Collaborative Research: Local Newforms for GSp(4) Grant uri icon



  • 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).

date/time interval

  • August 1, 2004 - July 31, 2008

total award amount

  • 106,797