Arithmetic, Geometry, and Coding Theory: Homage to Gilles Lachaud
Sudhir R. Ghorpade, Christophe Ritzenthaler, François Rodier, Michael A. Tsfasman
AARITHMETIC, GEOMETRY, AND CODING THEORY: HOMAGETO GILLES LACHAUD
SUDHIR R. GHORPADE, CHRISTOPHE RITZENTHALER, FRANC¸ OIS RODIER,AND MICHAEL A. TSFASMAN
Abstract.
We give an overview of several of the mathematical works of GillesLachaud and provide a historical context. This is interspersed with somepersonal anecdotes highlighting many facets of his personality.
The AGC T Conference at CIRM in June 2019 was dedicated to Gilles Lachaud(26 July 1946 – 21 February 2018) who was a founder of this series of conferenceswhich began in 1987. Two speakers spoke exclusively about the work and personaof Gilles Lachaud, while many others paid rich tributes to him during their talks .This article is a homage to Gilles and it attempts to give an outline of his math-ematical journey and glimpses of his work and personality. It is divided in fourparts. The first part, written by Rodier, gives an account of Gilles’ early works in-cluding his Ph.D. thesis related to automorphic forms, works on Warings problemand real quadratic fields, and some of his earlier articles on coding theory. Next,Tsfasman recounts the beginnings of algebraic geometric codes, his interactionswith Gilles Lachaud, and some of their joint work. The third part, by Ghorpade,describes some of Gilles’ work on linear codes and on counting the number of ra-tional points of varieties over finite fields, as well as some related developments.Finally, Ritzenthaler gives a personalized account of Gilles’ work that deals mainlywith non-hyperelliptic curves and abelian varieties.1. Gilles Lachaud’s early works
Fran¸cois RodierI met Gilles Lachaud in 1969 when he had just been appointed to what wasstill called the University of Paris, as Roger Godement’s assistant. He was a pure
Mathematics Subject Classification. The slides of most of the talks in AGC T-2019, and videos of some of them, are available at:https://conferences.cirm-math.fr/1921.html a r X i v : . [ m a t h . HO ] A p r GHORPADE, RITZENTHALER, RODIER, AND TSFASMAN product of the university. He did not want to go to the so-called “Grandes ´Ecoles”.Gilles had always refused this royal road.During the academic year 1969-70, Roger Godement had left Paris for a six-months stay at the Institute for Advanced Study in Princeton, USA, to work with,among others, Robert Langlands. Meanwhile, Godement’s assistants organized thesame DEA (Masters degree) course that he offered each year. With Gilles, therewere Miss Chambon, Jean-Pierre Labesse, and Michel Duflo, assisted by Fran¸coisBruhat. It was by attending these classes on
Representations of Locally CompactGroups that I first met Gilles. He had lectured on spectral theory, a field in whichhe specialized.Having started a thesis myself under the direction of Godement, I met Gilles quiteoften, especially during the weekly sessions of seminars. He came to the symposiumin 1971 in Budapest where we had the opportunity to cross the Iron Curtain tomeet the Soviet School of Group Theory, and first Israel Gelfand, who had beenauthorized to cross the border of the USSR to come to Hungary. Then, in theevening, Gilles organized some parties with the younger people in the conference.In 1972, an International Summer School was organized by the University ofAntwerp, on
Modular functions of one variable . But it was financed by NATO.The dislike of the army by Godement which was well known, made him prefer asymposium organized by the AMS in Williamston, in the United States on
Har-monic analysis on homogeneous spaces . Gilles was one of the mathematicians whofollowed Godement instead of attending the symposium in Antwerp. This was thefirst time that I did my presentation in English in front of an amphitheater filledwith people that I considered more capable than me. So Gilles offered me to makeme rehearse it. Gilles was more sure of himself in English, but he still asked me toassist in exchange for his own rehearsal.After publishing two articles in French in the journal Inventiones:
Analyse spec-trale des formes automorphes et s´eries d’Eisenstein [19] in 1978, and
Variations surun th`eme de Mahler [21] in 1979, Gilles defended his thesis of “Doctorat d’´Etat”at Paris Diderot University - Paris 7 in June 1979 on
Analyse spectrale et pro-longement analytique: S´eries d’Eisenstein, Fonctions Zˆeta et nombre de solutionsd’´equations diophantiennes [22] under the direction of Roger Godement. In this the-sis, Gilles reproves the results of Selberg on the meromorphic prolongation of theEisenstein series by a method of perturbing the continuous spectra of self-adjointoperators. He also obtains the analytical extension of a type of generalized zetafunctions, and applies these results to diophantine equations. The Rivoire prizefrom the University of Clermont-Ferrand was awarded to him in 1979 for his thesis.More specifically, in the article
Spectral analysis of automorphic forms and Eisen-stein series , Gilles is concerned with the spectral decomposition of the left regularrepresentation π of G on K \ G/ Γ for G a real, linear, connected simple Lie group ofrank 1, K a maximal compact subgroup, Γ a discrete subgroup such that G/ Γ hasfinite invariant volume. Then, by a general principle of functional analysis, thereexists a spectral measure m on the spectrum of the algebra L ( G, K ) of complexintegrable functions on G bi-invariant under K . This spectrum is a closed subset GCT: HOMAGE TO GILLES LACHAUD 3 of the complex plane, such that we have( π ( F ) f, g ) = ∫ ˜ F ( λ ) dm f,g ( λ )by noting ˜ F the Gel’fand transform of the function F ∈ L ( G, K ), and dm f,g thecomplex measure associated with the spectral measure m and the elements f, g of L ( K \ G/ Γ). Some eigenfunctions of π ( F ) are given by Eisenstein’s series. In thecase of G = SL (2 , R ), Γ = SL (2 , Z ), K = SO (2) they are written E s ( z ) = 12 y s ∑ ( c,d )=1 | cz + d | − s , ℜ e s > , z = x + iy, y > , since one can identify the symmetric space K \ G with the upper half-plane. Theyextend analytically to the entire complex plane, and their value on the spectrumof the algebra L ( G, K ) allows to calculate the continuous part of the measures dm f,g . It was A. Selberg who stated for the first time these results in the case thegroup G is of rank one on the field of rationals, following his own method.The aim of Gilles is to use another method based on the theory of perturbationsof the continuous spectra of self-adjoint operators. It consist in finding a represen-tation π ′ of G in L ( K \ G/ Γ) fulfilling two properties. First its spectral analysis isknown, second it is close to the representation π in a suitable sense, namely that thedifference π ( F ) − π ′ ( F ) must satisfy a certain compactness criterion for sufficientlynumerous F ∈ L ( G ; K ). This method, known from Selberg and Gel’fand, wasdeveloped by Faddeev in the case of the group SL (2 , R ). Hence, the ideas used byGilles in proving his main theorems are not completely different from known ones.But, after using many general theorems, the theory has been given a smooth andsystematic look.In the article Variations on a theme of Mahler , he studied the function N ( t ) = { x ∈ Z n || F ( x ) | ≤ t } for multivariate polynomials F with n variables on Z . More precisely let S = { p , . . . , p s } be a finite set of prime numbers, P = p . . . p s and Z n ( S ) the set ofvectors x = ( x , . . . , x n ) of Z n whose coordinates are prime to P as a whole. Forany prime p , we denote | x | p the absolute value of x ∈ Q associated with the placedefined by p . Denote | x | the Archimedean absolute value of Q . If x ∈ Q n , wewrite: F S ( x ) = | F ( x ) | ∏ p ∈ S | F ( x ) | p . Gilles proves the following theorem about N S ( t ) = { x ∈ Z n || F S ( x ) | ≤ t } . Theorem 1.1.
