On the Special Values of Certain L-functions: The case G2

In this thesis, we prove the rationality results for the ratio of the critical values of certain L-functions, which appear in the constant term of Eisenstein series associated with the exceptional group G₂ over a totally imaginary field. Our methodology builds upon the works of Harder and Raghuram, who established rationality results for special values of Rankin-Selberg L-functions for GL_n× GL_n' by studying the rank-one Eisenstein cohomology of the ambient group GL_n+n' over a totally real field, as well as its generalization by Raghuram [35] for the case over a totally imaginary field.

The L-functions in this thesis were constructed using the Langlands-Shahidi method for G₂ over a totally imaginary field, attached to maximal parabolic subgroups. This is the first instance of applying the Harder-Raghuram method to an exceptional group, and the first case involving more than one function appearing in the constant term. Our results demonstrate the relationship between the rationality of different L-functions appearing in the constant term, allowing one to prove the rationality of one L-function based on the known rationality result of another L-functions.