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Proof with Computer and Algebraic Geometry
We give a human-computer-aided proof of the Theorem .
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Construction: |
Fix the word w =
X-1 Y X Y-1 X
and consider the matrices x , y in
PSL(2,p):
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| x = |
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0 -1 | |
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y = |
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1
b | |
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1 t | |
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c 1+bc | |
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Note: |
U1 ( x , y ) = 1 has no solution
over Fp .
The entries of U1 ( x , y ) -
U2 ( x , y )
create a polynomial ideal I
in Z[b,c,t] .
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Statement: |
The theorem holds for PSL(2,p) <==>
V(I)
has a rational point
in (Fp)3.
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We show:
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V(I) has rational points
over Fp
for all prime numbers p>3.
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What is needed.
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