With the previous notations, let us suppose the forme F anisotropicon R and on Q p when p ∈ S . Then N S ( t ) = V S t n/d + O ( t ( n − θ ) /d ) , where V S is a constant (which he computes) and where θ > . GHORPADE, RITZENTHALER, RODIER, AND TSFASMAN
He deduced the analytic continuation of certain Zeta functions.Strangely enough, because the theory of automorphic forms has developed strongly,under the impetus of Langlands among others, these publications have had littleecho. But Gilles’s interest in this theme has not disappeared since, as we can seein several of his later publications, and perhaps one of the last
The distributionof the trace in the compact group of type G [36], where he used the theory ofrepresentations of Lie groups. This article responds to a request from J.-P. Serreconcerning the calculation of a certain exponential sum of degree 7, involving the G group as in the theory of Sato-Tate.Gilles was interested in other things than the theory of automorphic forms. Hewas also concerned about number theory and history of mathematics.He gave lectures at various universities about some points of history of math-ematics, the foundations of arithmetic and the connections between mathematicsand philosophy, mysticism and magic. In the article Nombres et arch´etypes: duvieux et du neuf [Number and archetypes: something old, something new] [20], hesuggested: mathematics and philosophy have been estranged since the time of Dio-phantos, whose number theory can be viewed as a succession of relations betweenobjects whose status is determined axiomatically and mathematical truth is deter-mined internally, without reference to external reality; there is another tradition ofmathematics working under mystical influences, e.g., Euclid’s Pythagorean empha-sis on perfect numbers and solids. This led Gilles to speculate on the connectionsbetween mathematics and magic with the philosophers like Plato, Iamblichus andFerdinand Gonseth.He contributed to the publication of
Les Arithm´etiques by Diophantus in 1984(
R. Rashed, Diophante. Les Arithm´etiques, t. Ill : Livre IV, ccvi + 162 p. ; Paris,Les Belles Lettres, 1984 ). In [6, p. 367], Karine Chemla writes about this book: “R.Rashed in the publication of third book of Diophantus makes in his commentary,after having gathered in an abstract of algebraic geometry, written in conjunctionwith the number theorist Lachaud, the set of techniques that the translation ofthe text of Diophantus supposes. This abstract thus presents the double virtue ofbeing a well-made mathematical introduction and of being adapted to the text ofDiophantus. We have here a most remarkable example of what the combination ofmathematics and history can produce.” Indeed, Gilles wrote about fifty pages ofintroduction to modern algebraic geometry, including B´ezout theorem, a discussionof birationnal equivalence, and the quartic associated to the double equations ofdegree 2. He gives some examples of applications of elementary algebraic geometryto diophantine analysis avoiding the megamachines often needed in these kind ofresults.In Number Theory, he worked on Waring’s problem. He wrote an article
Unepr´esentation ad´elique de la s´erie singuli`ere et du probl`eme de Waring [An adelicpresentation of the singular series and the Waring problem] [23] to consider Waring’sproblem and the corresponding singular series in the adelic setting from ideas ofIgusa, Ono and Weil. The local singular series in the place p is expressed in the formof counting points. Later it is this expression that applies in the demonstration of GCT: HOMAGE TO GILLES LACHAUD 5 some examples of Yuri Manin’s conjectures about the asymptotic behaviour of thenumber of points of bounded height on an algebraic variety defined over a numberfield when the height goes to infinity by some people who tackled that subject suchas Antoine Chambert-Loir, Yuri Tschinkel, R´egis de la Bret`eche, Tim Browning,Emmanuel Peyre.He worked also on real quadratic fields. In the article
On real quadratic fields [25],he studies the caliber of a real quadratic field Q ( √ d ) (the number k ( d ) of reducedprimitive binary quadratic forms of discriminant d ). A theorem of Siegel saysthere are but a finite number of real quadratic fields with a given caliber but itis not effective. Gilles shows that under GRH, the only real quadratic fields withcaliber one are the seven fields Q ( √ t ) with t = 2 , , , , , , d = r + 1 ≡ Q ( √ t ) with t = 5 , , , , , h ( d ) and the fundamentalunit e : h ( d ) log e ( d ) < c · k ( d ) where c < . Caliber number ofreal quadratic fields [17], where they find, without GRH, the fields with caliber 1and also, the fields of caliber 2 and discriminant ̸≡ the Geometric Goppa codes [24] where he explained these works. I wasthere, especially since at that time, I wanted to change my field of research fromthe Theory of Group Representations to the Theory of Codes.It was also in 1985 that Gilles took up the position of Directeur de RecherchesCNRS and moved from Nice to Luminy. He became the second director of CIRM,from 1986 to 1991. The CIRM was inaugurated in 1991, and Gilles knew it already,as vice-president of SMF, i.e.,
Soci´et´e Math´ematique de France (French Mathemat-ical Society), during 1982–1984, and as the president of the scientific council ofCIRM. His stewardship was around the beginning of CIRM. With the technicalteam, he had a pioneering spirit. During that period, there was a rise in scientificactivities: from 20 symposia in 1986, it reached 37 in 1991.Already, at that time, a certain number of symposia foreshadowed in part themajor currents of mathematical research. There were some big conferences, suchas the 300-person symposium organized by Mohammed Mebkhout, or the
Journ´eesArithm´etiques in 1989, with a similar number of participants, organised by Gilles.The highlight of this period was the construction of the new library. The formerwas housed in the premises of the University library, it was open at the same hoursand it was far from CIRM. Obtaining the necessary funding to build new library forCIRM was not easy, but finally he was able to launch an architectural competition.Of course there was some complication during the works. For instance he wantedto build a beautiful amphitheater in the new building, but while digging, they fellon the rock of a creek (that’s why the conference room at that time was horizontal).
