We start by recalling the main theorem of this seminar (Weil II for curves) and illustrating some of its arithmetic consequences. Then we introduce the theory of Lefschetz pencils and deduce the last part (Riemann hypothesis) of the Weil conjectures. Finally, we prove the geometric semisimplicity of lisse pure sheaves, and use it to deduce the hard Lefschetz theorem.
This is an expanded note prepared for a STAGE talk, Spring 2014. Our main references are ,  and .
Weil II for curves
Recall that we have proved the following target theorem.
be a smooth compactification over
- : By Leray or Gysin, we have , which consists of the -invariants, is -pure of weight . (Note: cohomology groups are always understood as taking the cohomology of the base change ).
- : By Leray or Gysin, we have . Since is -mixed of weight , and by Poincare duality, is -mixed of weight , it follows is -pure of weight .
- : by birational invariance of top (or long exact sequence associated to ), we have , which is -pure of weight .
Several remarkable arithmetic consequences follows.
(Riemann Hypothesis over finite fields)
be a smooth projective geometrically connected variety. Then for any
-pure of weight
We will deduce this theorem in the next section. The Riemann hypothesis illustrates the following surprising slogan: the arithmetic of a smooth projective variety
over a finite field
is controlled by the topology of the corresponding complex manifold
. If we factorize the zeta function of
) are the
eigenvalues on the odd (resp. even) degree cohomology groups
, then by the Lefschetz trace formula,
The Riemann hypothesis provides the key to understand these
: it allows us to obtain very good estimates of
as long as we know enough about the cohomology of
. For example, when
is an elliptic curve over
, we obtain
By Corollary 1
, we know that
, has pure weight 1. In this way we recover the classical Hasse-Weil bound,
-function is defined to be the coefficients of the
-expansion of the weight 12 cusp eigenform
Ramanujan famously observed (!) without proof that
. This turns out to be a general phenomenon for coefficients of a cusp eigenform
, as a consequence of Weil II together with the algebro-geometric incarnation of cusp eigenforms. Let
be the modular curves over
with the universal family of elliptic curves
. By the Eichler-Shimura relation, the Hecke eigenvalue
matches up with the trace of
on the (2-dimensional)
-isotypic component of
is lisse on
of pure of weight 1, by Corollary 1
, the above
is pure of weight
. Therefore both
eigenvalues have absolute values
Notice in this example, it is convenient to work with non-constant coefficient systems. Here is another typical example.
. The classical Kloosterman sum is defined to be
This can be interpreted geometrically as follows. Let
be the Artin-Schrier sheaf on
associated to the additive character
be the smooth affine variety of dimension
defined by the equation
Deligne computed that
Hence by the strong version of Weil II, we obtain the estimate
, which is certainly not easy to obtain using elementary methods.
Lefschetz pencils and Riemann Hypothesis
When is a curve, the Riemann hypothesis follows from Corollary 1, since is -pure of weight 0 for any . For the general case, we induct on via the theory of Lefschetz pencils.
Using incidence correspondences and the Bertini theorem, one can show the existence of Lefschetz pencils.
There exists a Lefschetz pencil of hyperplanes defined over
, after possibly a finite extension of the base field and possibly replacing the projective embedding from
Let be the blow up of along , then we obtain a projective morphism with smooth fibers over , where is a finite set of points. After possibly a finite base extension, we may assume consists of -rational points. The Leray spectral sequence for the blow up implies that So it suffices to prove the purity statement for .
Due to the simple nature of singularities, it is possible to describe both the local and global monodromy actions on the cohomology. In the complex setting, this is classically known as the Picard-Lefschetz theory. In the -adic setting, this is done in SGA 7 and is briefly summarized as follows.
Now we can finish the induction step. For simplicity let us assume is odd (the even case is similar). Let and . Then by the property of the Lefschetz pencil (Theorem 3 a), b)), we have . The Leray spectral sequence implies that By induction and proper base change, is lisse on , pure of weight . So Corollary 1 implies that each term is pure of weight . Hence is pure of weight as well. This completes the proof of Theorem 2.
The hard Lefschetz theorem
be a finite field and
be a prime number. Let
be a smooth geometrically connected variety. Let
be a lisse
. Then the representation
As we have already seen in Koji's talk, we can replace
by a space filling curve without changing the geometric monodromy group, the smallest algebraic group containing the image of
. So it suffices to treat the curve case. We are going to induct on the length of
is irreducible as a
-representation, as in Kestutis's talk,
is semisimple as a
is normal). Now suppose
is an extension of lisse
-pure sheaves on
we would like to show that there is a section (as
-representations). In other words, we would like to show that the element
corresponding to the identity morphism
lies in the image of the first map in the following sequence,
The crucial thing is the mismatch of weights
in the second map: by Theorem 1
, the source has weight 0 but the target has weight 1. Since
is fixed by
, it must die in
and hence comes from some element of
Next we will see how the geometric semisimplicity grew out of the arithmetic consideration of weights can help us to understand the fundamental geometric structure of smooth projective varieties.
be a projective smooth connected variety over an algebraically closed field (of any characteristic). Let
be an ample line bundle on
. Then for any
-th iterated cup product
is an isomorphism.
are defined over a finitely generated subfield
, using the defining equations, we obtain a morphism of scheme
and an ample line bundle on
with the generic fiber
is scheme of a finite type over
. After possibly shrinking
, we may assume
has projective smooth connected fibers. To show
is an isomorphism on over generic point, it suffices to check it is an isomorphism on each closed point. This puts us in the situation where
is the algebraic closure of a finite field
(even if we work with
at the beginning).
The case is trivial. We are going to induct on the dimension of . Take a Lefschetz pencil on . Let be a smooth hyperplane section. By the Lefschetz hyperplane theorem, is an isomorphism when and is an injection when (I am going to omit all the Tate twists due to my laziness). Taking Poincare dual, is an isomorphism when and is surjective when . Now by the projection formula, can be decomposed as When , and are isomorphisms. By induction hypothesis, is an isomorphism and it follows that is also an isomorphism. It remains to treat the key case . In this case, is an injection and is a surjection. By Poincare duality, is an isomorphism is equivalent to that the pairing is non-degenerate. Using the injection , it is equivalent to that the pairing is non-degenerate on the image of . By the property of the Lefschetz pencil (Theorem 3 c)), we have Now we use Theorem 5: the -action on is semisimple (this is the only place we use Weil II). So we obtain a -equivariant decomposition for some without trivial -constituents. Hence the non-degenerate cup product pairing on decomposes accordingly and in particular restricts to a non-degenerate pairing on , as desired.
We mention one immediate geometric consequence of the hard Lefschetz theorem to end this talk.
-th Betti number
is even for odd
The hard Lefschetz together with the Poincare duality provides a non-degenerate pairing on
, which is alternating
L-functions and monodromy: four lectures on Weil II, 2000, http://web.math.princeton.edu/~nmk/arizona34.pdf.
Lectures on Etale Cohomology (v2.10), Available at www.jmilne.org/math/.
La conjecture de Weil. II, Inst. Hautes &EACUTE;tudes Sci. Publ. Math. (1980), no.52, 137--252.