nLab Global analytic geometry

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.

Motivation

Global analytic geometry

• combines non-archimedean and archimedean analytic geometry;

• contains algebraic geometry as a sub-theory;

• treats all places on equal footing, contrary to scheme theory. This implies that it is closer in spirit to the automorphic philosophy and Langlands program, and more historically to Tate’s proof of the functional equation.

One aim of the theory is to define, using global analytic tools, a good Hodge theory for arithmetic varieties.

Possible set of initial constraints

For a relaxed approach to global Hodge theory: it is not an easy task to find a good set of constraints on such a global Hodge theory, but they are useful to understand better the motivations underlying the construction of global analytic spaces.

1. having a good theory of linear coefficients on global analytic spaces, with the Grothendieck six operations (this should be done by the use of the étale sub-analytic? topology in characteristic $0$, and by a “probably quite hard to develop in characteristic p, but easy to develop in characteristic 0” model theoretical description of definable? sets for the étale G-topology on strict and non-strict overconvergent analytic spaces). It seems that global analytic motivic spectral coefficients (given by imposing homotopy invariance with respect to the unit disc are not so well adapted to the study of torsion phenomena in the characteristic p situation).
2. having a good theory of higher and derived global analytic geometry, with a well-behaved notion of de Rham type cohomology theory and a Chern character. The constraints on such a theory would be:

• get back (or be isomorphic to) the usual algebraic de Rham Chern character when one works with usual schemes.
• get back the p-adic analytic de Rham Chern character (on Ayoub’s motivic cohomology) of dagger spaces when one works with dagger p-adic spaces.
• get back the usual de Rham Chern character when one works over $\C$.

This first set of constraints is worked out in the theory of overconvergent global analytic geometry.

The problems related to the torsion phenomena for coefficients in characteristic $p$ may be perhaps be solved by the development of a really non-linear analytic homotopy theory, meaning a proper theory of spectral global analytic geometry.

A more optimistic set of constraints

1. having a theory of derived analytic microlocalization of sheaves and differential equations, allowing the proper settlement of a global analytic index theory.

2. being able to prove the functional equation of zeta functions of arbitrary arithmetic varieties;

3. being able to settle down an analytic langlands program, giving a correspondence between general (non-algebraic) automorphic representations and a sort of global analytic motives. The p-adic Langlands program should be a particular case of this general construction when the base Banach ring is $\mathbb{Z}_p$.

4. being able to devise a robust and simple enough arithmetic cryptography protocol based on a discrete logarithm problem? or on a cohomological product problem? on a given geometric cohomology? theory for global analytic spaces.

Argument in favor of its use are:

• the fact that archimedean factors are deeply related to (real and complex 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.

Related concepts

• Berkovich space, Huber space, perfectoid space

• Arakelov geometry

• function field analogy

• Arithmetic cryptography

• global analytic index theory

• overconvergent global analytic geometry

• spectral global analytic geometry

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

• Jérôme Poineau, La droite de Berkovich sur Z, Asterisque 334 (2010).

  (on fundamental properties of the affine line)

• Jérôme Poineau, Espaces de Berkovich sur Z: etude locale arXiv:1202.0799

  (on the coherence of the sheaf of analytic functions on higher dimensional affine spaces)

• Frédéric Paugam, Overconvergent global analytic geometry

  (preprint)

• Frédéric Paugam, Global analytic geometry (arXiv:0803.0148)

• Frédéric Paugam, Global analytic geometry and the functional equation, lecture notes (pdf)

• Oren Ben-Bassat, Kobi Kremnizer, section 7 of Non-Archimedean analytic geometry as relative algebraic geometry (arXiv:1312.0338)