GHORPADE, RITZENTHALER, RODIER, AND TSFASMAN
The library was finally inaugurated in 1991, at the time of the 10th anniversary ofCIRM. It had been a really short time (5 years) since the granting of the creditsuntil the inauguration. Gilles showed his ecological feeling and ended this periodwith a long-term wish: CIRM has 4 hectares of a forest that is quite exceptional,since it is a northern forest in the middle of the southern forest of creeks. Theflora is well supplied, but he complained that the forest areas are not restored atall. He wished those who follow him to achieve preservation and enrichment of thisprecious ecological heritage.In 1987, while Gilles was the head of CIRM, he organised with Jacques Wolfmannfrom the University of Toulon, a conference named Algebraic Geometry and CodingTheory (AGCT). At that time, one did not imagine that this conference would stillexist now under a slightly modified name because of the renewal of the subjects:Arithmetic, Geometry, Cryptography, and Coding Theory (AGC T).He did several works concerning popularization of Mathematics. He wrote severalarticles in the general public journal
La Recherche (Les codes correcteurs d’erreurs)or for the
Encyclopedia Universalis (Qu’est ce qu’une ´equation).In the article
The weights of the orthogonals of the extended quadratic binaryGoppa codes [46] with Jacques Wolfmann they started from results on elliptic curvesand Kloosterman sums over the finite field F t , and they determined the weights ofthe orthogonals of some binary linear codes. The function f ( x ) = x − on F t resiststo differential cryptanalysis and it is used in the “Rijndael” protocol retained inthe call for proposals for the Advanced Encryption Standard (AES) launched bythe National Institute of Standards and Technology (NIST). They could computethe Walsh spectrum of that function from their result giving a relation between thenumber of points of ordinary elliptic curve and Kloosterman sums. This article hasbecome the most quoted article of his.A little before 1990, Gilles created a working group on the Theory of Informationthat he set up at CIRM, that included Guy Chass´e, Yves Driencourt, DominiqueLe Brigand, Jean-Francis Michon, Marc Perret, Robert Rolland, and myself. Thisgroup was very active and organised a meeting every month.At that time, he was also working with G´erard Rauzy and Jean-Yves Girard andwith the head of CNRS to create a CNRS proper laboratory dedicated to DiscreteMathematics. With CIRM, CNRS wanted to create a “mathematics campus” inLuminy. The LMD (Discrete Mathematics Laboratory) was created in Marseille inOctober 1992 with a team ATI (Arithmetic and Theory of Information) that he ledand that I joined naturally.Gilles was generous, always willing to help others. He helped me a lot to makemy moving easier, because I was disabled at the time. In order that I get used togiving talks to students he left me a course of DEA/Master2 to teach in his place.He became the Director of IML (Mathematics Institute of Luminy) from 2006 to2011, and then gave me the place of Director. At that time, I saw the investmentin time and energy that this appointment required, and wondered how he hadmanaged to occupy this position twice while he kept on doing mathematics. GCT: HOMAGE TO GILLES LACHAUD 7
He was optimistic as shows the last letter I received from him, one month beforehe died. “Pour ma sant´e, ¸ca va bien, je suis normal, j’ai un excellent traite-ment, [...] Le r´esultat est que je deviens casanier et paresseux.Donc je fais des maths `a la maison, sur les statistiques de tracesde repr´esentations, ce qui est une activit´e qui se fait assis.J’arrive quand mˆeme `a sortir dans le quartier, et j’ai pu aller `aBesan¸con pr´esider la th`ese de Philippe Lebacque.Il faudrait tout de mˆeme qu’on se voie un de ces jours. Quel estton jour? Le jeudi?” Besides mathematics, Gilles’s interests included philosophy, literature, art, his-tory, and so on. We miss Gilles a lot. He breathed tranquility. I retain his vastculture and his humanism. He leaves a great void.2.
