On the work and persona of Gilles Lachaud

Sudhir R. Ghorpade
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai 400076, India
http://www.math.iitb.ac.in/∼srg/

AGC2 T – 16
CIRM, Luminy, France, May 11, 2019
Sudhir Ghorpade (IIT Bombay)

On Gilles LACHAUD

1 / 20Gilles LACHAUD (26 July 1946 – 21 February 2018)

Sudhir Ghorpade (IIT Bombay)

On Gilles LACHAUD

2 / 20Gilles LACHAUD (26 July 1946 – 21 February 2018)

Sudhir Ghorpade (IIT Bombay)

On Gilles LACHAUD

2 / 20Gilles LACHAUD (26 July 1946 – 21 February 2018)[Source: Google Images and the article by Y. Aubry in Gaz. Math. 157 (2018), 74–75.]

Career in Brief
Doctorat d’Etat, Univ. Paris 7, 1979 [Advisor: Roger GODEMENT. Thesison Analyse spectrale et prolongement analytique: Séries d’Eisenstein,
Fonctions Zeta et nombre de solutions d’équations diophantiennes]
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2 / 20Gilles LACHAUD: Career in Brief (Contd.)
Prix Rivoire, 1979
Held a position with the CNRS and was for most part at IML, MarseilleDirector of CIRM, September 1986 – August 1991
Director (Responsable), Jan 2000 – August 2011
Founder and a strong driving force behind the AGCT meetings
13 Ph.D. students: Renault DANSET (1983), Bernadette DESHOMMES (1983),
Franck WIELONSY (1983), Jean-Pierre CHERDIEU (1985), Marc PERRET(1990), Yves AUBRY (1993), Robert ROLLAND (1995), Didier ALQUIE (1996),
Antoine EDOUARD (1998), Cédric CORNUS (2000), François-Régis BLACHE(2000), Alexandre TEMPKINE (2000), and Iman ISLIM (2001).

Guided Habilitations of: Iwan DUURSMA (2000), Yves AUBRY (2002).
Conference in honour of his 60th birthday: SAGA-1, Tahiti, May 2007.
Proceedings published by World Scientific, Singapore, 2008.
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3 / 20Some Numbers

Sudhir Ghorpade (IIT Bombay)

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4 / 20Major Themes of Work (and some representative papers)
Automorphic Forms.
Spectral analysis of automorphic forms on rank one groups by perturbation
methods, in: Proc. Sympos. Pure Math., Vol. XXVI, AMS, 1973, 441–450.
Analyse spectrale des formes automorphes et séries d’Eisenstein. Invent.
Math. 46 (1978), 39–79.
Variations sur un thème de Mahler, Invent. Math. 52 (1979), 149–162.
The distribution of the trace in the compact group of type G2 , Contemp.
Math. 722 (2019), 79–103.

Curves and Abelian Varieties over Finite Fields.
Sommes d’Eisenstein et nombre de points de certaines courbes algébriques
sur les corps finis, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 729–732.
(with M. Martin-Deschamps) Nombre de points des jacobiennes sur uncorps fini, Acta Arith. 56 (1990), 329–340.
Ramanujan modular forms and the Klein quartic, Mosc. Math. J. 5 (2005),
829–856.
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5 / 20Major Themes of Work (and some representative papers) Contd.
(with C. Ritzenthaler) On some questions of Serre on abelian threefolds, in:Algebraic Geometry and its Applications, World Scientific, 2008, 88–115.
(with C. Ritzenthaler and A. Zykin) Jacobians among abelian threefolds: aformula of Klein and a question of Serre, Math. Res. Lett. 17 (2010),
323–333.
(with Y. Aubry and S. Haloui) On the number of points on abelian andJacobian varieties over finite fields. Acta Arith. 160 (2013), 201–241.

Algebraic Varieties and Algebraic Sets over Finite Fields.
(with M. A. Tsfasman) Formules explicites pour le nombre de points desvariétés sur un corps fini, J. Reine Angew. Math. 493 (1997), 1–60.
(with S. R. Ghorpade) Étale cohomology, Lefschetz theorems and numberof points of singular varieties over finite fields, Mosc. Math. J. 2 (2002),
589–631.
(with R. Rolland) On the number of points of algebraic sets over finite fields,
J. Pure Appl. Algebra 219 (2015), 5117–5136.
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6 / 20Major Themes of Work (and some representative papers) Contd.
Continued Fractions, Sails and Klein Polyhedra.
Polyèdre d’Arnold et voile d’un cne simplicial: analogues du théorème deLagrange, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 711–716.
Klein polygons & geometric diagrams, Contemp. Math. 210 (1998), 365-372.
Sails and Klein polyhedra, Contemp. Math. 210 (1998), 373–385.

