Combinatorial Aspects of the Ikeda Lift via Extended Gross-Keating Data

Abstract

This thesis is motivated by the problem of studying the Ikeda lift via a converse theorem due to R. Weissauer. We investigate a certain function; denoted by 𝜙(𝑎; 𝐵), where 𝑎 is a positive integer and 𝐵 is a symmetric positive-definite half-integral matrix; appearing in the Fourier coefficient formulas of a linear version of the Ikeda lift due to W. Kohnen. We develop new methods for computing 𝜙(𝑎; 𝐵) via the extended Gross-Keating (EGK) datum of a quadratic form and develop novel combinatorial interpretations for 𝜙(𝑎; 𝐵) which involve integer partitions with restrictions depending on the EGK datum attached to 𝐵 at each prime.