nLab
global analytic geometry

Contents

Idea

Global analytic geometry is a developing subject that gives an alternative/complementary approach to scheme theory in arithmetic geometry and analytic number theory. The starting point of this theory is in Vladimir Berkovich’s book about spectral theory and non-archimedean analytic geometry. It was then developped further by Jérôme Poineau.

Many interesting results on polynomial equations can be proved using the mysterious interactions between algebraic, complex analytic and p-adic analytic geometry. The aim of global analytic geometry is to construct a category of spaces which contains these three geometries. (Paugam08)

Motivation

Global analytic geometry

The main aim of the theory is to define, using global analytic tools, a good category of analytic motivic coefficients, that would help in proving naturally:

  • the functional equation of zeta functions of arbitrary arithmetic varieties?;
  • an analytic langlands program, giving a correspondence between general (non-algebraic) automorphic representations? and global analytic motives?.

Argument in favor of its use are:

  • the fact that archimedean factors are deeply related to (analytic) Hodge theory;

  • the fact that all proofs of parts of local Langlands program use deeply non-archimedean analytic spaces that are out of the scope of classical algebraic scheme theory.

References

A short introduction for large audience is in

  • Jérôme Poineau, Global analytic geometry, pages 20-23 in EMS newsletter September 2007 (pdf)

For more see

Revised on July 9, 2014 00:42:30 by Urs Schreiber (37.205.56.247)