Proceedings of the International Congress of Mathematicians
Helsinki, 1978

The Work of Pierre Deligne
N. M. Katz

My purpose here is to convey to you some idea of the scope and the depth of the
work for which we are today honoring Pierre Deligne with the Fields Medal.
Deligne's work centers around the remarkable relations, first envisioned by Weil,
which exist between the cohomological structure of algebraic varieties over the
complex numbers, and the diophantine structure of algebraic varieties over finite
fields.
I. The Weil conjectures. Let us first consider an algebraic variety Y over a finite
field Fq. For each integer n^l there is a unique field extension Fqil of degree
n over Fq. We denote by Y(Fqn) the (finite) set of points of Y with coordinates
in Fqli, and by # Y(Fqn) the cardinality of this set. The zeta function of Y over
Fg is the formal series defined by
Z(Y/Fq, D = exp f 2 v *

Y

(FÀ •

Knowledge of the zeta function is equivalent to knowledge of the numbers { # Y(Fqn)}.
After the pioneering work of E. Artin, W. K. Schmid, H. Hasse, M. Deuring and
A. Weil on the zeta functions of curves and abelian varieties, Weil in 1949 made
the following conjectures about the zeta function of a projective non-singular
w-dimensional variety Y over a finite field ¥q.
(1) The zeta function is a rational function of T, i.e. it lies in Q(T).
(2) There exists a factorization of the zeta function as an alternating product
of polynomials P0(T)9..., P2H(T)9
Z(Y/Fq,T)=r

P1(T)P>(T)...P*-1(T)

P0(T)P2(T)...P2tl(T)

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N. M. Katz

of the form

Pi(T)=

nV-aijT),

such that the map oL*^qnloL carries the a u bijectively to the a2n-i,j(3) The polynomials Pt(T) lie in Z[T]9 and their reciprocal roots <xitj are
algebraic integers which, together with all their conjugates, satisfy

This is the "Riemann Hypothesis" for varieties over finite fields.
(4) If Y is the "reduction mod/?" of a projective smooth variety F i n characteristic zero, then the degree bt of Pf is the ith Betti number of Y as complex
manifold.
Underlying these conjectures was Weil's belief in the existence of a "cohomology
theory", with a coefficient field of characteristic zero, for varieties over finite fields.
In this theory, the polynomial Pi(T) would be the "inverse" characteristic polynomial det (1 — TF) of the "Frobenius endomorphism" acting on H\ Conjectures
(1) and (2) would then follow from a Lefschetz trace formula for F and its iterates,
and from a suitable form of Poincaré duality. Conjecture (4) would follow if the
cohomology of the "reduction mod/?" Y ot a projective smooth variety Ï in
characteristic zero were (essentially) equal to the topological cohomology of Y as
complex manifold.
The next years saw the systematic introduction of sheaf-theoretic and cohomological methods into algebraic geometry. By the mid-1960s, M. Artin and A. Grothendieck had developed the étale cohomology theory of arbitrary schemes, along the
lines foreseen in Grothendieck's 1958 Edinburgh address. For each prime number /,
this gives a cohomology theory, '7-adic cohomology", with coefficients in the field
Ql of /-adic numbers, which is adequate to give parts (1), (2) and (4) of the Weil
conjectures for projective smooth varieties over finite fields of characteristic p^l.
In their theory, the cohomology of the "reduction mod /?" Y of a projective smooth
Y in characteristic zero is just the singular cohomology, with coefficients in Qh
of "Y as complex manifold". In the case of curves and abelian varieties, these
constructions agree with those already given by Weil.
For a given projective smooth Y/Fq, we now have, for each l^p, a factorization of the zeta function as an alternating product of /-adic polynomials Pi§l(T).
There is, however, no assurance that the Pitl have coefficients in Q rather than
in Ql9 much less that their reciprocal zeros are algebraic integers with the predicted
absolute values. Of course, if one could prove directly that the reciprocal zeros of
Pitl were algebraic integers which, together with all their conjugates, had the correct
absolute value qi/2, then the polynomials Pu could be described intrinsically
in terms of the zeros and poles of the zeta function itself, and hence would have
rational coefficients independent of /. But how could one even introduce archi-