Gilles Lachaud: friend and mathematician
Michael A. TsfasmanAlgebraic geometry codes were discovered by Goppa at the beginning of 1980sand became widely known to mathematicians after 1981 when our preprint of [52]—with the proof that there exist codes over the Gilbert–Varshamov bound—wassmuggled through the iron curtain. In 1985, Gilles Lachaud gave a talk on thesubject [24] at the Bourbaki seminar. Most likely he was attracted by the subjectbecause our proof was based on modular curves, and his basic education was inmodular and automorphic forms. In 1987, he organised the first AGCT meeting atCIRM and invited me to come. Strong mutual dislike between me and the Sovietregime naturally led to the refusal of the leave permit needed at the time to crossthe border. Two years later—the regime being much weakened by perestroika—Icame for AGCT-2, and we saw each other for the first time. Gilles met me at theentrance to CIRM, and we became close friends ever after.I was impressed by the scope of his interests with quite an intersection withmine: literature, philosophy, history, religion, art, and science, not to speak aboutmathematics. I appreciated it even more when I learned French enough to switchto it in our communication.Gilles was the director of CIRM for 6 years and his influence is felt there evennow. He built the library that should bear his name. And he reorganized CIRMin such a way as to make it maximally friendly for the mathematical communityin everything from perfect blackboards to wonderful cuisine. Then he organised “For my health, I’m fine, I’m normal, I have an excellent treatment, [...] The result is thatI become a homebody and lazy. So I do maths at home, on statistics of traces of representations,which is an activity that is done sitting.I can still go out in the neighborhood, and I went to Besan¸con to preside over the thesis ofPhilippe Lebacque.We should still see each other one of these days. What is your day? Thursday?” GHORPADE, RITZENTHALER, RODIER, AND TSFASMAN the LMD (Discrete Mathematics Laboratory) that later became the Institute ofMathematics of Luminy.It is because of him that among the countries to work outside Russia I havechosen France.Now let me describe the part of his legacy that is close to my interests. I meanmostly geometry over finite fields and a little bit coding theory.Gilles’s work on codes and curves over finite fields started with questions wherehe could use his expertise in exponential sums, Kloostermann sums, Eisensteinsums, and so on [26, 45, 28, 31, 32, 29].He also did some nice work, joint with J. Stern, on purely coding theory questions[41, 42]. In the first of these papers they prove that there are families of asymptoti-cally good codes over F q with polynomial complexity of construction whose relativeweights are as close to ( q − /q as we want. In the second one they prove a niceasymptotic lower bound for polynomially constructed spherical codes and for the(polynomial) kissing number.How many F q -points can there be on an abelian variety? On the Jacobian of acurve? Since h X = | J X ( F q ) | = P X (1) the obvious bound is( √ q − g ≤ h ≤ ( √ q + 1) g . In his paper with Martin–Deschamps [37], they prove much better bounds:
Theorem.
For a smooth geometrically irreducible algebraic curve X of genus g ,the number h X = | J X ( F q ) | satisfies the lower bounds h X ≥ q g − ( q − ( q + 1)( g + 1) and h X ≥ ( √ q − q g − − g | X ( F q ) | + q − q − . If g > √ q and if X ( F q ) is nonempty, then h X ≥ ( q g − q − q + g + gq . Much later, in joint papers with Aubry and Haloui [3, 2], they amelioratedthese bounds. In the same paper they proved an analogue of Serre’s bound for anarbitrary abelian variety.His paper with S. Vlˇadut¸ [44] treats the function field analogue of the famousGauss problem asking whether the number of real quadratic fields with class number1 is infinite, or not. Its weak version asks whether the number of principal algebraicnumber fields (i.e., with class number 1) is infinite. Both are wide open. In thefunction field case we have two types of their analogues, the obvious one asks justfor infinity of such fields and is easy to solve in the affirmative, the more reasonableone asks for infinity of such fields when the constant field F q is fixed.Here is an illustrious specimen of their results: GCT: HOMAGE TO GILLES LACHAUD 9
Theorem.
Assume q = 4 , , , , or . Let P ∈ P ( F q ) . Then there areinfinitely many extensions ( K, S ) of the pair ( F q ( T ) , { P } ) such that: (1) The place P completely splits in K . (2) The field K is a Galois extension of F q ( T ) . (3) The ring A S is principal. Now let me discuss in more details our only joint paper
Formules explicites pourle nombre de points des vari´et´es sur un corps fini [Explicit formulas for the numberof points of varieties over a finite field] [43]. In mid-nineties, I realised that thoughthere are plenty of mathematicians working on curves over finite fields and theirnumber of points, actually no-one had ever studied the number of points on surfaces.After a while I had decided to write down my first results on surfaces and multi-dimensional varieties. It was the moment of my intensive attempts to study French(Gilles’ influence, of course). So I started to write my only French language paper[53]. Quite naturally, the French of it was awful. So I asked Gilles to correct it. Hedid it perfectly and little by little we started to think on the subject together.For curves the asymptotic problem is that of the behaviour of the ratio | X ( F q ) | /g X .For multidimensional varieties I suggested to divide by the sum of Betti numbers.The problem becomes mutidimensional, say for surfaces we have two parametersinstead of one, x = b / ( b + b ) and y = N/ ( b + b ), here N = | X ( F q ) | . Therefore,the answer is not a number as in the case of curves (where, at least when q is asquare, it is equal to √ q − x, y )-plane.One of the results of that paper of mine was the generalization of the Drinfeld–Vlˇadut¸ asymptotic upper bound to the multi-dimensional case. I did that usingthe elementary approach of Drinfeld–Vlˇadut¸. It was clear for both of us that theexplicit formulae approach could be better.The process of our work showed the difference between the Russian and Frenchmathematical schools, as well as Gilles’ style of doing mathematics. I would givea vague and often incorrect idea and try to explain it to him. Next day he wouldcome with a couple of pages: precise definitions and clear results. I would read andcriticize it, and then in the morning a better version was given to me.After you write down explicit formulae, to get bounds you need to choose goodtest functions. These test functions should be doubly positive, meaning that boththeir values and their Fourier coefficients are positive. Gilles noticed that thesefunctions happen to be related with (doubly) positive kernels used in sophisticatedanalysis starting from 19th century. After handing the previous night results to meto read, he would go to CIRM library to look through classical books to find newand new ones for us to use.Classics mainly considered doubly positive kernels (densities of the correspondingmeasures) such as, for example, Fej´er, Johnson, and de la Vall´ee-Poussin kernels,and the Jacobi kernel.There is also a more recent source of doubly positive sequences. Namely, J. Oesterl´e,who studied for the case of curves a more subtle question on the minimum genusof a curve having a prescribed number N of points, used explicit formulas andsubstituted into them doubly positive sequences optimal for precisely this problem, WSLM
