Beyond Endoscopy and the stable trace formula

点击翻译按钮获取中文摘要

One of the great problems of present day mathematics is Langlands' conjectural Principle of Functoriality. It represents the centre of what is now called the Langlands program. There has been considerable progress towards functoriality in a number of cases since the conjecture was posed fifty years ago. However, the most interesting and fundamental cases have remained well beyond reach. Around the year 2000, Langlands proposed a strategy for attacking the general Principle of Functoriality, which he called Beyond Endoscopy. Endoscopy itself is an earlier theory, which had been proposed as a conjecture by Langlands. It contains as a biproduct much of the progress on functoriality we have achieved so far. However, its limitations are clear. Both Endoscopy and Beyond Endoscopy are based on the stable trace formula. But while one of the main goals of Endoscopy has been to construct the more refined stable trace formula from the pre-existing invariant trace formula, Beyond Endoscopy entails a deeper analysis of the stable trace formula itself. In particular, one of its main goals is to construct an explicit partition of the automorphic representations that occur in the stable trace formula of a given group, according to how they are expected to behave under the Principle of Functoriality. I am proposing to continue my work on Beyond Endoscopy. I have been thinking seriously about the problem for the past four years, as I proposed for my last NSERC grant, and I have written three general papers on it so far. I have also almost completed a longer paper on the properties of elliptic orbital integrals on the geometric side of the stable trace formula. The goal would be to apply the Poisson summation formula to these elliptic terms, following the methods introduced by Ali Altug for the group GL(2). There are indications that the resulting Fourier transforms of the these terms will display the contributions to the geometric side of the nontempered representations on the spectral side. As has been pointed out by Frenkel, Langlands and Ngo, one would have to modify the geometric side by subtracting these contributions before one could begin to look for a geometric characterization of the desired functorial partition of automorphic representations mentioned above. I have studied the question for general groups. While I am not yet in a position to state a precise conjecture, it seems clear to me that the geometric contributions of the nontempered automorphic representations will be both suggestive and striking. I propose to formulate a general conjecture, and to prove it for groups of small rank.