The Work of Pierre Deligne

49

medean considerations into the /-adic theory without first knowing the rationality
of the cofficients of the Pitl ?
Let me now try to indicate the brilliant synthesis of ideas involved in Deligne's
solution of these problems.
Initially, he tries to prove à priori that the Pul have rational coefficients independent of /. The idea is to proceed by induction on the dimension of Y. If Y is
77-dimensional, then Poincaré duality and the fact that the zeta function is itself
rational and independent of / reduce us to treating the polynomials PUi for
i<77 —1. Now let Z be a smooth hyperplane section of Y. The "weak" Lefschetz
theorem assures us that Y and Z have the same Pul for i^n—2, and that the
Pn-jt1 for Y divides that for Z. This alone is enough to show inductively that the
reciprocal zeroes of the Pitl are algebraic integers.
In order to go further, and show that the Pitl actually have rational coefficients
independent of /, the idea is to show that P H _ lfI for Y is a generalized "greatest
common divisor" of the P„-u of all possible smooth hyperplane sections. Unfortunately, this "g.c.d." argument, which itself depends on the full strength of the
monodromy theory of Lefschetz pencils, works only when Y satisfies the "hard"
Lefschetz theorem (existence of the "primitive decomposition" on its cohomology),
otherwise the "g.c.d." will be too big at some stage of the induction. But Deligne will
later prove the hard Lefschetz theorem in arbitrary characteristic as a consequence
of the Weil conjectures. What is to be done?
With characteristic daring, Deligne simply ignores the preliminary problem of
establishing independence of /. Fixing one I^p, he turns to a direct attack on the
absolute values of the algebraic integers which occur as the reciprocal roots of the Pitl.
Consider a smooth projective even dimensional Y, and a Lefschetz pencil Zt of
hyperplane sections, "fibering" Y over the /-line. Factor the P„-lti of each
Zt as the product of the "g.c.d." of all of them, and of the "variable" part. Deligne
shows à priori that these "variable" parts are each polynomials with rational
coefficients whose reciprocal zeroes all satisfy the Riemann Hypothesis
K - l , variable | = ^ ( " " l ) / 2 .

Deligne's proof of this is simply spectacular; no other word will do. He first uses
a theorem of Kazdan-Margoulis, according to which the monodromy group of
a Lefschetz pencil of odd fibre dimension is "as big as possible", to establish the
rationality of the coefficients of the "variable" parts. Then he considers the L-function
over the /-line whose Euler factors are the reciprocals of the "variable" parts. This
L-function has rational Dirichlet coefficients. Deligne realizes that Rankin's method
of estimating Ramanujan's function T(W) by "squaring" might be applied in this
context to estimate the reciprocal poles of the individual Euler factors (i.e. the
reciprocal zeroes of the "variable" parts!). The problem is to control the poles
of all the L-functions obtained from this one by passing to even tensor powers
("squaring"). Deligne gains this control by ingeniously combining Grothendieck's

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N. M. Katz

cohomological theory of such L-functions, the Kazdan-Margoulis theorem, and the
classical invariant theory of the symplectic group!
Once he has this à priori estimate for the variable parts of the P„-ltl of the
hyperplane sections Z,, a Leray spectral sequence argument shows that in the
P„tl for Y itself, all the reciprocal zeroes are algebraic integers which, together
with all their conjugates, satisfy the apparently too weak estimate
Kj\ < 0 (n+1)/2 (instead of qn%
But this estimate is valid for Y of any even dimension n. The actual Riemann
hypothesis for any projective smooth variety X follows by applying this estimate
to all the even cartesian powers of X.
II. Consequences for number theory. That there are many spectacular consequences
for number theory comes as no surprise. Let us indicate a few of them.
(1) Estimation of # Y(Fq) when Y has a "simple" cohomological structure.
For example, if Y is a smooth «-dimensional hypersurface of degree d, we get

i*rW-o+,+...+rti -= f - ^ r « - y ,
(2) Estimates for exponential sums in several variables, e,g
(a) if / is a polynomial over Fp in n variables of degree d prime to /?, whose
part of highest degree defines a nonsingular projective hypersurface, then

2 expßjp/fe,..., xn))\ ^ (rf-l)V/2;

*iEFp

v

p

(b) "multiple Kloosterman sums":
xt£F*

VP \

x

l~-xn)J\

(3) The Ramanujan-Petersson conjecture. Already in 1968 Deligne had combined
techniques of /-adic cohomology and the arithmetic moduli of elliptic curves with
earlier ideas of Kuga, Sato, Shimura and Ihara to reduce this conjecture to the Weil
conjectures. Thus if 2a(n^n *s ^ e ^-expansion of a normalized (a(l) = l) cusp
form on rx(N) of weight k^2 which is a simultaneous eigenfunction of all Hecke
operators, then
\a(p)\ < 2p (ft - l)/2 for all primes p\N.
III. Cohomological consequences; weights. As Grothendieck foresaw in the 1960s
with his "yoga of weights", the truth of the Weil conjectures for varieties over
finite fields would have important consequences for the cohomological structure of
varieties over the complex numbers. The idea is that any reasonable algebro-geometric situation over C is actually defined over a subring of C which, as a ring,
is finitely generated over Z. Reducing modulo a maximal ideal m of this ring,

