Analytic Properties of Automorphic L-functions

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Abstract: Piatetski-Shapiro DMS-0072450The basic question described in the proposal is the problem of Langlands lifting. The definition of Langlands lifting can be done the following way. First one can define the local lifting. For an archimedean place one can define Langlands lifting in the following way. By Langlands theorem any irreducible admissible representation of a classical group one can attach a homomorphism of the Weil group to the L-group of this classical group. After that, for any classical group there is a natural embedding to the L-group of a general linear group. In this way one can attach a representation of the Weil group over a archimedean field a representation of the general linear group over that field. Similarly one can attach to any unramified representation of a classical group over a non-archimedean field we attach a semisimple conjugacy class of the L-group of the given classical group. The image of this conjugacy class in the corresponding general linear group L-group determines an unramified representation of the general linear group over the non-archimedean field. In this way to an automorphic representation of the classical group one can attach a collection of local representations of the general linear group defined at all archimedean places and at all unramified places. It is a well know statement that if a cuspidal automorphic representation of the general linear group exists with this data then it is unique. So the proposal is devoted to the study of Langlands lifting.Langlands lifting is a part of Langlands philosophy. I was asked by the program director for algebra and number theory to explain what the Langlands program is. I have thought about this a lot of time and I can say the following. Langlands philosophy gives a connection between automorphic forms and Dirichlet series. The most important example of a Dirichlet series is the famous Riemann zeta function. The automorphic form is the classical theta function of Jacobi. A second example the automorphic function is the so called Delta function which is a modular form of weight twelve. The corresponding Dirichlet series was studied first by Mordell who showed it had an Euler product. The coefficients of the Delta function are usually called the Ramanujan numbers. The basic properties of these numbers is that they are multiplicative. So the corresponding L-function has an Euler product and similar properties to the Riemann zeta function, namely holomorphic continuation the the whole complex plane and satisfies a functional equation. This connection can be extended to any algebraic reductive group. Let me explain it for the general linear group. Let's consider the space of functions which consists on all functions on the group of adeles invariant on the left side with respect to the group of principal adeles. Any smooth function of this type will be called an automorphic form. There is a way how one can attach to any automorphic form on the general linear group a Dirichlet series which has an Euler product and nice analytic properties, a nice functional equation, and holomorphic continuation to the whole complex plane.