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SUMMARY:Resolution of singularities on the Lubin-Tate tower - Jared Weinst
 ein (UCLA)
DTSTART:20100518T133000Z
DTEND:20100518T143000Z
UID:TALK23458@talks.cam.ac.uk
CONTACT:Tom Fisher
DESCRIPTION:A fundamental result in local class field theory is the 1965 p
 aper of Lubin and Tate\, which classifies the abelian extensions of a nona
 rchimedean local field in terms of an algebraic structure known as a one-d
 imensional formal module.  We'll review this result\,\nand show how the qu
 estion of constructing nonabelian extensions leads to the study of the Lub
 in-Tate tower\, which can be viewed as an infinitesimal version of the cla
 ssical tower of modular curves _X(p^n^)_.\n\nBy results of Harris-Taylor a
 nd Boyer\, the cohomology of the Lubin-Tate tower encodes precise informat
 ion about non-abelian extensions of the local field (namely\, it realizes 
 the local Langlands correspondence). The Lubin-Tate tower has a horribly s
 ingular special fiber\, which hinders any direct study of its cohomology\,
  but we will show that after blowing up a singularity there is a\nmodel fo
 r the tower whose reduction contains a very curious nonsingular hypersurfa
 ce defined over a finite field -- curious because it seems to have the max
 imum number of rational points relative to its topology. We will write dow
 n the equation for this hypersurface and formulate a conjecture (alas\, st
 ill unproved) regarding its zeta function.
LOCATION:MR13
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