#### Alladi Ramakrishnan Hall

#### Hecke algebras and the Langlands program

#### Manish Mishra

##### University of Heidelberg

*Given an irreducible polynomial f(x) with integer coefficients*

and a prime number p, one wishes to determine whether f(x) is a product of

distinct linear factors modulo p. When f(x) is a solvable polynomial, this

question is satisfactorily answered by the Class Field Theory. Attempts to

find a non-abelian Class Field Theory lead to the development of an area of

mathematics called the Langlands program.

The Langlands program, roughly speaking, predicts a natural correspondence

between the finite dimensional complex representations of the Galois group

of a local or a number field and the infinite dimensional representations

of real, p-adic and adelic reductive groups. I will give an outline of the

statement of the local Langlands correspondence. I will then briefly talk

about two of the main approaches towards the Langlands program - the type

theoretic approach relying on the theory of types developed by

Bushnell-Kutzko and others; and the endoscopic approach relying on the

trace formula and endoscopy. I will then state some of my results involving

these two approaches.

Done