Asymptotically exact sequences of algebraic function fields defined over F q and application

Transcription

1 Asymptotically exact sequences of algebraic function fields defined over F q and application Stéphane Ballet and Robert Rolland Institut de Mathématiques de Luminy UMR C.N.R.S. / Université de la Méditerranée Luminy Case 930, F13288 Marseille CEDEX 9 FRANCE Stéphane Ballet and Robert Rolland () Asymptotically exact sequences 1 / 14

2 Asymptotically exact sequences Definition Definition Consider a sequence F /F q = (F k /F q ) k 1 of algebraic function fields F k /F q defined over F q of genus g k = g(f k /F q ). We suppose that the sequence of genus g k is an increasing sequence growing to infinity. The sequence F /F q is called asymptotically exact if for all m 1 the following limit exists : B m (F k /F q ) β m (F /F q ) = lim g k g k where B m (F k /F q ) is the number of places of degree m on F k /F q. Stéphane Ballet and Robert Rolland () Asymptotically exact sequences 2 / 14

3 Asymptotically exact sequences Explicit families of asymptotically exact sequences Proposition Let r be an integer and F /F q = (F k /F q ) k 1 be a sequence of algebraic function fields defined over F q such that β r (F /F q ) = 1 r (q r 2 1). Then β m (F /F q ) = 0 for any integer m r. In particular, the sequence F /F q is asymptotically exact. Stéphane Ballet and Robert Rolland () Asymptotically exact sequences 3 / 14

4 Asymptotically exact sequences Explicit families of asymptotically exact sequences Example of explicit asymptotically exact sequences Let F q 2 with q = p r and r an integer. The Garcia-Stichtenoth tower T 0 over F q 2 : the sequence (F 0, F 1,...) where and z k+1 satisfies the equation : F k+1 := F k (z k+1 ) z q k+1 + z k+1 = x q+1 k with x k := z k /x k 1 in F k (for k 1). Stéphane Ballet and Robert Rolland () Asymptotically exact sequences 4 / 14

5 Asymptotically exact sequences Explicit families of asymptotically exact sequences If r > 1 : The completed Garcia-Stichtenoth towers : T 1 /F q 2 = F 0,0 F 0,1... F 0,r F 1,0 F 1,1... F 1,r... such that F k F k,s F k+1 for any integer s such that s = 0,..., r, with F k,0 = F k and F k,r = F k+1. T 2 /F q = G 0,0 G 0,1... G 0,r G 1,0 G 1,1... G 1,r,... defined over the constant fied F q and related to the tower T 1 by F k,s = F q 2G k,s for all k and s, Stéphane Ballet and Robert Rolland () Asymptotically exact sequences 5 / 14

6 Asymptotically exact sequences Explicit families of asymptotically exact sequences Proposition Let p = 2. If q = p 2, the descent of the definition field of the tower T 1 from F q 2 to F p is possible. More precisely, there exists a tower T 3 defined over F p given by a sequence : T 3 /F p = H 0,0 H 0,1 H 0,2 H 1,0 H 1,1 H 1,2,... defined over the constant fied F p and related to the towers T 1 and T 2 by F k,s = F q 2H k,s for all k and s = 0, 1, 2, G k,s = F q H k,s for all k and s = 0, 1, 2, namely F k,s /F q 2 is the constant field extension of G k,s /F q and H k,s /F q and G k,s /F q is the constant field extension of H k,s /F p. Stéphane Ballet and Robert Rolland () Asymptotically exact sequences 6 / 14

7 Asymptotically exact sequences Explicit families of asymptotically exact sequences q = p 2 = 4 F k+1 /F q 2 G k+1 /F q H k+1 /F q F k,1 /F q 2 G k,1 /F q H k,1 /F q F k /F q 2 G k /F q H k /F q F q 2(x) F q (x) F q(x) Stéphane Ballet and Robert Rolland () Asymptotically exact sequences 7 / 14

