Intertwinned towers of Shimura curves and
Bilinear multiplication
Matthieu Rambaud
Telecom ParisTech
JC2, la Bresse Apr. 28, 2017
Bilinear multiplication
f and g in Fp [X] of degree m, compute f.g
1
2
3
Choose P1 , . . . , P2m+1 in Fp
Evaluate f(Pi )i=1..2m+1 and g(Pi )i=1..2m+1
{
}
Compute f.g(Pi ) = f(Pi )•g(Pi ) : 2m + 1 multiplications
i
4
Lagrange’s interpolation: recover f.g.
M. Rambaud – Shimura curves and Multiplication
JC2 la Bresse
Apr. 28, 2017
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Chudnovky 2’s improvement
set:
Before
After
Fp
curve X/Fp
rational functions
f and g in L(D)
points
P1 , . . . P2m+g+1
in X(Fp )
f and g in Fp [X]:
polynomials
evaluation on:
points
P1 , . . . P2m+1
in Fp
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JC2 la Bresse
Apr. 28, 2017
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Small genera, small fields,
many thick points
the Graal
Conjecture
Let p prime and 2t ⩾ 4. Does there exist a family (Xs /Fp )s⩾1 of
curves defined over Fp , with genera gs such that:
1
gs −→ ∞
2
gs+1 /gs −→ 1 (density of (Xs )s )
3
|Xs (Fp2t )| /gs −−−→ pt − 1 (Optimality over Fp2t ) ?
s→∞
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JC2 la Bresse
Apr. 28, 2017
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the Graal
Conjecture
Let p prime and 2t ⩾ 4. Does there exist a family (Xs /Fp )s⩾1 of
curves defined over Fp , with genera gs such that:
1
gs −→ ∞
2
gs+1 /gs −→ 1 (density of (Xs )s )
3
|Xs (Fp2t )| /gs −−−→ pt − 1 (Optimality over Fp2t ) ?
s→∞
1
2
3
! 2t = 2
Classical modular curves X0 (N) ? △
! 2t ⩾ 4 ⇒ defined over Fpt
Shimura curves X0 (N ) ? △
! gs+1 /gs ∼ p2t
Garcia–Stichtenoth’s towers Fs ? △
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JC2 la Bresse
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Solution proposed
for p=3 and 2t=6
1
Hard work: compute two towers of Shimura curves over F36 !
f4
f3
f2
f1
g2
g1
... −
→ X0 (73 ) −
→ X0 (72 ) −
→ X0 (71 ) −
→X0 (1)
g4
g3
→ X0 (82 ) −
→ X0 (81 ) −
→X0 (1)
→ X0 (83 ) −
... −
2
Descend everything over F3 .
M. Rambaud – Shimura curves and Multiplication
JC2 la Bresse
Apr. 28, 2017
5 / 8
Solution proposed
for p=3 and 2t=6
1
Hard work: compute two towers of Shimura curves over F36 !
f4
f3
f2
f1
g2
g1
... −
→ X0 (73 ) −
→ X0 (72 ) −
→ X0 (71 ) −
→X0 (1)
g4
g3
→ X0 (82 ) −
→ X0 (81 ) −
→X0 (1)
→ X0 (83 ) −
... −
2
3
Descend everything over F3 .
Then for the density...
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JC2 la Bresse
Apr. 28, 2017
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Elkies’ Trick
X0 (74 ) o
X0 (73 ) o
X0 (72 ) o
X0 (71 ) o
X0 (1) o
X0 (74 81 ) o
X0 (73 81 ) o
X0 (72 81 ) o
X0 (71 81 ) o
X0 (81 ) o
X0 (74 82 ) o
X0 (73 82 ) o
X0 (72 82 ) o
X0 (71 82 ) o
X0 (82 ) o
M. Rambaud – Shimura curves and Multiplication
X0 (74 83 ) o
X0 (73 83 ) o
X0 (72 83 ) o
X0 (71 83 ) o
X0 (83 ) o
JC2 la Bresse
z
X0 (74 84 ) o
X0 (73 84 ) o
X0 (72 84 ) o
X0 (71 84 ) o
X0 (84 ) o
Apr. 28, 2017
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Example: starting a tower
X0 (82 ) = Elliptic/C
f2 8
X0 (81 ) = P1C
f1 9
(7)
P
7
AA
AA
A
1
7 AA (7) B
Q7
BB
BB 1
7 BB
X0 (1) = P1C
M. Rambaud – Shimura curves and Multiplication
P ′7
{
{{
{
{
}{ 1
{
1
z
zz
zz7
z
}zz
′
Q7
R7
JC2 la Bresse
(2)4
(7)
8
(3)3
33
Q2 , (2)4
1 24
R3
R2
Apr. 28, 2017
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Example: ...and a recursion
X0 (83 ) =
X0 (82 )
ω1 ◦f2 ◦ω2
×
, X (81 ) r
0
X0 (82 )
f2
X0 (82 )F3 : y2 = x3 + x2 + 2
X0 (81 )F3 = P1F3
1 + x2 + x3 + x4 + (x + 2x2 )y
f2 (x, y) =
2 + x2 + x3 + x4 + x2 y
ω2 : X0 (82 )F3 ∋ P −→ (1, 2, 1) − P
ω1 : t ∈ P1F3 ∋ t −→ −t
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JC2 la Bresse
Apr. 28, 2017
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