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Proof with Computer and Algebraic Geometry
We give a human-computer-aided proof of the Theorem
.
Here, we concentrate on case 1 (PSL(2,p))
-- cases 2 and 3 are similar, case 4 trivial, while case 5 is a bit more
involved.
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Construction: |
Fix the word w =
X-2 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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