20080912, 16:36  #1 
(loop (#_fork))
Feb 2006
Cambridge, England
13×491 Posts 
Small problem in sixdimensional space
I'm considering the codimension1inP5object defined by
x^6+y^6+z^6 = u^6+v^6+w^6 with the obvious solutionsbypermutation 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<N seems to be slightly more than linear in N, so I suspect there are some families of points lurking. Suspiciously many of the points I've found have x+y+u=z+v+w, for example. http://www.jstor.org/pss/2005335 gives a set of homogeneous quartics parameterising some solutions, but clearly not all solutions since the quartics happen also to satisfy x^2+y^2+z^2=u^2+v^2+w^2. What are the right sort of questions to ask about the set of points on a highdimensional variety? 
20080913, 06:03  #2  
Nov 2003
1D24_{16} Posts 
Quote:
What does the jinvariant look like? Is the variety even Abelian? How many localizations yield singularities? (i.e. how many primes have bad reduction) 

20080913, 06:07  #3  
Nov 2003
1D24_{16} Posts 
Quote:
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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