Stephen C. Milne

Affiliation: The Ohio State University

Email: milne@math.ohio-state.edu

Title Of Talk: Sums of squares, the Leech lattice, and new formulas for Ramanujan's tau function and other classical cusp forms

Abstract: As motivation, we first recall our infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's (1829) $4$ and $8$ squares identities to $4n^2$ or $4n(n+1)$ squares, respectively, without using cusp forms such as those of Glaisher or Ramanujan for $16$ and $24$ squares. We derive our formulas by utilizing combinatorics to combine a variety of methods and observations from the theory of Jacobi elliptic functions, continued fractions, Hankel or Turanian determinants, Lie algebras, Schur functions, and multiple basic hypergeometric series related to the classical groups. We also note our derivation proof of the two Kac and Wakimoto (1994) conjectured identities concerning representations of a positive integer by sums of $4n^2$ or $4n(n+1)$ triangular numbers, respectively. These conjectures arose in the study of Lie algebras and have also recently been proved by Zagier using modular forms. Related and subsequent work of Don Zagier, Ken Ono, Getz and Mahlburg, Rosengren, Imamo\=glu and Kohnen, H.-H. Chan and K. S. Chua, and, H.-H. Chan and C. Krattenthaler is very briefly reviewed. We conclude with a discussion of our new formulas for Ramanujan's tau function, including one in terms of the Leech lattice. Utilizing classical elliptic function invariants, we first sketch our derivation of several useful new formulas for Ramanujan's $ \tau$ function. This work includes: the main pair of new formulas for the $ \tau$ function that ``separate'' the two terms in the classical formula for the modular discriminant, a generating function form for both of these formulas, a Leech lattice form of one of these formulas, and a triangular numbers form. We then present analogous new formulas for several other classical cusp forms that appear in quadratic forms, sphere-packings, lattices and groups. If time allows, an additional application to the theory of quadratic forms is also given.

Last update made Sat Feb 21 10:27:30 EST 2009.
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