Collaborative Research: Local Newforms for GSp(4)

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This research is about a theory of new- and oldforms for irreducibleadmissible representations of the algebraic group PGSp(4,F), for F anonarchimedean local field of characteristic zero. The first objectiveof this research is to prove the main conjecture, supported byextensive evidence, which asserts that a generic representation admitsa nonzero vector fixed by a paramodular group of some level; that atthe minimal level such a vector is unique up to nonzero scalars; andthat at the minimal level a suitable normalization of such a vector,called a newform, computes the L-factor of the representation. Thesecond objective is to prove a precise conjecture about the structureof the spaces of oldforms in a generic representation, i.e., the spacesof vectors fixed by paramodular groups of level exceeding the minimallevel. The third, and final, objective is to investigate whether aversion of the main conjecture holds for nongeneric representations atthe level of L- or A-packets.Briefly summarized, number theory is the part of mathematics concernedwith the study of integral solutions to algebraic equations. Though theproblems of number theory often can be simply stated, solutions mayrequire extensive theoretical development and input from other parts ofmathematics. Broadly characterized, this research seeks to tietogether, in a useful and new way, two parts of number theory. Thefirst part, the theory of automorphic representations, allows theformulation of comprehensive conjectures and proofs; however,automorphic representations are abstract. The second part, the theoryof Siegel modular forms, is important for examples and applications; onthe other hand, Siegel modular forms can hide important generalphenomena. By developing the theory of local new- and oldforms forPGSp(4) this research will make it easier to systematically andconceptually construct Siegel modular forms from automorphicrepresentations, and thus may lead to work that draws on the strengthsof both theories. This research crosses several subdisciplines ofmathematics, and to facilitate this and other work this project willcreate a freely accessible database on the World Wide Web about theexisting literature on GSp(4).