alpha_j^2 T]_{T=1}\\ &= [1 + 61\cdot T + 3\cdot 587\cdot T^2 + 61\cdot 41^2\cdot T^3 + 41^4\cdot T^4]_{T=1} = 2930125 = 5^3\cdot 11\cdot 2131. \end{alignat}</math> The inverses <math>\alpha_{i,j}</math> of the zeros of <math>P_i(T)</math> do have the expected absolute value of <math>41^{i/2}</math> (Riemann hypothesis). Moreover, the maps <math>\alpha_{i,j}\longmapsto 41^2/\alpha_{i,j},</math> <math>j=1,\ldots,\deg P_i,</math> correlate the inverses of the zeros of <math>P_i(T)</math> and the inverses of the zeros of <math>P_{4-i}(T)</math>. A non-singular, complex, projective, algebraic variety <math>Y</math> with good reduction at the prime 41 to <math>X=\text{Jac}(C/{\bf F}_{41})</math> must have Betti numbers <math>B_0=B_4=1, B_1=B_3=4, B_2=6</math>, since these are the degrees of the polynomials <math>P_i(T).</math> The Euler characteristic <math>E</math> of <math>X</math> is given by the alternating sum of these degrees/Betti numbers: <math>E=1-4+6-4+1=0</math>.

By taking the logarithm of

<math>

\zeta(\text{Jac}(C/{\bf F}_{41}), s)\,=\, \exp\left(\sum_{m = 1}^\infty \frac{M_m}{m} (41^{-s})^m\right)\,=\,\prod_{i=0}^{4} \, P_i (41^{-s})^{(-1)^{i+1}} =\frac{P_1(T)\cdot P_3(T)}{P_0(T)\cdot P_2(T)\cdot P_4(T)}, </math> it follows that

<math>\begin{alignat}{2}

\sum_{m = 1}^\infty & \frac{M_m}{m} (41^{-s})^m\,=\,\log\left(\frac{P_1(T)\cdot P_3(T)}{P_0(T)\cdot P_2(T)\cdot P_4(T)}\right)\\ &=1375\cdot T + 2930125/2\cdot T^2 + 4755796375/3\cdot T^3 + 7984359145125/4 \cdot T^4 + 13426146538750000/5\cdot T^5 + O(T^6). \end{alignat}</math>

Aside from the values <math>M_1</math> and <math>M_2</math> already known, you can read off from this Taylor series all other numbers <math>M_m</math>, <math>m\in\mathbb{N}</math>, of <math>{\bf F}_{41^m}</math>-rational elements of the Jacobian variety, defined over <math>{\bf F}_{41}</math>, of the curve <math>C/{\bf F}_{41}</math>: for instance, <math>M_3=4755796375=5^3\cdot 11\cdot 61\cdot 56701</math> and <math>M_4=7984359145125=3^4\cdot 5^3\cdot 11\cdot 2131\cdot 33641</math>. In doing so, <math>m_1|m_2</math> always implies <math>M_{m_1}|M_{m_2}</math> since then, <math>\text{Jac}(C/{\bf F}_{41^{m_1}})</math> is a subgroup of <math>\text{Jac}(C/{\bf F}_{41^{m_2}})</math>.

Weil cohomologyEdit

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 Template:Mvar is the Frobenius automorphism over the finite field, then the number of points of the variety Template:Mvar over the field of order Template:Math is the number of fixed points of Template:Math (acting on all points of the variety Template:Mvar 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 Template:Mvar. The endomorphism ring of this is an order in a quaternion algebra over the rationals, and should act on the first cohomology group, which should be 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 Template:Mvar-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 Template:Mvar-adic numbers for some prime Template:Math, 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 Template:Mvar-adic numbers for each prime Template:Math, called [[l-adic cohomology|Template:Mvar-adic cohomology]].

Grothendieck's proofs of three of the four conjecturesEdit

By the end of 1964 Grothendieck together with Artin and Jean-Louis Verdier (and the earlier 1960 work by Dwork) proved the Weil conjectures apart from the most difficult third conjecture above (the "Riemann hypothesis" conjecture) (Grothendieck 1965). The general theorems about étale cohomology allowed Grothendieck to prove an analogue of the Lefschetz fixed-point formula for the ℓ-adic cohomology theory, and by applying it to the Frobenius automorphism F he was able to prove the conjectured formula for the zeta function:

<math>\zeta(s)=\frac{P_1(T)\cdots P_{2n-1}(T)}{P_0(T)P_2(T)\cdots P_{2n}(T)}</math>

where each polynomial P_i is the determinant of I − TF on the ℓ-adic cohomology group Hⁱ.

The rationality of the zeta function follows immediately. The functional equation for the zeta function follows from Poincaré duality for ℓ-adic cohomology, and the relation with complex Betti numbers of a lift follows from a comparison theorem between ℓ-adic and ordinary cohomology for complex varieties.

