Poincaré s invariants arithmétiques

Transcription

1 Poincaré s invariants arithmétiques

2 A. Weil : Les écrits de Poincaré qui touchent à l Arithmétique occupent un volume entier (tome V des Œuvres). On ne saurait nier qu ils sont de valeur inégale [...] Poincaré s writings that relate to number theory occupy an entire volume (Volume V of it complete works). One can not deny that they are unequal in value [...]

3 Weil highlights two pieces of Poincaré s number theoretical works. Relations between non-euclidean geometry and indefinite ternary quadratic forms leading to arithmetic (fuchsian) groups. The long 1901 paper entitled Sur les propriétés arithmétiques des courbes algébriques where Poincaré introduit la notion de rang d une [...] courbe [algébrique rationnelle de genre 1]; ce rang est à peu près le plus petit nombre de points rationnels sur la courbe à partir desquels on puisse obtenir tous les autres par addition des arguments elliptiques attachés à ces points lorsqu on uniformise la courbe par des fonctions elliptiques.

4 Weil highlights two pieces of Poincaré s number theoretical works. Relations between non-euclidean geometry and indefinite ternary quadratic forms leading to arithmetic (fuchsian) groups. The long 1901 paper entitled Sur les propriétés arithmétiques des courbes algébriques where Poincaré introduces the rank of a rational elliptic curve; this rank is roughly the smallest number of rational points on the curve from which we can get all the others by adding the elliptical arguments attached to these points using the elliptic functions uniformizing the curve.

5 Weil s conclusion is: J espère vous avoir montré que l œuvre de Poincaré n appartient pas seulement à l histoire de notre science; elle appartient aussi à la plus vivante actualité. I hope I have shown that Poincaré s work not only belongs to the history of our science, but also belongs to its most vivid events.

6 I want to discuss two papers of Poincaré which have been quite neglected (not only for bad reasons) but which contain quite surprising results: Sur les invariants arithmétiques, published in Crelle s journal in 1905 on the occasion of the 100th anniversary of Dirichlet s birth. Fonctions modulaires et fonctions fuchsiennes, Poincaré s very last paper which he sent to print the day before he left for the hospital where he would die. In doing so I am motivated by the following quotation from L avenir des mathématiques (1908).

7 Un domaine arithmétique où l unité semble faire absolument défaut, c est la théorie des nombres premiers; on n a trouvé que des lois asymptotiques et l on n en doit pas espérer d autres; mais ces lois sont isolées et l on n y peut parvenir que par des chemins différents qui ne semblent pas pouvoir communiquer entre eux. Je crois entrevoir d où sortira l unité souhaitée, mais je ne l entrevois que vaguement ; tout se ramènera sans doute à l étude d une famille de fonctions transcendantes qui permettront, par l étude de leurs points singuliers et l application de la méthode de Darboux, de calculer asymptotiquement certaines fonctions de très grands nombres.

8 A domain of arithmetic where unity seems completely missing, is the theory of prime numbers; only asymptotic laws have been found, and one can not hope for others; but these laws are isolated and one can only reach them by different paths which do not seem to be able to communicate. I believe I can glimpse where the desired unity will come from, but I see this only vaguely; all will probably be reduced to the study of a family of transcendental functions which will permit, through the study of their singular points and application of Darboux s method, the asymptotic computation of certain functions of very large numbers.

9 Thesis: The transcendental functions automorphic forms. The singular points cups. In his papers on analytic number theory Poincaré tries to relate all sorts of arithmetical functions to the automorphic forms he just created. In 1905 he already knew all the different kinds of constructions of automorphic forms on GL(2) that Gelbart mentions in the first chapter of his 1973 book. Poincaré had developped the tools but he was still in search for truly new number theoretical problems to which he could apply them. Il est clair que les propriétés générales des fonctions fuchsiennes s appliquent aux fonctions modulaires [...] It is obvious that general properties of fuchsian functions apply to modular functions [...]

10

11 Arithmetical invariants of linear forms These are functions (ω 1, ω 2 ) F (ω 1, ω 2 ) invariant under SL 2 (Z): F (aω 1 +bω 2, cω 1 +dω 2 ) = F (ω 1, ω 2 ) (a, b, c, d Z, ad bc = 1). If moreover F is supposed to be homogeneous of weight k: F (λω 1, λω 2 ) = λ k F (ω 1, ω 2 ), then f (z) = F (z, 1) is modular of weight k: f (z) = f k g (z) := (cz + d) k f ( ) a b for all g = SL c d 2 (Z) and z H. ( az + b cz + d ),

12 Example (Eisenstein) Consider G k (ω 1, ω 2 ) = m,n 1 (mω 1 + nω 2 ) 2k, where means that (m, n) (0, 0). If k > 1 then G k is absolutely convergent; it defines a modular function of weight 2k. Note that G k ( ) = 2ζ(2k) 0.

