Dr Anil Kumar Chhangani

14 Pages·2012·0.18 MB·English

Asymptotically exact sequences of algebraic function
fields defined over F and application
q
St´ephane Ballet and Robert Rolland
Institut de Math´ematiques de Luminy
UMR C.N.R.S. / Universit´e de la M´editerran´ee
Luminy Case 930, F13288 Marseille CEDEX 9
FRANCE
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Asymptoticallyexactsequences Definition
Definition
Consider a sequence F/F = (F /F ) of algebraic function fields
q k q k≥1
F /F defined over F of genus g = g(F /F ). We suppose that the
k q q k k q
sequence of genus g is an increasing sequence growing to infinity. The
k
sequence F/F is called asymptotically exact if for all m ≥ 1 the following
q
limit exists :
B (F /F )
β (F/F ) = lim m k q
m q
gk→∞ gk
where B (F /F ) is the number of places of degree m on F /F .
m k q k q
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Asymptoticallyexactsequences Explicitfamiliesofasymptoticallyexactsequences
Proposition
Let r be an integer and F/F = (F /F ) be a sequence of algebraic
q k q k≥1
function fields defined over Fq such that βr(F/Fq) = 1r(q2r −1). Then
β (F/F ) = 0 for any integer m (cid:54)= r. In particular, the sequence F/F is
m q q
asymptotically exact.
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Asymptoticallyexactsequences Explicitfamiliesofasymptoticallyexactsequences
Example of explicit asymptotically exact sequences
Let F with q = pr and r an integer.
q2
The Garcia-Stichtenoth tower T over F :
0 q2
the sequence (F ,F ,...) where
0 1
F := F (z )
k+1 k k+1
and z satisfies the equation :
k+1
zq +z = xq+1
k+1 k+1 k
with
x := z /x in F (for k ≥ 1).
k k k−1 k
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Asymptoticallyexactsequences Explicitfamiliesofasymptoticallyexactsequences
If r > 1 :
The completed Garcia-Stichtenoth towers :
T /F = F ⊆ F ⊆ ... ⊆ F ⊆ F ⊆ F ⊆ ... ⊆ F ...
1 q2 0,0 0,1 0,r 1,0 1,1 1,r
such that F ⊆ F ⊆ F for any integer s such that s = 0,...,r, with
k k,s k+1
F = F and F = F .
k,0 k k,r k+1
T /F = G ⊆ G ⊆ ... ⊆ G ⊆ G ⊆ G ⊆ ... ⊆ G ,...
2 q 0,0 0,1 0,r 1,0 1,1 1,r
defined over the constant fied F and related to the tower T by
q 1
F = F G for all k and s,
k,s q2 k,s
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Asymptoticallyexactsequences Explicitfamiliesofasymptoticallyexactsequences
Proposition
Let p = 2. If q = p2, the descent of the definition field of the tower T
1
from F to F is possible. More precisely, there exists a tower T defined
q2 p 3
over F given by a sequence :
p
T /F = H ⊆ H ⊆ H ⊆ H ⊆ H ⊆ H ,...
3 p 0,0 0,1 0,2 1,0 1,1 1,2
defined over the constant fied F and related to the towers T and T by
p 1 2
F = F H for all k and s = 0,1,2,
k,s q2 k,s
G = F H for all k and s = 0,1,2,
k,s q k,s
namely F /F is the constant field extension of G /F and H /F
k,s q2 k,s q k,s q
and G /F is the constant field extension of H /F .
k,s q k,s p
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Asymptoticallyexactsequences Explicitfamiliesofasymptoticallyexactsequences
q = p2 = 4
F /F G /F H /F√
k+1 q2 k+1 q k+1 q
F /F G /F H /F√
k,1 q2 k,1 q k,1 q
F /F G /F H /F√
k (cid:31) q2 k (cid:31) q k (cid:31) q
(cid:31) (cid:31) (cid:31)
(cid:31) (cid:31) (cid:31)
F (x) F (x) F√ (x)
q2 q q
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Asymptoticallyexactsequences Explicitfamiliesofasymptoticallyexactsequences
Let g the genus of F in T and g the genus of F in T .
k k 0 k,s k,s 1
Proposition
Let q = p2 = 4. For any integer k ≥ 1, for any integer s such that
s = 0,1,2, the algebraic function field H /F in the tower T has a
k,s p 3
genus g(H ) = g with B (H ) places of degree one, B (H ) places
k,s k,s 1 k,s 2 k,s
of degree two and B (H ) places of degree 4 such that :
4 k,s
1) H ⊆ H ⊆ H with H = H and H = H .
k k,s k+1 k,0 k k,2 k+1
2) g(H ) ≤ g(Hk+1) +1 with g(H ) = g ≤ qk+1+qk .
k,s pr−s k+1 k+1
3) B (H )+2B (H )+4B (H ) ≥ (q2−1)qk−1ps.
1 k,s 2 k,s 4 k,s
4) β (T /F ) = lim B4(Hk,s/Fp) = 1(p2−1).
4 3 p gk,s→∞ gk 4
g(H )
5) lim l+1 = 2 where g(H ) and g(H ) denote the genus of two
l→∞ g(H) l l+1
l
consecutive algebraic function fields in T .
3
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Application:multiplicationinfinitefields
Tensor rank
M : F ⊗F −→ F
qn qn qn
t ∈ F∗ ⊗F∗ ⊗F
M qn qn qn
If
λ
(cid:88)
t = a ⊗b ⊗c (1)
M l l l
l=1
where a ∈ F∗ , b ∈ F∗ , c ∈ F , then
l qn l qn l qn
λ
(cid:88)
x.y = t (x ⊗y) = a (x)b (y)c . (2)
M l l l
l=1
Definition :
µ (n) = Rank(t ) = minλ
q M
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Application:multiplicationinfinitefields Knownresult
Known result
Shparlinski-Tsfasman-Vladut (1992) :
µ (n)
q
M = limsup
q
n
n→∞
Asymptotical bound for the tensor rank of the multiplication in F
2n
M ≤ 27
2
Remarks :
1 The algorithm of D.V. and G.V. Chudnovsky (1987).
2 Asymptotically exact sequences of algebraic function fields defined
over F and the embedding of F into F :
q2=16 2 16
µ (n) ≤ µ (4n) ≤ µ (4)×µ (n).
2 2 2 24
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