More generally, Grothendieck proved a similar formula for the zeta function (or "generalized L-function") of a sheaf F₀:

as a product over cohomology groups:

The special case of the constant sheaf gives the usual zeta function.

Deligne's first proof of the Riemann hypothesis conjectureEdit

Template:Harvtxt, Template:Harvtxt, Template:Harvtxt and Template:Harvtxt gave expository accounts of the first proof of Template:Harvtxt. Much of the background in ℓ-adic cohomology is described in Template:Harv.

Deligne's first proof of the remaining third Weil conjecture (the "Riemann hypothesis conjecture") used the following steps:

Use of Lefschetz pencilsEdit

Grothendieck expressed the zeta function in terms of the trace of Frobenius on ℓ-adic cohomology groups, so the Weil conjectures for a d-dimensional variety V over a finite field with q elements depend on showing that the eigenvalues α of Frobenius acting on the ith ℓ-adic cohomology group Hⁱ(V) of V have absolute values Template:Abs = q^i/2 (for an embedding of the algebraic elements of Q_ℓ into the complex numbers).
After blowing up V and extending the base field, one may assume that the variety V has a morphism onto the projective line P¹, with a finite number of singular fibers with very mild (quadratic) singularities. The theory of monodromy of Lefschetz pencils, introduced for complex varieties (and ordinary cohomology) by Template:Harvtxt, and extended by Template:Harvtxt and Template:Harvtxt to ℓ-adic cohomology, relates the cohomology of V to that of its fibers. The relation depends on the space E_x of vanishing cycles, the subspace of the cohomology H^d−1(V_x) of a non-singular fiber V_x, spanned by classes that vanish on singular fibers.
The Leray spectral sequence relates the middle cohomology group of V to the cohomology of the fiber and base. The hard part to deal with is more or less a group H¹(P¹, j_*E) = HTemplate:Su(U,E), where U is the points the projective line with non-singular fibers, and j is the inclusion of U into the projective line, and E is the sheaf with fibers the spaces E_x of vanishing cycles.

The key estimateEdit

The heart of Deligne's proof is to show that the sheaf E over U is pure, in other words to find the absolute values of the eigenvalues of Frobenius on its stalks. This is done by studying the zeta functions of the even powers E^k of E and applying Grothendieck's formula for the zeta functions as alternating products over cohomology groups. The crucial idea of considering even k powers of E was inspired by the paper Template:Harvs, who used a similar idea with k = 2 for bounding the Ramanujan tau function. Template:Harvtxt pointed out that a generalization of Rankin's result for higher even values of k would imply the Ramanujan conjecture, and Deligne realized that in the case of zeta functions of varieties, Grothendieck's theory of zeta functions of sheaves provided an analogue of this generalization.

The poles of the zeta function of E^k are found using Grothendieck's formula

and calculating the cohomology groups in the denominator explicitly. The HTemplate:Su term is usually just 1 as U is usually not compact, and the HTemplate:Su can be calculated explicitly as follows. Poincaré duality relates HTemplate:Su(E^k) to HTemplate:Su(E^k), which is in turn the space of covariants of the monodromy group, which is the geometric fundamental group of U acting on the fiber of E^k at a point. The fiber of E has a bilinear form induced by cup product, which is antisymmetric if d is even, and makes E into a symplectic space. (This is a little inaccurate: Deligne did later show that E∩E^⊥ = 0 by using the hard Lefschetz theorem, this requires the Weil conjectures, and the proof of the Weil conjectures really has to use a slightly more complicated argument with E/E∩E^⊥ rather than E.) An argument of Kazhdan and Margulis shows that the image of the monodromy group acting on E, given by the Picard–Lefschetz formula, is Zariski dense in a symplectic group and therefore has the same invariants, which are well known from classical invariant theory. Keeping track of the action of Frobenius in this calculation shows that its eigenvalues are all q^k(d−1)/2+1, so the zeta function of Z(E^k,T) has poles only at T = 1/q^k(d−1)/2+1.

The Euler product for the zeta function of E^k is

If k is even then all the coefficients of the factors on the right (considered as power series in T) are non-negative; this follows by writing

<math>\frac{1}{\det(1-T^{\deg(x)}F_x\mid E^k)} =\exp\left(\sum_{n>0}\frac{T^n}{n} \operatorname{Trace}(F_x^n\mid E)^k\right) </math>

and using the fact that the traces of powers of F are rational, so their k powers are non-negative as k is even. Deligne proves the rationality of the traces by relating them to numbers of points of varieties, which are always (rational) integers.

