2018/05/11 by Yiannis Sakellaridis, Sakellaridis, Yiannis · 1 citation
Mathematics · Medicine · #11F70 #Advanced Algebra and Geometry #FOS: Mathematics #Mathematical Analysis and Transform Methods #Medical Imaging Techniques and Applications #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1805.04640
openalex publication_date 2018/05/11 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
The goal of this article and its precursor is to demonstrate, by example, the\nexistence of "transfer operators" betweeen relative trace formulas, which\ngeneralize the scalar transfer factors of endoscopy. These transfer operators\nhave all properties that one could expect from a trace formula comparison:\nmatching, fundamental lemma for the Hecke algebra, transfer of (relative)\ncharacters. Most importantly, and quite surprisingly, they appear to be of\nabelian nature (at least, in the low-rank examples considered in this paper),\neven though they encompass functoriality relations of non-abelian harmonic\nanalysis. Thus, they are amenable to application of the Poisson summation\nformula in order to perform the global comparison. Moreover, we show that these\nabelian transforms have some structure -- which presently escapes our\nunderstanding in its entirety -- as deformations of well-understood operators\nwhen the spaces under consideration are replaced by their "asymptotic cones".\n In this second paper we use Rankin-Selberg theory to prove the local transfer\nbehind Rudnick's 1990 thesis (comparing the stable trace formula for\n\SL2 with the Kuznetsov formula) and Venkatesh's 2002 thesis\n(providing a "beyond endoscopy" proof of functorial transfer from tori to\n\GL2). As it turns out, the latter is not completely disjoint\nfrom endoscopic transfer -- in fact, our proof "factors" through endoscopic\ntransfer. We also study the functional equation of the symmetric-square\nL-function for \GL2, and show that it is governed by an\nexplicit "Hankel operator" at the level of the Kuznetsov formula, which is also\nof abelian nature. A similar theory for the standard L-function was\npreviously developed (in a different language) by Jacquet.\n