13 A function f : H C is modular of weight 2k iff the form f (z)(dz) k is SL 2 (Z)-invariant, in otherwords goes down on the modular surface:

14 Definitions A function f : H C belongs to A k if it is meromorphic, modular of weight 2k and meromorphic at infinity, i.e. f (z) = n n 0 a n q n with q = e 2πiz. It belongs to M k if it is holomorphic on H and at infinity, i.e. n 0 0. It belongs to S k if it moreover vanishes at infinity, i.e. n 0 1. Note that G k belongs to M k but not to S k.

15 The starting point of Poincaré s theory of Fuchsian functions is the following. Let k > 1. Proposition Let f : CP 1 CP 1 be a rational function without poles on R and with a zero of order at least 2k at infinity. Then g SL 2 (Z) f 2k g (z) = a b SL c d 2 (Z) (cz + d) 2k f ( ) az + b cz + d is absolutely convergent for z H; its sum Θ f is modular of weight 2k and vanishes at infinity. Moreover: S k = span{θ f f is holomorphic on H}.

16 Consider h(z, τ) = 1 (τz 1) 2k, (z, τ H). Then Θ(z, τ) = g SL 2 (Z) h(, τ) 2k g (z) is modular in both variables and the asymptotic expansion as τ is Θ(z, τ) = (2iπ)2k (2k 1)!τ 2k + m=1 m 2k 1 e 2πi m τ Pm (z, k) where the P m (z, k) = c,d Z c d=1 (cz + d) 2k az+b 2πim e cz+d are now called Poincaré series.

17 Decomposing rational fractions into simple elements, Poincaré proves that Poincaré series P m (z, k), (m > 0, k > 1), span S k. This is where he stops on that matter in his 1905 paper. In his last paper he computes the Fourier coefficients of the P m (z, k) introducing for the first time the Kloosterman sums. I don t know what was Poincaré s motivation to do so. But the tools are here ready to be used by others (Kloosterman, Hecke, Selberg...). In his 1905 Memoir, Poincaré then switches to another subject which is less known and quite surprising.

18 Polynomial periods Poincaré proposes to extend the classical theory of Abelian integrals and their periods on a compact Riemann surface to integrals of higher order, but now having polynomial periods. Definition A generalized Abelian differential is a meromorphic map F : H C such that f := D 2k 1 F is modular of weight 2k. We say that F is of the first kind if f S k. Here D = d dz.

19 Using Bol s identity D 2k 1 (F 2 2k g ) = (D 2k 1 F ) 2k g g SL 2 (Z), Poincaré then shows that for all g = ( ) a b c d SL2 (Z), D 2k 1 ((cz + d) 2k 2 F ( az + b cz + d ) ) F (z) = 0. So that there exists a polynomial P g C 2k 2 [X ] such that ( ) az + b F = (cz + d) 2 2k F (z) + (cz + d) 2 2k P g (z). cz + d

20 Les fonctions [F ] jouent le rôle des intégrales abéliennes [...] les polynômes P jouent le rôle des périodes. The functions [F ] play the role of Abelian integrals [...] the polynomials P play the role of the periods.

21 Le nombre des coefficients arbitraires des périodes est double de celui des intégrales de première espèce. The number of arbitrary coefficients of the periods is twice the one of integrals of the first kind.

22 The map c : g P g (= F g F ) satisfies the cocyle relation P g1 g 2 = (P g1 ) g2 + P g2 w.r.t. the action of SL 2 (Z) on C 2k 2 [X ] given by So that P a b c d (g, P) P g = P 2 2k g. ( ) (X ) = (cx + d) 2k 2 P ax +b cx +d. The class of c modulo coboundaries g P g P (P C 2k 2 [X ]) only depends of f = D 2k 1 F.

23 Poincaré then essentially proves the following theorem rediscovered by Eichler about 50 years later. Theorem We have an explit isomorphism: H 1 0 (SL 2 (Z), C 2k 2 [X ]) = S k S k. Here again Poincaré does not pursue this direction. Later Eichler applied this theory to obtain the trace formula for Hecke operators really opening the arithmetic side of this story.

24 Arithmetical invariants of positive definite quadratic forms Notations Quadratic forms Q(x, y) = ax 2 + bxy + cy 2 = [a, b, c] with d = b 2 4ac < 0 and a > 0. The modular group SL 2 (Z) acts on quadratic forms by (g, Q) Q t g. Definition An arithmetical invariant is a map Q F (Q) C which is SL 2 (Z)-invariant.

