<p dir="ltr">In this thesis, we prove the rationality results for the ratio of the critical values of certain <i>L</i>-functions, which appear in the constant term of Eisenstein series associated with the exceptional group <i>G</i><sub><em>2</em></sub> 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 <i>L</i>-functions for<i> </i><i>GL</i><sub><em>n</em></sub><i>× GL</i><sub><em>n'</em></sub> by studying the rank-one Eisenstein cohomology of the ambient group <i>GL</i><sub>n+n'</sub> over a totally real field, as well as its generalization by Raghuram [35] for the case over a totally imaginary field.</p><p dir="ltr">The <i>L</i>-functions in this thesis were constructed using the Langlands-Shahidi method for <i>G</i><sub><em>2</em></sub> 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 <i>L</i>-functions appearing in the constant term, allowing one to prove the rationality of one <i>L</i>-function based on the known rationality result of another <i>L</i>-functions.</p>
| Identifer | oai:union.ndltd.org:purdue.edu/oai:figshare.com:article/26058469 |
| Date | 24 June 2024 |
| Creators | Farid Hosseinijafari (18846826) |
| Source Sets | Purdue University |
| Detected Language | English |
| Type | Text, Thesis |
| Rights | CC BY 4.0 |
| Relation | https://figshare.com/articles/thesis/On_the_Special_Values_of_Certain_L-functions_The_case_G2/26058469 |
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