Fig. 1. Fig. 2. which involves two free parameters, N and g . However, in the asymptotic problem,two parameters appear not for curves but for surfaces, so Oesterl´e sequences yieldgood bounds for the case of surfaces too.Each of the above-mentioned doubly positive kernels or sequences gives a specificbound on the number of points. There are countably many such bounds, theirgraphs intersect each other, and the best known bound is given by their envelopecurve.The result (one of, the paper being large enough) is depicted on the followingfigures for the case of q = 2 (on these plots, the abscissa axis corresponds to theparameter b b + b , and the ordinate axis, to the parameter Nb + b ): Fig. 1 showsbounds that correspond to the above-described doubly positive objects and theirenvelope; Fig. 2, besides the envelope, shows the Weil bound W , Serre bound S ,generalized elementary bound L , and the existence bound M from [53].3. A Tribute to Gilles Lachaud
Sudhir R. GhorpadeDuring August 1993, there was a 3-week CIMPA School on Discrete Mathemat-ics at Lanzhou, China. The speakers included Gilles Lachaud, Maurice Mignotte,Robert Rolland, and Wu Wen-Tsun. I was one of the participants in this school,and it was here that I first met Gilles Lachaud. As it turned out, that was thebeginning of a long and cherished mathematical association and close friendship.In what follows, we will discuss some of Gilles’ work on topics such as countingpoints of algebraic varieties over finite fields, and linear error correcting codes. CIMPA =
Centre International de Math´ematiques Pures et Appliqu´ees = International Cen-tre of Pure and Applied Mathematics
GCT: HOMAGE TO GILLES LACHAUD 11
In the previous two sections, the Bourbaki seminar [24] and some of the earlierworks of Lachaud on coding theory, including his much cited paper with Wolfman[46], have been mentioned. I will begin with an innocuous little paper on projectiveReed-Muller codes that first appeared in the Cachan proceedings [27]. Recall thatgiven any positive integers d, m , the (generalized) Reed-Muller code RM q ( d, m ) oforder d and length n = q m can be defined as the image of the evaluation mapEv : F q [ x , . . . , x m ] ≤ d → F nq given by Ev( f ) = ( f ( P ) , . . . , f ( P n )) , where F q [ x , . . . , x m ] ≤ d denotes the F q -vector space of all polynomials of (total)degree ≤ d in m variables x , . . . , x m with coefficients in the finite field F q , andwhere P , . . . , P n is an ordered listing of distinct points in A m ( F q ). This has beenwidely studied in coding theory, and we cite in particular, the work of Delsarte,Goethals, and MacWilliams [10]. Gilles proposes a natural generalization of thiswhere the affine space A m is replaced by the projective space P m and quite natu-rally, F q [ x , . . . , x m ] ≤ d is replaced by the space F q [ x , x , . . . , x m ] d of homogeneouspolynomials of degree d in m + 1 variables. Further, he takes fixed representativesin F m +1 q of the points of P m ( F q ) (for instance, those where the first nonzero co-ordinate is 1). One can then define a similar evaluation map and its image is theprojective Reed-Muller code PRM q ( d, m ) of order d . It has length(3.1) p m := | P m ( F q ) | = q m + q m − + · · · + q + 1 . Moreover, if d ≤ q , then the evaluation map is injective and so the dimension ofPRM q ( d, m ) is ( m + dd ) . In [27], Gilles gives a lower bound on the minimum distanceof PRM q ( d, m ), and computes the exact value in the special cases d = 1 (wherethe Plotkin bound is attained) and d = 2. The latter uses classical facts aboutzeros of quadrics in finite projective spaces. For the general case, one needs tightbounds on the maximum number of F q -rational points on a hypersurface in P m or inother words, the maximum number, say N d , of zeros in P m ( F q ) that a homogeneouspolynomial of degree d in m +1 variables with coefficients in F q can have. Tsfasmanconjectured that if d ≤ q +1, then N d = dq m − + p m − , where p j is as in (3.1) if j is anonnegative integer and p j := 0 if j <
0. Tsfasman’s conjecture was quickly provedin the affirmative by J.-P. Serre, who communicated it to Tsfasman in a letter whichwas published some two years later [50]. Gilles was then in a position to completethe results of [27] and give a formula for the minimum distance in the general casein [30], where he argues that from coding theoretic viewpoint, projective Reed-Muller codes are somewhat better than (affine, or generalized) Reed-Muller codes,and as such, they deserve to be studied further. A beginning was already made bySorensen [51] whose paper, published in the same year as [50], and gives, in effect,an alternative proof of Tsfasman’s conjecture and additional results of particularinterest to coding theory.The introduction of projective Reed-Muller codes by Gilles and the questionabout the minimum distance of these codes, which is essentially equivalent to Ts-fasman’s conjecture mentioned above, has had many offshoots, and we will brieflydescribe some of them.
Tsfasman’s conjecture is about projective hypersurfaces defined over F q of degree d ≤ q + 1. Now a hypersurface is the simplest example of a complete intersectionin P m of high dimension (vis-`a-vis m ). Keeping this in view, Gilles formulated thefollowing conjecture (which appeared in print much later, in our joint paper [13]). Conjecture. If X is a complete intersection in P m defined over F q of dimen-sion n ≥ m/ and degree d ≤ q + 1 , then | X ( F q ) | ≤ dp n − ( d − p n − m = d ( p n − p n − m ) + p n − m . Observe that if X is a hypersurface (so that n = m − d ≤ q + 1 to intersection, say S d,r of r hypersurfaces, each defined by a homogeneous polynomial of degree d < q ,such that these r polynomials are linearly independent. Tsfasman-BoguslavskyConjecture proposes an intricate, but explicit, formula T r ( d, m ) for the maximumnumber of F q -rational points of algebraic sets such as S d,r . Again, if r = 1, thenthis reduces to Tsfasman’s Conjecture about hypersurfaces.In 2015-2017, that is, roughly about two decades after these conjecture weremade, two interesting developments took place almost simultaneously. On the onehand, Alain Couvreur [7] showed that Lachaud’s Conjecture, and in fact, a moregeneral result, holds in the affirmative. On the other hand, Mrinmoy Datta andI showed that the Tsfasman-Boguslavsky Conjecture holds if r ≤ m + 1, but itis false, in general (cf. [8], [9]). I think Gilles was rather happy and proud thathis guess turned out to be correct. Shortly before these developments took place,Gilles himself returned to the topic and together with Robert Rolland, he wrotea nice article [40] giving various bounds for the number of F q -rational points ofaffine as well as projective varieties over finite fields. In AGC T-2015, Lachaud,Couvreur and I gave talks in a single session to explain these developments, and Iremember how Gilles provided a nice perspective on the topic in the first talk of thatsession. I might mention in passing that a refined version of Tsfasman-BoguslavskyConjecture has been proposed in [9] and [5]. The newer conjectures remain openin general although they are known to be valid in many cases (cf. [4] and [5]).Another interesting offshoot of the work of Gilles on projective Reed-Muller codeand the inequality Serre proved to answer Tsfasman’s Conjecture is a developmentwhere Gilles himself was involved. It is the subject matter of his paper [1], whichwas published barely six months before he passed away. Perhaps it is worthwhileto mention a background to this. In February 2016, Everett Howe, Kristin Lauter,and Judy Walker organized a workshop on Algebraic Geometry for Coding Theoryand Cryptography at the Institute for Pure and Applied Mathematics (IPAM) onthe campus of the University of California, Los Angeles. They said that this willnot be a standard “five talks per day with a break for lunch” type of conference,but instead an activity focused on research projects that will be started during theworkshop with a hope that it will result in ongoing collaboration among partici-pants. Gilles and I were asked to be co-leaders of a working group. Both of uswere apprehensive about this experiment. Nonetheless, we put together a group