25 Example (Dirichlet) The series ζ Q (s) = m,n 1 Q(m, n) s = 1 (am 2 + bmn + cn 2 ) s m,n are absolutely convergent for Re(s) > 1; the map Q ζ Q (s) then defines an arithmetical invariant. Poincaré rather considers Θ Q (z) = m,n q am2 +bmn+cn 2, with q = e 2πiz (z H). Note that (2π) s Γ(s)ζ Q (s) = 0 y s (Θ Q (iy) 1) dy y.

26 The reason why Poincaré rather likes Θ Q is because he proves: Theorem The map f : z H Θ Q (z) 2 satisfies f 2 g (z) = f (z) z H, g Γ 0 (4d). Here Γ 0 (4d) = {( ) a b c d SL2 (Z) } c 0 (mod 4d).

27 L étude de ce groupe fuchsien jetterait sans doute quelque lumière sur les propriétés arithmétiques des formes quadratiques. The study of this Fuchsian group would certainly shed some light on the arithmetical properties of quadratic forms.

28 Proof Poincaré gives a physical proof of a generalized form of Poisson summation formula: e 2πiz(am2 +bmn+cn 2) e i(mu+nv) m,n i = 2 e 2πi 4 d z [a(v n)2 b(v n)(u m)+c(u m) 2 ]. d z m,n Setting u = v = 0, he concludes that ( d 0 Θ Q (z) = i 2 d z Θ Q ( 1 4 d z ) (z H). Since Θ Q (z + 1) = Θ Q (z), the map f : z H Θ Q (z) 2 is modular ) of weight 2 w.r.t. to the group generated by ( ) and.

29 Here again Poincaré does not pursue this direction of research the theory will have to wait for Siegel. He only explains how to recover Dirichlet s class number formula from a weak version of the theorem, namely: Θ Q (iy) 1 2, as y 0. d y

30 Let h be the number of SL 2 (Z)-classes of positive definite quadratic forms Q = [a, b, c] with a, b, c Z, (a, b, c) = 1 and b 2 4ac = p with p prime > 3 and p 3 mod 4. Dirichlet s class number formula h = 2 p π + n=1 ( ) Here n p is the Legendre symbol: ( ) n 1 p n. ( ) n 1 if n is a non-zero square mod p, = 1 if n is not a square mod p, and p 0 if p divides n.

31 Proof For every posive integer n we let R(n, Q) = #{(x, y) Z 2 Q(x, y) = n} and R(n) = Q S R(n, Q) where S is a set of h representatives of classes of primitive positive definite quadratic forms. Classical Lemma R(n) = 2 m n ( ) m. p

32 Q S Θ Q (z) = Q S = h + (x,y) Z 2 q ax2+bxy+cy 2 R(n)q n n=1 = h + 2 n=1 m n ( m = h + 2 p m=1 k=1 ( m = h + 2 p m=1 ( ) m q n p ) q km ) q m 1 q m.

33 The LHS is have: h 2 py q m 1 q m = 1 2πy as z = iy 0 (in H). On the other side we ( ) 1 1 m + O y 2 with q = e 2πy and y 0 +. We conclude: h 2 p = 1 π m=1 ( ) m 1 p m.

34 The End Finally Poincaré considers the case of indefinite forms Q = [a, b, c] with (a, b, c) = 1, b 2 4ac > 0 and a > 0. Then Q has an infinite group of automorphs and the above Dirichlet series diverges. One has to consider 1 Q(m, n) s m,n where correspond to a sum in a certain cone. Poincaré fails to relate these series to automorphic forms. But he relates them to the L-function of the associated real quadratic field. Moreover: he considers (for the first time?) more general L-functions associated to particular Hecke Grössencharacters and essentially give all the ingredients to prove their meromorphic continuation to the whole complex plane.

35

36 The series he considered are studied along the same lines by Zagier in a 1977 paper Valeurs des fonctions zêta des corps quadratiques réels aux entiers négatifs without any mention to Poincaré. His work on analytic number theory seems to have been forgotten by the (french) community. Quoting Godement (2003): J ai rencontré Serre l autre jour, il m a dit: Tiens, je me rappelle, c est dans tes exposés au Séminaire Bourbaki [ ] que j ai appris ce qu étaient les formes modulaires. Non, j ai été le premier à en parler. C était totalement inconnu. Les gens avaient complètement oublié qu il y avait Poincaré qui... et Hecke...

37 THANKS FOR YOUR ATTENTION!