Weil conjectures

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

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

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.

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

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

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.

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

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

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

As an example, consider the hyperelliptic curve[1]

${\displaystyle C:y^{2}+y=x^{5},}$

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

${\displaystyle \zeta (C/{\bf {F}}_{41},s)={\frac {P_{1}(T)}{P_{0}(T)\cdot P_{2}(T)}}={\frac {1-9\cdot T+71\cdot T^{2}-9\cdot 41\cdot T^{3}+41^{2}\cdot T^{4}}{(1-T)(1-41\cdot T)}}.}$

The values ${\displaystyle c_{1}=-9}$ and ${\displaystyle c_{2}=71}$ can be determined by counting the numbers of solutions ${\displaystyle (x,y)}$ of ${\displaystyle y^{2}+y=x^{5}}$ over ${\displaystyle {\bf {F}}_{41}}$ and ${\displaystyle {\bf {F}}_{41^{2}}}$, respectively, and adding 1 to each of these two numbers to allow for the point at infinity ${\displaystyle \infty }$. This counting yields ${\displaystyle N_{1}=33}$ and ${\displaystyle N_{2}=1743}$. It follows:[2]

${\displaystyle c_{1}=N_{1}-1-q=33-1-41=-9}$   and

${\displaystyle c_{2}=(N_{2}-1-q^{2}+c_{1}^{2})/2=(1743-1-41^{2}+(-9)^{2})/2=71.}$

The zeros of ${\displaystyle P_{1}(T)}$ are ${\displaystyle z_{1}:=0,12305+{\sqrt {-1}}\cdot 0,09617}$ and ${\displaystyle z_{2}:=-0,01329+{\sqrt {-1}}\cdot 0,15560}$ (the decimal expansions of these real and imaginary parts are cut off after the fifth decimal place) together with their complex conjugates ${\displaystyle z_{3}:={\bar {z}}_{1}}$ and ${\displaystyle z_{4}:={\bar {z}}_{2}}$. So, in the factorisation ${\displaystyle P_{1}(T)=\prod _{j=1}^{4}(1-\alpha _{1,j}T)}$, we have ${\displaystyle \alpha _{1,j}=1/z_{j}}$ . As stated in the third part (Riemann hypothesis) of the Weil conjectures, ${\displaystyle |\alpha _{1,j}|={\sqrt {41}}}$ for ${\displaystyle j=1,2,3,4}$.

The non-singular, projective, complex manifold that belongs to ${\displaystyle C/\mathbb {Q} }$ has the Betti numbers ${\displaystyle B_{0}=1,B_{1}=2g=4,B_{2}=1}$.[3] As described in part four of the Weil conjectures, the (topologically defined!) Betti numbers coincide with the degrees of the Weil polynomials ${\displaystyle P_{i}(T)}$, for all primes ${\displaystyle q\neq 5}$: ${\displaystyle {\rm {deg}}(P_{i})=B_{i},\,i=0,1,2}$.

• 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 |α| = 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 Lefschetz (1924), and extended by Grothendieck (1972) and Deligne & Katz (1973) 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) = H¹
  _c(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 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 Rankin (1939), who used a similar idea with k = 2 for bounding the Ramanujan tau function. Langlands (1970, section 8) 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

${\displaystyle Z(U,E^{k},T)={\frac {\det(1-F^{*}T\mid H_{c}^{1}(E^{k}))}{\det(1-F^{*}T\mid H_{c}^{0}(E^{k}))\det(1-F^{*}T\mid H_{c}^{2}(E^{k}))}}}$

and calculating the cohomology groups in the denominator explicitly. The H⁰
_c term is usually just 1 as U is usually not compact, and the H²
_c can be calculated explicitly as follows. Poincaré duality relates H²
_c(E^k) to H⁰
(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

${\displaystyle Z(E^{k},T)=\prod _{x}{\frac {1}{Z(E_{x}^{k},T)}}}$

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

${\displaystyle {\frac {1}{\det(1-T^{\deg(x)}F_{x}\mid E^{k})}}=\exp \left(\sum _{n>0}{\frac {T^{n}}{n}}\operatorname {Trace} (F_{x}^{n}\mid E)^{k}\right)}$

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(E^k
  _x, 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

${\displaystyle |\alpha ^{k}|\leq q^{k(d-1)/2+1}}$

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

${\displaystyle |\alpha |\leq q^{(d-1)/2}.}$

Poincaré duality then implies that

${\displaystyle |\alpha |=q^{(d-1)/2}.}$

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 H¹
  _c(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 |T| < 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

${\displaystyle |\alpha |\leq q^{d/2+1/2}}$

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

${\displaystyle |\alpha ^{k}|\leq q^{kd/2+1/2}}$

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

${\displaystyle |\alpha |\leq q^{d/2}}$

Poincaré duality then implies that

${\displaystyle |\alpha |=q^{d/2}.}$

• 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.