GCT: HOMAGE TO GILLES LACHAUD 13 of six participants including Yves Aubry, Wouter Castryck, Mike O’Sullivan, andSamrith Ram. The broad topic for our group was “Number of points of algebraicsets over finite fields”. After some introductory talks, we discussed every day ofthe week (and sometimes late in the evenings as well) and bounced ideas off eachother. As a result of this brainstorming, and continued correspondence thereafter,we wrote an article concerning an analogue of Serre’s inequality for hypersurfaces inweighted projective space, and weighted projective Reed-Muller codes. Gilles wasa very active participant in this, and wrote almost single handedly a compendiumon the (geometry of) weighted projective spaces as a handy manual for us, whichformed an appendix to our article [1]. Although there are many partial results inthis article, the problem of determining the maximum number of F q -rational pointson a hypersurface in weighted projective spaces is open, in general, and a conjec-ture stated here could be a pointer for further research on the topic. Before endingthe discussion of projective Reed-Muller codes introudced by Gilles, I would like tometion an article of well-known finite geometers and group theorists Bill Kantorand Ernie Shult [18], which was published in 2013. Here they remark that theirfirst main theorem can be viewed as a statement about a certain code C havinga check matrix whose columns consist of one nonzero vector in each Veroneseanpoint, Then they write: We have not been able to find any reference to this code inthe literature. It is probably worth studying, at least from a geometric perspective.
In fact, the code C that they talk about is nothing but the dual of the projectiveReed-Muller code studied by Gilles about 25 years ago!I would now like to go back to my first meeting with Gilles at the 1993 CIMPAworkshop in China, which was also the first time I learned about error correctingcodes from the lecture courses of Robert Rolland and Gilles Lachaud. In the courseof lectures or possibly, during the discussions over meals, I learned about the seem-ingly intractable open problem of determining an explicit formula for the number,say γ ( q ; k, n ), of q -ary MDS codes of length n and dimension k . The answer wasknown for k ≤ k = 3 and n ≤
8. I could see that the problem wasequivalent to determining the number of F q -rational points of the open subset U k,n of the Grassmannian G k,n consisting of points of G k,n for which all the Pl¨uckercoordinates are nonzero. Based on this observation, I could make some observa-tions, which interested Gilles and he encouraged me to think further. The activitypicked up when Gilles arranged for me to get a fellowship from CNRS to visit himMarseille for a month in June 1996. This was the first of my numerous visits toMarseille in subsequent years. Using the connection with Grassmannians togetherwith the work of Nogin [49] on Grassmann codes, and various combinatorial inputs,we could obtain explicit lower and upper bounds for γ ( q ; k, n ) and an asymptoticformula in the general case. Further, we could use some of the auxiliary resultstogether with Grothendieck-Lefschetz trace formula to derive some geometric prop-erties of certain linear sections of Grassmannians. These results are given in ourarticle Hyperplane sections of Grassmannians and the number of MDS linear codes [12]. Originally, there was to be a remark in this paper indicating how some re-sults of Nogin on higher weights of Grassmann codes could also be derived as a consequence of the combinatorial results we had in preparation for our results on γ ( q ; k, n ). But then there was to be a conference in Guanajuato, Mexico in 1998,which Gilles encouraged me to attend. We decided to expand this remark and usedthe opportunity to introduce what seemed a natural generalization of Grassmanncodes, called Schubert codes. This paper Higher weights of Grassmann codes [11]was published in the proceedings of the Guanajuato conference and it contained aconjecture about the minimum distance of Schubert codes. As it turned out, [11]led to a lot of further research and it has many more citations than [12]. The moralof the story for me (and I think Gilles agreed with that) was: A good problem issometimes more valuable than a good theorem!In the course of our work on MDS codes, we thought it would be nice to haveconcrete estimates for the number of F q -rational points of arbitrary algebraic va-rieties, especially when the variety is not necessarily smooth. The best generalestimate we had at our disposal was the following inequality proved by Lang andWeil in 1954: Lang-Weil Inequality.
Let X be an (absolutely) irreducible projective variety in P N defined over F q . If X has dimension n and degree d , then || X ( F q ) | − p n | ≤ ( d − d − q n − (1 / + Cq n − , where C is a constant depending only on N , n , and d . When the variety is smooth, and better still, a complete intersection, muchsharper estimates are available such as the one obtained by Deligne in 1973 as aconsequence of his seminal work on Weil conjectures.
Deligne’s Inequality for Smooth Complete Intersections. If X is anonsingular complete intersection in P N over F q of dimension n = N − r , then || X ( F q ) | − p n | ≤ b ′ n q n/ . Here b ′ n = b n − ϵ n is the primitive n th Betti number of X (where ϵ n = 1 if n is evenand ϵ n = 0 if n is odd), and p n := | P n ( F q ) | = q n + q n − + · · · + q + 1 . We remark that if X has multidegree d = ( d , . . . , d r ), then a formula of Hirze-bruch shows that b ′ n = b ′ n ( N, d ) equals( − n +1 ( n + 1) + N ∑ c = r ( − N + c ( N + 1 c + 1 ) ∑ ν + ··· + ν r = cν i ≥ ∀ i d ν · · · d ν r r . From a practical viewpoint, one likes to have some control on the constant C appearing in Lang-Weil inequality and have a Deligne-like inequality for completeintersections (in particular, hypersurfaces) that are not necessarily smooth. We hadan idea how this might be achieved, and we worked for years together to prove thefollowing result ([13, Theorem 6.1]), which is one of the main results of our paper ´Etale cohomology, Lefschetz theorems and number of points of singular varietiesover finite fields [13], published in a special volume of Moscow Mathematical Journal dedicated to Yuri I. Manin on the occasion of his 65th birthday.