The powers series for Z(E^k, T) converges for T less than the absolute value 1/q^k(d−1)/2+1 of its only possible pole. When k is even the coefficients of all its Euler factors are non-negative, so that each of the Euler factors has coefficients bounded by a constant times the coefficients of Z(E^k, T) and therefore converges on the same region and has no poles in this region. So for k even the polynomials Z(ETemplate:Su, T) have no zeros in this region, or in other words the eigenvalues of Frobenius on the stalks of E^k have absolute value at most q^k(d−1)/2+1.
This estimate can be used to find the absolute value of any eigenvalue α of Frobenius on a fiber of E as follows. For any integer k, α^k is an eigenvalue of Frobenius on a stalk of E^k, which for k even is bounded by q^1+k(d−1)/2. So

<math>|\alpha^k|\le q^{k(d-1)/2 +1}</math>

As this is true for arbitrarily large even k, this implies that

<math>|\alpha|\le q^{(d-1)/2}.</math>

Poincaré duality then implies that

<math>|\alpha|=q^{(d-1)/2}.</math>

Completion of the proofEdit

The deduction of the Riemann hypothesis from this estimate is mostly a fairly straightforward use of standard techniques and is done as follows.

The eigenvalues of Frobenius on HTemplate:Su(U,E) can now be estimated as they are the zeros of the zeta function of the sheaf E. This zeta function can be written as an Euler product of zeta functions of the stalks of E, and using the estimate for the eigenvalues on these stalks shows that this product converges for Template:Abs < q^−d/2−1/2, so that there are no zeros of the zeta function in this region. This implies that the eigenvalues of Frobenius on E are at most q^d/2+1/2 in absolute value (in fact it will soon be seen that they have absolute value exactly q^d/2). This step of the argument is very similar to the usual proof that the Riemann zeta function has no zeros with real part greater than 1, by writing it as an Euler product.
The conclusion of this is that the eigenvalues α of the Frobenius of a variety of even dimension d on the middle cohomology group satisfy

<math> |\alpha| \le q^{d/2+1/2}</math>

To obtain the Riemann hypothesis one needs to eliminate the 1/2 from the exponent. This can be done as follows. Applying this estimate to any even power V^k of V and using the Künneth formula shows that the eigenvalues of Frobenius on the middle cohomology of a variety V of any dimension d satisfy

<math> |\alpha^k| \le q^{kd/2+1/2}</math>

As this is true for arbitrarily large even k, this implies that

<math>|\alpha| \le q^{d/2}</math>

Poincaré duality then implies that

<math>|\alpha| = q^{d/2}.</math>

This proves the Weil conjectures for the middle cohomology of a variety. The Weil conjectures for the cohomology below the middle dimension follow from this by applying the weak Lefschetz theorem, and the conjectures for cohomology above the middle dimension then follow from Poincaré duality.

Deligne's second proofEdit

Template:Harvtxt found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf. In practice it is this generalization rather than the original Weil conjectures that is mostly used in applications, such as the hard Lefschetz theorem. Much of the second proof is a rearrangement of the ideas of his first proof. The main extra idea needed is an argument closely related to the theorem of Jacques Hadamard and Charles Jean de la Vallée Poussin, used by Deligne to show that various L-series do not have zeros with real part 1.

A constructible sheaf on a variety over a finite field is called pure of weight β if for all points x the eigenvalues of the Frobenius at x all have absolute value N(x)^β/2, and is called mixed of weight ≤ β if it can be written as repeated extensions by pure sheaves with weights ≤ β.

Deligne's theorem states that if f is a morphism of schemes of finite type over a finite field, then Rⁱf_! takes mixed sheaves of weight ≤ β to mixed sheaves of weight ≤ β + i.

The original Weil conjectures follow by taking f to be a morphism from a smooth projective variety to a point and considering the constant sheaf Q_ℓ on the variety. This gives an upper bound on the absolute values of the eigenvalues of Frobenius, and Poincaré duality then shows that this is also a lower bound.

In general Rⁱf_! does not take pure sheaves to pure sheaves. However it does when a suitable form of Poincaré duality holds, for example if f is smooth and proper, or if one works with perverse sheaves rather than sheaves as in Template:Harvtxt.

Inspired by the work of Template:Harvtxt on Morse theory, Template:Harvtxt found another proof, using Deligne's ℓ-adic Fourier transform, which allowed him to simplify Deligne's proof by avoiding the use of the method of Hadamard and de la Vallée Poussin. His proof generalizes the classical calculation of the absolute value of Gauss sums using the fact that the norm of a Fourier transform has a simple relation to the norm of the original function. Template:Harvtxt used Laumon's proof as the basis for their exposition of Deligne's theorem. Template:Harvtxt gave a further simplification of Laumon's proof, using monodromy in the spirit of Deligne's first proof. Template:Harvtxt gave another proof using the Fourier transform, replacing etale cohomology with rigid cohomology.

ApplicationsEdit

Template:Harvtxt was able to prove the hard Lefschetz theorem over finite fields using his second proof of the Weil conjectures.
Template:Harvtxt had previously shown that the Ramanujan–Petersson conjecture follows from the Weil conjectures.
Template:Harvtxt used the Weil conjectures to prove estimates for exponential sums.
Template:Harvs were able to prove the Künneth type standard conjecture over finite fields using Deligne's proof of the Weil conjectures.

ReferencesEdit

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External linksEdit

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ReferencesEdit

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