Linear Codes and Related Varieties
Les codes géométriques de Goppa, Séminare Bourbaki, no. 641, 1984/85,
Astérisque 133-134 (1986), 189–207.
(with J. Wolfmann), Sommes de Kloosterman, courbes elliptiques et codes
cycliques en caractéristique 2, C. R. Acad. Sci. Paris Sér. I Math. 305(1987), 881–883.
(with J. Wolfmann), The weights of the orthogonals of the extended quadratic
binary Goppa codes, IEEE Trans. Inform. Theory 36 (1990), 686-692.
The parameters of projective Reed-Muller codes, Discrete Math. 81 (1990),
217–221.
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7 / 20Major Themes of Work (and some representative papers) Contd.
Linear Codes and Related Varieties (Contd.)
Artin-Schreier curves, exponential sums, and the Carlitz-Uchiyama boundfor geometric codes. J. Number Theory 39 (1991), 18–40.
Number of points of plane sections and linear codes defined on algebraic
varieties, in: Arithmetic, geometry and coding theory (Luminy, 1993), de
Gruyter, 1996, 77–104.
(with S. R. Ghorpade) Higher weights of Grassmann codes, in: Coding
theory, cryptography and related areas. Springer, Berlin, 2000, 122–131.
(with S. R. Ghorpade) Hyperplane sections of Grassmannians and thenumber of MDS linear codes, Finite Fields Appl. 7 (2001), 468–506.
(with Y. Aubry, W. Castryck, S. R. Ghorpade, M. E. O’Sullivan, and S. Ram)
Hypersurfaces in weighted projective spaces over finite fields withapplications to coding theory, in: Algebraic geometry for coding theory andcryptography, Springer, 2017, 25–61.
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8 / 20A Sampling of the work of Gilles Lachaud
MOSCOW MATHEMATICAL JOURNAL
Volume 2, Number 3, July–September 2002, Pages 589–631

ÉTALE COHOMOLOGY, LEFSCHETZ THEOREMSAND NUMBER OF POINTS OF SINGULAR VARIETIESOVER FINITE FIELDSSUDHIR R. GHORPADE AND GILLES LACHAUD
Dedicated to Professor Yuri Manin for his 65th birthday

Abstract. We prove a general inequality for estimating the numberof points of arbitrary complete intersections over a finite field. Thisextends a result of Deligne for nonsingular complete intersections. For
normal complete intersections, this inequality generalizes also the classical Lang–Weil inequality. Moreover, we prove the Lang–Weil inequality for affine, as well as projective, varieties with an explicit description and a bound for the constant appearing therein. We also prove a
conjecture of Lang and Weil concerning the Picard varieties and étale
cohomology spaces of projective varieties. The general inequality forcomplete intersections may be viewed as a more precise version of theestimates given by Hooley and Katz. The proof is primarily based on asuitable generalization of the Weak Lefschetz Theorem to singular varieties together with some Bertini-type arguments and the Grothendieck–
Lefschetz Trace Formula. We also describe some auxiliary results concerning the étale cohomology spaces and Betti numbers of projective
varieties over finite fields, and a conjecture along with some partial results concerning the number of points of projective algebraic sets overfinite fields.

The quotation from Rg Veda (X, 129)
meaning “Their cord was extended across”
that appears in this paper owes its
presence to Gilles Lachaud.

2000 Math. Subj. Class. 11G25, 14F20, 14G15, 14M10.
Key words and phrases. Étale cohomology, varieties over finite fields, complete intersections, Trace Formula, Betti numbers, zeta functions, Weak Lefschetz Theorems, hyperplane sections, motives, Lang–Weil inquality, Albanesevariety.