8 Asymptotically exact sequences Explicit families of asymptotically exact sequences Let g k the genus of F k in T 0 and g k,s the genus of F k,s in T 1. Proposition Let q = p 2 = 4. For any integer k 1, for any integer s such that s = 0, 1, 2, the algebraic function field H k,s /F p in the tower T 3 has a genus g(h k,s ) = g k,s with B 1 (H k,s ) places of degree one, B 2 (H k,s ) places of degree two and B 4 (H k,s ) places of degree 4 such that : 1) H k H k,s H k+1 with H k,0 = H k and H k,2 = H k+1. 2) g(h k,s ) g(h k+1) p r s + 1 with g(h k+1 ) = g k+1 q k+1 + q k. 3) B 1 (H k,s ) + 2B 2 (H k,s ) + 4B 4 (H k,s ) (q 2 1)q k 1 p s. 4) β 4 (T 3 /F p ) = lim gk,s B 4(H k,s /F p) g k = 1 4 (p2 1). g(h 5) lim l+1 ) l g(h l ) = 2 where g(h l ) and g(h l+1 ) denote the genus of two consecutive algebraic function fields in T 3. Stéphane Ballet and Robert Rolland () Asymptotically exact sequences 8 / 14

9 Application : multiplication in finite fields Tensor rank M : F q n F q n F q n t M F q n F q n F q n If λ t M = a l b l c l (1) l=1 where a l F q n, b l F q n, c l F q n, then Definition : λ x.y = t M (x y) = a l (x)b l (y)c l. (2) l=1 µ q (n) = Rank(t M ) = minλ Stéphane Ballet and Robert Rolland () Asymptotically exact sequences 9 / 14

10 Known result Application : multiplication in finite fields Known result Shparlinski-Tsfasman-Vladut (1992) : M q = lim sup n µ q (n) n Asymptotical bound for the tensor rank of the multiplication in F 2 n Remarks : M The algorithm of D.V. and G.V. Chudnovsky (1987). 2 Asymptotically exact sequences of algebraic function fields defined over F q 2 =16 and the embedding of F 2 into F 16 : µ 2 (n) µ 2 (4n) µ 2 (4) µ 2 4(n). Stéphane Ballet and Robert Rolland () Asymptotically exact sequences 10 / 14

11 New result The Generalized Chudnovsky algorithm Theorem F /F q be an algebraic function field, Q be a degree n place of F /F q, D be a divisor of F /F q, P = {P 11,..., P 1N1, P 21,..., P 2N2,..., P r1,..., P rnr } a) the application Ev Q : L(D) F q n F Q b) the application Ev P : { L(2D) F N 1 q F N 2... F Nr q 2 q r f (f (P 11 ),..., f (P 1N1 ),..., f (P r1 ),..., f (P rnr )) Then µ q (n) N 1 + µ q (2)N 2 + µ q (3)N µ q (r)n r. Stéphane Ballet and Robert Rolland () Asymptotically exact sequences 11 / 14

12 New result Let q be a prime power and n be an integer. To obtain a very efficient Chudnovsky algorithm for multiplication in F q n : Good properties An algebraic function field F /F q of genus g such that 1) there exists a place of degree n. 2) r i=1 in i 2n + g 1 3) For any real number ɛ > 0, 2D g(1 ɛ) +1 < h where D m is the number of the effective divisors of degree m of F /F q and h = h(f /F q ) is the cardinal of the Jacobian of F /F q. Stéphane Ballet and Robert Rolland () Asymptotically exact sequences 12 / 14

13 New result Proposition Let l be a real number and let F /F q = (F k /F q ) k 1 be an asymptotically exact sequence of algebraic function fields defined over F q. Then, for any real number ɛ > 0, there exists an integer k 0 such that for any integer k k 0, we get : ld gk (ɛ 1)+o(g k ) < h k where D m is the number of the effective divisors of degree m of F k /F q and h k = h(f k /F q ) is the cardinal of the Jacobian of F k /F q. Stéphane Ballet and Robert Rolland () Asymptotically exact sequences 13 / 14

14 New bound New result Asymptotical bound for the tensor rank of the multiplication in F 2 n Remarks : M The generalized algorithm of D.V. and G.V. Chudnovsky applied with algebraic function fields defined over F 2 having many places of degree r = 4. 2 The asymptotically exact sequence T 3 of algebraic function fields defined over F 2. Stéphane Ballet and Robert Rolland () Asymptotically exact sequences 14 / 14