The Work of Pierre Deligne

51

wefinda situation over afinitefield,and the corresponding Frobenius endomorphism
F(m) operating on this situation. This Frobenius operates by functoriality on the
/-adic cohomology, which is none other than the singular cohomology, with Qr
coefficients, of our original situation over C. This natural operation of Frobenius
imposes a previously unsuspected structure on the cohomology of complex algebraic
varieties, the so-called "weight filtration", or filtration by the magnitude of the
eigenvalues of Frobenius.
In a remarkable tour de force in the late 1960's and early 1970's, Deligne developed,
independently of the Weil conjectures, a complete theory of the weight filtration
of complex algebraic varieties, by making systematic use of Hironaka's resolution of
singularities, of the notion of differential forms with "logarithmic poles" (i.e. products of fi?///'s), and of his own earlier work on cohomological descent. The resulting theory, which Deligne named "mixed Hodge theory", should be seen as a farreaching generalization of the classical theory of "differentials of the second kind"
on algebraic varieties, as well as of "usual" Hodge theory.
Consider, for example, a smooth affine variety U over C. By one of Hironaka's
fundamental results, we can find a projective smooth variety X and a collection of
smooth divisors Dj in X which cross tran sver sally, suchthat U^X—UDi. Deligne
shows that the Leray spectral sequence, in rational cohomology, of the inclusion
map Ua+X9 degenerates at £ 3 , and that the filtration it defines on the cohomology
of U is independent of the choice of the compactification. This is the desired weight
filtration; its smallest filtrant is the image of Hm(X) in H*(U)9 i.e. the space of
"differentials of the second kind on £/".
One of the many applications of this theory is to the global monodromy of families
of projective smooth varieties. Given a projective smooth map X-+ S of smooth
complex varieties, and a point s£S9 the fundamental group n^S, s) acts on each
of the cohomology groups Hl(XS9 C) of the fibre Xs. Deligne shows that these
representations are all completely reducible, and that each of their isotypical components, especially the space of invariants, is stable under the Hodge decomposition
into (p9 ^-components.
By means of an extremely ingenious and difficult argument drawing upon Grothendieck's cohomological theory of L-functions and the ideas of the Hadamard-de la
Vallée Poussin proof of the prime number theorem, Deligne later established an
/-adic analogue of this theorem of complete reducibility for /-adic "local systems"
in characteristic /; over open subsets of the projective /-line P1, provided that all
of the "fibres" of the local system satisfy the Riemann Hypothesis (with a fixed
power of |/#). Once he had proven the Riemann Hypothesis for varieties over
finitefields,he could apply this theorem to the local system coming from a Lefschetz
pencil on a projective smooth variety over a finite field. The resulting complete
reducibility is easily seen to imply the hard Lefschetz theorem. This theorem, previously known only over C, and there by Hodge's theory of harmonic integrals,
is thus established, in all characteristics as a consequence of the Weil conjectures.

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N. M. Katz: The Work of Pierre Deligne

Other outgrowths of Deligne's work on weights include his theory of differential
equations with regular singular points on smooth complex varieties of arbitrary
dimension, which yields a new solution of Hubert's 21st problem, and his affirmative
solution of the "local invariant cycle problem" in the local monodromy theory of
families of projective smooth varieties.
Still another application of "weights" is to homotopy theory. Deligne, Griffiths,
Morgan and Sullivan jointly apply mixed Hodge theory to prove that the homotopy
theory ((g)Q) of a projective smooth complex variety is a "formal consequence"
of its cohomology,
IV. Other work. We have passed over in silence a considerable body of Deligne's
work, which alone would be sufficient to mark him as a truly exceptional mathematician; duality in coherent cohomology, moduli of curves (jointly with Mumford),
arithmetic moduli (jointly with Rapoport), the Ramanujan-Petersson conjecture
for forms of weight one (jointly with Serre), the Macdonald conjecture (jointly with
Lusztig), local "roots numbers", /?-adic L-functions (jointly with Ribet), motivic
L-functions, "Hodge cycles" on abelian varieties, and much more.
I hope that I have conveyed to you some sense, not only of Deligne's accomplishments, but also of the combination of incredible technical power, brilliant clarity,
and sheer mathematical daring which so characterizes his work.
PRINCETON UNIVERSITY

PRINCETON, N.J. 08540, U.S.A.