*
i
trË
no

ivta
to

ri
Xmre
zam(

Introduction
This paper has roughly a threefold aim. The first is to prove the following inequality for estimating the number of points of complete intersections (in particular,
Received March 26, 2001; in revised form April 17, 2002.
The first named author supported in part by a ‘Career Award’ grant from AICTE, New Delhi
and an IRCC grant from IIT Bombay.
* “Their cord was extended across” (R. g Veda X.129).
c 2002 Independent University of Moscow

589

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9 / 20Some Background:Lang-Weil Inequality (1954). If X is an irreducible projective variety in PN
defined over Fq and of dimension n and degree d, then|X(Fq )| − pn ≤ (d − 1)(d − 2)qn−(1/2) + Cqn−1 ,where C is a constant depending only on N, n and d.
Deligne’s Inequality for Smooth Complete Intersections (1973). If X is anonsingular complete intersection in PN over Fq of dimension n = N − r, then|X(Fq )| − pn ≤ b0n qn/2 .
Here b0n = bn − n is its primitive nth Betti number of X (where n = 1 if n iseven and n = 0 if n is odd), and pn := |Pn (Fq )| = qn + qn−1 + · · · + q + 1.

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Some Background:Lang-Weil Inequality (1954). If X is an irreducible projective variety in PN
defined over Fq and of dimension n and degree d, then|X(Fq )| − pn ≤ (d − 1)(d − 2)qn−(1/2) + Cqn−1 ,where C is a constant depending only on N, n and d.
Deligne’s Inequality for Smooth Complete Intersections (1973). If X is anonsingular complete intersection in PN over Fq of dimension n = N − r, then|X(Fq )| − pn ≤ b0n qn/2 .
Here b0n = bn − n is its primitive nth Betti number of X (where n = 1 if n iseven and n = 0 if n is odd), and pn := |Pn (Fq )| = qn + qn−1 + · · · + q + 1.
We remark that if X has multidegree d = (d1 , . . . , dr ), then b0n = b0n (N, d) equals


N
X
X
n+1
N+c N + 1(−1) (n + 1) +(−1)
d1ν1 · · · drνr
c
+
1
c=r
ν +···+ν =c1

r

νi ≥1 ∀i
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Estimates for singular complete intersections
Theorem (Deligne-type inequality for arbitrary complete intersections)
Let X be an irreducible complete intersection of dimension n in PNFq , defined byr = N − n equations, with multidegree d = (d1 , . . . , dr ), and let s ∈ Z with
dim Sing X ≤ s ≤ n − 1. Then
|X(Fq )| − pn ≤ b0n−s−1 (N − s − 1, d) q(n+s+1)/2 + Cs (X)q(n+s)/2 ,where Cs (X) is a constant independent of q. If X is nonsingular, thenC−1 (X) = 0. If s ≥ 0, thenCs (X) ≤ 9 × 2r × (rδ + 3)N+1

where δ = max{d1 , . . . , dr }.

