Functoriality and special values of L-functions

TL;DR

This paper explores the relationship between Langlands functoriality and Deligne's conjecture, focusing on symmetric power L-functions of cusp forms and leveraging recent advances by Mahnkopf.

Contribution

It provides a semi-expository analysis connecting functoriality with special value conjectures, emphasizing symmetric power L-functions and recent automorphic L-function results.

Findings

Clarifies the connection between functoriality and special value conjectures.

Highlights recent progress by Mahnkopf on automorphic L-functions.

Offers insights into symmetric power L-functions of cusp forms.

Abstract

This is a semi-expository article concerning Langlands functoriality and Deligne's conjecture on the special values of $L$-functions. The emphasis is on symmetric power $L$-functions associated to a holomorphic cusp form, while appealing to a recent work of Mahnkopf on the special values of automorphic $L$-functions.