GCT: HOMAGE TO GILLES LACHAUD 15
Deligne-type inequality for arbitrary complete intersections.
Let X be an irreducible complete intersection of dimension n in P N F q , defined by r = N − n equations, with multidegree d = ( d , . . . , d r ) , and let s be the dimension ofthe singular locus of X . Then || X ( F q ) | − p n | ≤ b ′ n − s − ( N − s − , d ) q ( n + s +1) / + C s ( X ) q ( n + s ) / , where C s ( X ) is a constant independent of q . If X is nonsingular, then C − ( X ) = 0 .If s ≥ , then C s ( X ) ≤ × r × ( rδ + 3) N +1 where δ = max { d , . . . , d r } . For normal complete intersections, this may be viewed as a common refinementof Deligne’s inequality and the Lang-Weil inequality. Corollaries include previousresults of Aubry and Perret (1996), Shparlinski˘ı and Skorobogatov (1990), as wellas Hooley and Katz (1991).We also used the power of Weil conjectures and some estimates of Katz (2001) toarrive at a version of Lang-Weil inequality with an explicit bound on the constant C appearing therein ([13, Theorem 11.1]). A precise statement is given below. Here,given any m -tuple d = ( d , . . . , d m ) of positive integers, we say that an affine (resp.projective) variety in A N F q (resp. P N F q ) is of type ( m, N, d ) if it can be defined by thevanishing of m polynomials in N variables (resp. m homogeneous polynomials in N + 1 variables) with coefficients in F q . Effective Lang-Weil inequality.
Suppose X is a projective variety in P N F q or an affine variety in A N F q defined over F q . Let n = dim X and d = deg X . Then || X ( F q ) | − p n | ≤ ( d − d − q n − (1 / + C + ( ¯ X ) q n − , where C + ( ¯ X ) is independent of q . Moreover, if X is of type ( m, N, d ) , with d =( d , . . . , d m ) , and if δ = max { d , . . . , d m } , then C + ( ¯ X ) ≤ { × m × ( mδ + 3) N +1 if X is projective × m × ( mδ + 3) N +1 if X is affine . As a corollary, one obtains an analogue of a result of Schmidt (1974) on a lowerbound for the number of points of irreducible hypersurfaces over F q .Finally, we showed that certain conjectural statements of Lang and Weil (1954),relating the Weil zeta function of a smooth projective variety X defined over F q and the “characteristic polynomial” of its Picard variety, hold true provided oneuses the “correct” Picard variety. For more details, one may refer to our paper[13] or the expository article Number of solutions of equations over finite fieldsand a conjecture of Lang and Weil [14] that appeared in the proceedings of aninternational conference on Number Theory and Discrete Mathematics, held atChandigarh, India, which Gilles attended.Although it took a long time to complete this work, it was quite satisfying in theend. What was particularly gratifying was to see many seemingly disparate topicsto which our results found applications (references to some of such applicationsas well as to extensions and generalizations of our results, especially by Cafure and Matera, can be found in the corrigenda and addenda [15] to [13]). For me, itwas a great learning experience to work with Gilles on this topic and to witnessfirsthand his tenacity and technical skills. At the beginning of this paper, there isa
Sanskrit verse from
Rg Veda that can roughly be translated as “Their cord wasextended across”. Now one might be tempted to think that this was put in thereby the Indian coauthor. But the fact is it was Gilles who insisted that we insertit. To him, it conveyed somehow that we are merely extending the key ideas of ourillustrious predecessors. Also, he felt it was pertinent because we frequently useBertini-type theorem to cut the variety by suitable linear sections!I would like to end by reproducing the concluding remarks in my talk on Gillesat AGCT-2019:Gilles Lachaud has made important and lasting contributions tomathematics, especially in the study of algebraic varieties over fi-nite fields and linear codes. His knowledge and interests were deepand wide. When he became interested in some topic, he wouldusually delve deeper and spend considerable time learning manyaspects of it. As far as I have seen, he would never be in a rushto publish quickly, but would prefer to take time and be thorough.Besides his contributions to mathematics, Gilles was an institu-tion builder. He helped nurture an institution like CIRM. Also,the continuing success of the AGCT conferences owes largely tohis vision and efforts.Other than scientific institutes and conferences, Gilles served asthe President of the French Pavilion at Auroville, near Pondicherry,India. He had read or had at least browsed through significantamount of ancient and modern 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 studentsand younger colleagues. His untimely demise last year is a greatloss to our subject and the community. Personally, it has beena pleasure and honor to have known him. He will certainly bemissed... 4.
Gilles Lachaud’s work
Christophe RitzenthalerAmong the four contributors, I am the one who met Gilles the latest, precisely in2006. He was then director of IML (Institut de Math´ematiques de Luminy) in Mar-seille and I was starting as Maˆıtre de Conf´erences there in the ATI (Arithm´etiqueet Th´eorie de l’Information) team. I think I owe to our common interest in genus3 non-hyperelliptic curves that he invited me to apply for this position.
