Geometrization of Harmonic Analysis on p-adic Manifolds

C. Papakyriakopoulos

1914-1976

André Weil (1906-1998) (I was told) used to say that the only thing we can do in mathematics is linear algebra.

Alexander Grothendieck

(1928-2014)

Idea: "Topology is encoded in covers of a space" (Grothendieck topology)

Vector spaces

Manifold X

Example:

The non-smooth case

Not always suitable!

Goresky (b.1950) and MacPherson (b.1944): Compute cohomology with a constructible local system called the intersection complex (IC).

Number Theory

Topology

Solomon Lefschetz (1884-1972)

fixed point theorem:

Grothendieck's sheaf-function dictionary

T

X =

(Non-smooth case: same,

when we replace by IC)

# fixed points (counted with multiplicity)

IC

What about the case when X is not smooth?

Pierre Deligne (b. 1944)

Extension of the Goresky-MacPherson

theory to the étale topology. Definition of a

constructible local system IC on X which

satisfies Grothendieck's trace formula.

The IC is a special case of a larger class of "good" complexes on X, called perverse.

A. Beilinson (b. 1957)

J. Bernstein (b. 1945)

O. Gabber (b. 1958)

The p-adics are not only used to solve diophantine equations, but also in harmonic analysis/representation theory.

Robert Langlands (b. 1936)

Arithmetic

objects (e.g.

diophantine

equations)

Analytic objects (Automorphic

forms)

etc.

Langlands is mainly concerned

with harmonic analytis on reductive

groups, but in practice (including his

"Beyond Endoscopy" program) one

needs to consider non-smooth real

or p-adic manifolds as well.

(L-functions)

Harmonic analysis

The formal arc space