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Local Langlands Correpondence for the twisted exterior and symmetric square epsilon-factors of GL(N)Dongming She (8782541) 02 May 2020 (has links)
In this paper, we prove the equality of the local arithmetic and analytic epsilon- and L-factors attached to the twisted exterior and symmetric square representations of GL(N). We will construct the twisted symmetric square local analytic gamma- and L-factor of GL(N) by applying Langlands-Shahidi method to odd GSpin groups. Then we reduce the problem to the stablity of local coefficients, and eventually prove the analytic stabitliy in this case by some analysis on the asymptotic behavior of certain partial Bessel functions.
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On the Special Values of Certain L-functions: The case G2Farid Hosseinijafari (18846826) 24 June 2024 (has links)
<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>
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