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 2008-09-12, 16:36 #1 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 13×491 Posts Small problem in six-dimensional space I'm considering the codimension-1-in-P5-object defined by x^6+y^6+z^6 = u^6+v^6+w^6 with the obvious solutions-by-permutation removed and GCD(x,y,z,u,v,w)=1. I've searched for z,w<2000, and am finding that the number of these points with z,w
2008-09-13, 06:03   #2
R.D. Silverman

Nov 2003

1D2416 Posts

Quote:
 Originally Posted by fivemack I'm considering the codimension-1-in-P5-object defined by x^6+y^6+z^6 = u^6+v^6+w^6 What are the right sort of questions to ask about the set of points on a high-dimensional variety?
Is there an associated L-function? Is there a motive?
What does the j-invariant look like? Is the variety even Abelian?
How many localizations yield singularities? (i.e. how many primes have

2008-09-13, 06:07   #3
R.D. Silverman

Nov 2003

1D2416 Posts

Quote:
 Originally Posted by R.D. Silverman Is there an associated L-function? Is there a motive? What does the j-invariant look like? Is the variety even Abelian? How many localizations yield singularities? (i.e. how many primes have bad reduction)
And I stupidly forgot perhaps the most obvious question: What is its
genus? i.e. if viewed as a Riemann surface over C, how many holes
does the surface have? Then one might study conformal maps of
that surface.

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