Automorphic forms for PGL_n

• Grant number: 22/09476-7
• Support Opportunities: Regular Research Grants
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Roberto Carlos Alvarenga da Silva Junior
• Grantee: Roberto Carlos Alvarenga da Silva Junior
• Host Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil
• Associated researchers: Ana Paula de Araújo Chaves ; Inder Kaur ; Oliver Lorscheid ; Valdir José Pereira Júnior

Abstract

The project deals with the theory of automorphic forms and is a natural continuation of the research already developed by the proponent. Automorphic forms are central objects in modern number theory. For instance, they play a key role both in Don Zagier's work in a search of a proof for the Classical Riemann Hypothesis, and in the famous Langlands Program, whose advent is the January-1967 handwritten letter by Robert Langlands to André Weil and is nowadays known as the unification theory. The main goal of this research plan is to probe for which periodical conditions in the space of automatic forms yield tempered representations. The automorphic forms considered are those defined over $\PGL_n$ ($n \geq 3$) for a global function field $F$ (elliptic or rational). Hence, shall be considered its geometric point of view, i.e. as function on the set of classes of isomorphism of projective vector bundles defined over the nonsingular projective curve whose function field is $F$. The reason is because, accordingly, it might be useful to consider geometric tolls, such as Hecke modifications and the graphs of Hecke operators, where the proponent has published works. (AU)