For normal complete intersections, this may be viewed as a common
refinement of Deligne’s inequality and the Lang-Weil inequality. Corollaries
include previous results of Aubry and Perret (1996), Shparlinskiı̆ and
Skorobogatov (1990), as well as Hooley and Katz (1991).
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Estimates for irreducible varieties over finite fields
Theorem (Effective Lang-Weil inequality)
Suppose X is a projective variety in PNFq or an affine variety in ANFq defined over
Fq . Let n = dim X and d = deg X. Then
|X(Fq )| − pn ≤ (d − 1)(d − 2)qn−(1/2) + C+ (X̄) qn−1 ,where C+ (X̄) is independent of q. Moreover if X is of type (m, N, d), withd = (d1 , . . . , dm ), and if δ = max{d1 , . . . , dm }, then we have
9 × 2m × (mδ + 3)N+1 if X is projective
C+ (X̄) ≤6 × 2m × (mδ + 3)N+1 if X is affine.
As a corollary, one obtains an analogue of a result of Schmidt (1974) on alower bound for the number of points of irreducible hypersurfaces over Fq .
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A Conjecture of Lang and Weil
When Lang and Weil proved the inequality , namely,|X(Fq )| − pn ≤ (d − 1)(d − 2)qn−(1/2) + Cqn−1 ,(1)they showed in the same paper that if K is an algebraic function field ofdimension n over k = Fq , then there is a constant γ for which (1) holds with(d − 1)(d − 2) replaced by γ, for any model X of K/k, and moreover, thesmallest such constant γ is a birational invariant. Subsequently, Lang andWeil went on to conjecture that this constant γ can be described algebraicallyas being twice the dimension of the associated Picard variety P, at least when
X is nonsingular. They made further conjectural statements relating the Weilzeta function of X and the “characteristic polynomial” of P when X is projective
and nonsingular.
In effect, we show that these conjectures hold in the affirmative provided one
uses the “correct” Picard variety.
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Tools and Techniques
Proofs of the above theorems use a variety of techniques from algebraic
geometry and topology, and to a lesser extent complex analysis and algebra.
These include
a variant of Bertini’s theorem to successively construct good hyperplanesections.
a suitable generalization of the Weak Lefschetz Theorem for singularvarieties, which is proved in the paper.
Grothendieck-Lefschetz trace formula and Deligne’s Main Theorem forgeneral varieties over finite fields.
Katz’s estimates for sums of Betti numbers.
Analysis of zeros and poles of the Weil zeta function and related objects.
Combinatorial methods to find suitable bounds using the formulae ofHirzebruch and Jouanolou for nonsingular complete intersections.
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Applications and Extensions
There have been several applications, some quite surprising. These include:
Work of T. Bandman, G.-M. Gruel, F. Grunewald, B. Kunyavskiı̆, G. Pfister and E.
Plotkin (2003 and 2006) and E. Ribnere (2009) on the characterization of finite
solvable groups by two-variable identities
topics in diophantine equations (Waring’s problem in function fields), by Y.-R. Liu
and T. Wooley (2007)
the study of Boolean functions by F. Rodier (2008), M. Delgado (2017)classification of hyperovals in planes by F. Caullery and K.-U. Schmidt (2015)arithmetic progressions over finite fields, by B. Cook and A. Magyar (2010)
the study of primitive semifields by R. Gow and J. Sheekey (2011)
to coding theory, by Nakashima (2009), F. Edoukou, S. Ling, and C. Xing (2009),
and also J. B. Little (2011).

There have also been several extensions and generalizations of some of theresults, mainly due to A. Cafure and G. Matera (2007-2012).
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A conjecture for algebraic sets over finite field
In the same paper, one can find the following conjecture due to Lachaud.
Conjecture.
If X is a complete intersection in Pm defined over Fq of dimension n ≥ m/2 anddegree d ≤ q + 1, then|X(Fq )| ≤ dpn − (d − 1)p2n−m = d(pn − p2n−m ) + p2n−m .

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A conjecture for algebraic sets over finite field
In the same paper, one can find the following conjecture due to Lachaud.
Conjecture.
If X is a complete intersection in Pm defined over Fq of dimension n ≥ m/2 anddegree d ≤ q + 1, then|X(Fq )| ≤ dpn − (d − 1)p2n−m = d(pn − p2n−m ) + p2n−m .
If X is a hypersurface (so that n = m − 1), then this is Tsfasman’s Conjecture or
Serre–Sørensen Inequality (1989/1991).
If X = V(F), where F ∈ Fq [x0 , . . . , xm ]d , with d ≤ q + 1, then|X(Fq )| ≤ dqm−1 + pm−2 .

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A conjecture for algebraic sets over finite field
In the same paper, one can find the following conjecture due to Lachaud.
Conjecture.
If X is a complete intersection in Pm defined over Fq of dimension n ≥ m/2 anddegree d ≤ q + 1, then|X(Fq )| ≤ dpn − (d − 1)p2n−m = d(pn − p2n−m ) + p2n−m .
If X is a hypersurface (so that n = m − 1), then this is Tsfasman’s Conjecture or
Serre–Sørensen Inequality (1989/1991).
If X = V(F), where F ∈ Fq [x0 , . . . , xm ]d , with d ≤ q + 1, then|X(Fq )| ≤ dqm−1 + pm−2 .
Recently, A. Couvreur (2016) has proved this conjecture in the affirmative andin fact, proved a more general result. See also: Lachaud and Rolland (2015).
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Another conjecture for algebraic sets over finite fields
Notation: pn := |Pn (Fq )| = qn + · · · + q + 1 if n ≥ 0 and pn := 0 if n < 0. Defineer (d, m) := max{|V(F1 , . . . , Fr )(Fq )| : F1 , . . . , Fr ∈ Fq [x0 , . . . , xm ]d lin. indep.}

