Executive Summary : | This project aims to obtain a strong multiplicity one theorem for the eigenvalues of Siegel eigenforms of degree two and higher levels. The multiplicity one theorem determines whether the set of Hecke eigenvalues determines the eigenform under consideration, a fundamental and interesting problem in the theory of automorphic forms. In 2018, Schmidt provided an affirmative answer to the long-standing problem of Siegel cuspidal eigenforms. Kumar et al. improved Schmidt's result by showing that any set of eigenvalues at primes of positive upper density is sufficient to distinguish the Siegel cuspidal eigenform. The project aims to prove that any polynomial relation between eigenvalues of degree two Siegel eigenforms at primes of positive upper density is sufficient to distinguish the forms. It is known that for any Siegel eigenform and for any prime l, there is a continuous, semisimple, l-adic Galois representation valued in GSp₄. To achieve this aim, the project will compute the images of product Galois representations attached to two Siegel eigenforms, which are valued in the group scheme G. The image of the product representations is expected to be as independent as possible, containing the direct product of Sp₄ with Sp₄. The Zariski closure of the image of the product representations will be connected, and the results can be applied to study other properties of eigenvalues of Siegel eigenforms, such as divisibility and improvements of Land-Trotter type results. |