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Program Description

The local Langlands correspondence (LLC) is of fundamental importance in number theory and representation theory and has triggered many new developments in modern number theory in the last few decades. The LLC for GL(n) over p-adic fields has been established a little more than 10 years ago by Harris-Taylor and Henniart. A crucial idea, originating from Deligne, Drinfeld and Carayol, is to exploit the l-adic etale cohomology theory for p-adic algebraic geometry (where l is a prime different from p) to study and establish LLC. The geometric objects in this story are certain moduli spaces of p-divisible groups and can be viewed as p-adic rigid analytic spaces. (They are called Lubin-Tate spaces, or Rapoport-Zink spaces in the more general context.) Thus their etale cohomology and its representation-theoretic description are at the core of interest for number theorists and representation theorists alike.

The l-adic cohomology theory for rigid spaces is exciting on its own right with diverse applications in algebraic geometry. Rigid geometry is appealing to complex algebraic geometers for its resemblance and intriguing for its unique features not present in the complex analogue.