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Another conjecture for algebraic sets over finite fields
Notation: pn := |Pn (Fq )| = qn + · · · + q + 1 if n ≥ 0 and pn := 0 if n < 0. Defineer (d, m) := max{|V(F1 , . . . , Fr )(Fq )| : F1 , . . . , Fr ∈ Fq [x0 , . . . , xm ]d lin. indep.}
Tsfasman-Boguslavsky Conjecture (TBC)
Assume that 1 ≤ r ≤ M and 1 ≤ d < q − 1. Let (ν1 , . . . , νm+1 ) be the r-th
element in the descending lexicographic order among (m + 1)-tuples(α1 , . . . , αm+1 ) of nonnegative intergers satisfying α1 + · · · + αm+1 = d. Thener (d, m) = pm−2j +mXνi (pm−i − pm−i−j ) where j := min{i : νi 6= 0}.
i=jSudhir Ghorpade (IIT Bombay)

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Another conjecture for algebraic sets over finite fields
Notation: pn := |Pn (Fq )| = qn + · · · + q + 1 if n ≥ 0 and pn := 0 if n < 0. Defineer (d, m) := max{|V(F1 , . . . , Fr )(Fq )| : F1 , . . . , Fr ∈ Fq [x0 , . . . , xm ]d lin. indep.}
Tsfasman-Boguslavsky Conjecture (TBC)
Assume that 1 ≤ r ≤ M and 1 ≤ d < q − 1. Let (ν1 , . . . , νm+1 ) be the r-th
element in the descending lexicographic order among (m + 1)-tuples(α1 , . . . , αm+1 ) of nonnegative intergers satisfying α1 + · · · + αm+1 = d. Thener (d, m) = pm−2j +mXνi (pm−i − pm−i−j ) where j := min{i : νi 6= 0}.
i=jExample: Suppose d > 1. The first m + 1 tuples ordered as above look like(d, 0, 0, . . . , 0), (d − 1, 1, 0, . . . , 0), . . . , (d − 1, 0, 0, . . . , 1). Hence for r ≤ m,er (d, m) = (d − 1)qm−1 + pm−2 + qm−r and em+1 (m, d) = (d − 1)qm−1 + pm−2 .
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Current Status of Tsfasman-Boguslavsky Conjecture
TBC follows from the Serre-Sørensen Inequality if r = 1. It is also triviallyvalid when d = 1 or m = 1. Note that in general, r ≤ m+dd
Boguslavsky (1997): TBC holds if r = 2.
Datta – G (2015): TBC holds when d = 2 and r ≤ m + 1, but it can befalse if r > m + 1.
Datta – G (2017): TBC holds for any d < q − 1, provided r ≤ m + 1.
Datta – G (2017): A new conjectural formula for er (d, m) proposed ifr ≤ m+d−1
d−1 . [the “incomplete conjecture”].
Beelen – Datta – G (2018): Incomplete conjecture established for
2 < d < q and r ≤ m+2
2 .
Beelen – Datta – G (2018-19): A “complete conjecture” proposed forer (d, m). It is established for several (but not all) values of r.
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Concluding Remarks
Gilles Lachaud has made important and lasting
contributions to mathematics, especially in the study ofalgebraic varieties over finite fields and linear codes. His
knowledge and interests were deep and wide. When hebecame interested in some topic, he would usually delve
deeper and spend considerable time learning many
aspects of it. As far as I have seen, he would never be ina rush to publish quickly, but would prefer to take timeand be thorough.
Besides his contributions to mathematics, Gilles was
an instiution builder. He helped nurture an institution likethe CIRM. Also, the continuing success of the AGCT
conferences owes largely to his vision and efforts.
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Concluding Remarks
Other than scientific institutes and conferences, Gilles
served as the President of the French Pavilion at
Auroville, near Pondicherry, India. He had read or had atleast browsed through significant amount of ancient andmodern Sanskrit works, including the Vedas,
Upanishadas, and the scholarly treatises of Sri
Aurobindo.
Above all, Gilles was a wonderful human being,
generous, warm-hearted, and kind, always willing to help
others, especially students and younger colleagues. His
untimely demise last year is a great loss to our subject
and the community. Personally, it has been a pleasureand honour to have known him. He will certainly be
missed.....
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