Eisenstein Cohomology and Critical Values of Certain $L$-Functions: The Case $G_2$

TL;DR

This paper proves rationality results for ratios of critical values of Langlands-Shahidi $L$-functions related to the exceptional group $G_2$, extending methods to complex cases involving multiple $L$-functions and aligning with Deligne's conjecture.

Contribution

It generalizes the Harder-Raghuram method to exceptional groups and multiple $L$-functions, establishing rationality of critical values and supporting Deligne's conjecture in this context.

Findings

Rationality of ratios of successive critical $L$-values established.

Rationality of critical values for product $L$-functions like symmetric cube shown.

Results align automorphic and motivic versions of Deligne's conjecture.

Abstract

We establish results on the rationality of ratios of successive critical values of Langlands-Shahidi $L$-functions, as they appear in the constant term of the Eisenstein series associated with the exceptional group of type $G_2$ over a totally imaginary number field. Furthermore, we prove the rationality of the critical values for each $L$-function in the products, such as the symmetric cube $L$-functions. Our method generalizes the Harder-Raghuram method to cases where multiple $L$-functions appear in the constant term and involve an exceptional group. Finally, our results on the automorphic version of Deligne's conjecture align with its motivic counterpart, as demonstrated in the recent work of Deligne and Raghuram.