GCT: HOMAGE TO GILLES LACHAUD 17
In 2005, Gilles had indeed published two articles [34, 33] on the Klein quartic(the smooth quartic X : x y + y z + z x = 0) in Moscow Mathematical Journal . Asmany other people (a full book “The eightfold way” [47] was published a few yearsearlier on this single curve), Gilles was fascinated by the conjunction of beautifulanalytic, algebraic and arithmetic geometry and group and representation theoryto address this particular genus 3 non-hyperelliptic curve. In the first and longestpaper Ramanujan modular forms and the Klein quartic , he conjugates his passionfor history of mathematics and research, unravelling Ramanujan’s contributions tosome identities between modular forms of level 7. The relation with modular formsboils down to the fact that X is isomorphic over C to the modular curve X (7),since they are, up to isomorphisms, the unique genus 3 curve with automorphismgroup isomorphic to PSL ( F ). He then gives four different expressions for an L -series which encodes the number of points of the Klein quartic over finite fields,and derives some relations between the Weil polynomial of X ⊗ F p and the one ofan elliptic curve E with complex multiplication by Z [(1 + √− / The Klein quartic as a cyclic group generator , Gilles comes back to this lastproblem and gives new expressions for the Weil polynomial. He does it not onlyfor the Klein quartic but also for some of its twists ax y + by z + cz x = 0 with a, b, c ∈ F × p . This is realized as a special case of a more general result on Faddeevcurves that he proves using Davenport-Hasse’s method. These results have beenexpanded later by Meagher and Top in [48] to all twists of the Klein quartic.When I arrived in Marseille, it was then natural that we started a collaborationon a tantalizing question of Serre on genus 3 curves. In his lectures Rational pointson curves over finite fields at Harvard in 1985, Serre explains that a principallypolarized abelian variety A of dimension g > k , whichis geometrically a Jacobian, is not necessarily a Jacobian over k (unlike in dimension1 or 2). The obstruction is given by a quadratic character of Gal(¯ k/k ) and is called Serre’s obstruction . When k ⊂ C and g = 3, he speculates in his lectures (and givesmore details in a letter to Top in 2003), that this obstruction can be computed interms of the value of a certain Siegel modular form called χ . At that time,it was pure magic to us as this Siegel modular form was a product of so-calledtheta constants that are purely transcendental objects and which only “see” thecurve over C . We could not imagine them containing arithmetic information. Inorder to understand Serre’s intuition, we decided to work out the formulae in theparticular case of Ciani quartic. They form a dimension 3 family of quartics withautomorphism group containing a group isomorphic to ( Z / Z ) . This structurehelps decomposing their Jacobians into a product of three explicit elliptic curvesup to isogeny. In 2000, Howe, Leprevost and Poonen in [16] had shown how,conversely, starting from elliptic curves E , E and E defined over a field k and a“good” k -rational maximal isotropic subgroup G ⊂ ( E × E × E )[2], the abelianthreefold A = ( E × E × E ) /G is the Jacobian over k of an explicit Ciani quarticif and only if a certain expression in terms of the coefficients of the E i is a non-zero From some manuscript notes I saw, it was part of a broader program to study the geometryand arithmetic of more general family of quartics with non-trivial automorphism groups. square in k . They had found an algebraic expression of Serre’s obstruction for A in this particular case! In On some questions of Serre on abelian threefolds [38],we worked out this expression in terms of the theta constants associated to the E i ,then to A and eventually in terms of χ as required. This first collaboration withGilles went very smoothly: I had the tools to manipulate theta functions, and heknew very well the geometry of this family of quartics and useful decompositionsof the symplectic group from his earlier works. This result was presented at theconference organized for Gilles’ 60th birthday in Tahiti in 2007.More than confirming Serre’s intuition for a dimension 3 family, this work madeus realize the correct normalization of the Siegel modular form we should look for.We used this knowledge to work out a complete proof of Serre’s formula in our nextarticle Jacobians among abelian threefolds: a formula of Klein and a question ofSerre [39], in collaboration with Zykin (simultaneously, Meagher got a similar resultin his PhD). In order to achieve that, the best formalism was to use the theory ofalgebraic stacks and geometric modular forms. It sounded scary at first but Gilleswas confident and calm as usual and we managed to get our way through. Thecherry on top of the cake was that we could make precise a formula due to Kleinlinking this modular form to the discriminant of plane quartics. This formula is thefirst beachhead that enables connecting very precisely the world of modular formsand invariants as it was done previously in genus 1 and 2. We used it recently withLercier to give the complete dictionary in genus 3.Three years later, I was leaving for Rennes, and we did not get the chanceto collaborate more. Driven by his tireless curiosity and maybe triggered by aconference and a winter school at CIRM in 2014 and discussions with Kohel, Gilles’interest moved towards arithmetic statistics. In his first article on the topic [35],
Onthe distribution of the trace in the unitary symplectic group and the distribution ofFrobenius , he considers the group USp g which corresponds to the “generic case” forcurves and abelian varieties in Katz-Sarnak theory. He recalls the link between thedistribution associated to the trace on USp g and the distribution of the numberof points for curves or abelian varieties over finite fields. Although this is well-known to experts, this is done with great care and with the purpose of making itexplicit enough for anybody who would need to do actual computations. When g = 2 and 3, he then gives some useful expressions for the distribution law in termsof hypergeometric series and related functions and draws some inspiring graphicsrepresenting the density of curves with a certain Weil polynomial. In 2018, in Thedistribution of the trace in the compact group of type G [36], he works out similarformulas for the case of a group (the only compact simple Lie group of type G )related to exponential sums and introduced by Katz. In the present volume, onecan find his last work on the case of SU (3).During my time in Marseille, Gilles, as director, managed to create such a pleas-ant working atmosphere that the years I spent there were extraordinary. Although,as I found out later, such a position is really time-consuming, his door was alwaysopen, and he never made anybody felt indebted for the facilities he provided. I GCT: HOMAGE TO GILLES LACHAUD 19 learned a lot from working with him and I still meditate about his way of address-ing new problems. When I consider them as “enemies” that should surrender afteryou have attacked them from every angles, Gilles never got upset by their resistanceand approached them peacefully. He was also a true humanist as Fran¸cois, Michaeland Sudhir mentioned, with many passions outside mathematics. In his car, stuckin the traffic on our way back from Luminy some evenings, I was always happy tolisten to his insights on Japanese culture, his collaborative projects with India orlately his study with his wife Patricia of the secrets of the painting
The Tempest by Giorgione. I guess we will all miss his friendly attitude towards life.
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Department of Mathematics, Indian Institute of Technology Bombay,Powai, Mumbai 400076, India
E-mail address : [email protected] Univ Rennes, IRMAR, Campus de Beaulieu, 35042 Rennes, France
E-mail address : [email protected] Aix Marseille Universit´e, CNRS, Centrale Marseille,Institut de Math´ematiques de Marseille, UMR 7373, 13288 Marseille, France
E-mail address : [email protected] CNRS, Laboratoire de Mathematiques de Versailles (UMR 8100), FranceInstitute for Information Transmission Problems, Moscow, RussiaIndependent University of Moscow, Russia
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