This research is about a theory of new- and oldforms for irreducible admissible representations of the algebraic group PGSp(4,F), for F a nonarchimedean local field of characteristic zero. The first objective of this research is to prove the main conjecture, supported by extensive evidence, which asserts that a generic representation admits a nonzero vector fixed by a paramodular group of some level; that at the minimal level such a vector is unique up to nonzero scalars; and that at the minimal level a suitable normalization of such a vector, called a newform, computes the L-factor of the representation. The second objective is to prove a precise conjecture about the structure of the spaces of oldforms in a generic representation, i.e., the spaces of vectors fixed by paramodular groups of level exceeding the minimal level. The third, and final, objective is to investigate whether a version of the main conjecture holds for nongeneric representations at the level of L- or A-packets.
Briefly summarized, number theory is the part of mathematics concerned with the study of integral solutions to algebraic equations. Though the problems of number theory often can be simply stated, solutions may require extensive theoretical development and input from other parts of mathematics. Broadly characterized, this research seeks to tie together, in a useful and new way, two parts of number theory. The first part, the theory of automorphic representations, allows the formulation of comprehensive conjectures and proofs; however, automorphic representations are abstract. The second part, the theory of Siegel modular forms, is important for examples and applications; on the other hand, Siegel modular forms can hide important general phenomena. By developing the theory of local new- and oldforms for PGSp(4) this research will make it easier to systematically and conceptually construct Siegel modular forms from automorphic representations, and thus may lead to work that draws on the strengths of both theories. This research crosses several subdisciplines of mathematics, and to facilitate this and other work this project will create a freely accessible database on the World Wide Web about the existing literature on GSp(4).
