[Paper Review] An automorphic variant of the Deligne conjecture

Jie Lin|arXiv (Cornell University)|Aug 27, 2016

Advanced Algebra and Geometry12 references3 citations

TL;DR

This paper formulates an automorphic variant of the Deligne conjecture for Rankin-Selberg L-functions of $GL_n \times GL_{n'}$ automorphic representations over a quadratic imaginary field. By defining automorphic periods and relating them to motivic periods, the authors establish a purely automorphic version of the conjecture, proving its compatibility with Deligne's original conjecture in several cases, including $n'=1$, $n>n'$ with parity and gap conditions, and $m=1$ with same parity.

ABSTRACT

In this paper we introduce an automorphic variant of the Deligne conjecture for tensor product of two motives over a quadratic imaginary field. On one hand, we define some motivic periods and rewrite the Deligne conjecture in terms of these periods. On the other hand, we define the automorphic analogue of these motivic periods and then give a purely automorphic variant of the Deligne conjecture. At last, we introduce some known results of this automorphic variant.

Motivation & Objective

• To formulate a purely automorphic version of the Deligne conjecture for tensor products of motives over a quadratic imaginary field.
• To define automorphic periods analogous to motivic periods and establish their compatibility with Deligne's conjecture.
• To prove the automorphic variant of the Deligne conjecture in specific cases, including $n'=1$, $n>n'$ with gap conditions, and $m=1$ with same parity.
• To generalize results to CM fields and establish functoriality of automorphic periods under base change.

Proposed method

• Define motivic periods for motives over a quadratic imaginary field and reformulate the Deligne conjecture in terms of these periods.
• Introduce automorphic periods for cuspidal representations of $GL_n(\mathbb{A}_K)$ over a quadratic imaginary field $K$.
• Establish a conjectural relation between automorphic periods and critical values of $L$-functions via the formula $L(m,\Pi \times \Pi') \sim_{E(\Pi)E(\Pi');K} (2\pi i)^{nn'm} \prod_j P^{(j)}(\Pi)^{sp(j,\Pi;\Pi')} \prod_k P^{(k)}(\Pi')^{sp(k,\Pi';\Pi)}$.
• Use base change theory to relate automorphic periods across different CM fields and establish their factorization at infinite places.
• Prove the automorphic variant of the Deligne conjecture in three key cases: $n'=1$, $n>n'$ with different parity and gap conditions, and $m=1$ with same parity.
• Leverage known results from Harris (1997), Gan-Gross-Prasad (2015), and the author’s thesis (2015) to verify the conjecture in these cases.

Experimental results

Research questions

• RQ1 Can the Deligne conjecture for tensor products of motives over a quadratic imaginary field be reformulated in terms of automorphic periods?
• RQ2 Is the automorphic variant of the Deligne conjecture compatible with the original motivic conjecture?
• RQ3 What are the critical values of Rankin-Selberg $L$-functions for $GL_n \times GL_{n'}$ and how are they related to automorphic periods?
• RQ4 How do automorphic periods behave under base change in CM fields, and can they be factorized at infinite places?
• RQ5 What is the functorial relationship between automorphic periods of a representation and its base change?

Key findings

• The automorphic variant of the Deligne conjecture is proven for $n'=1$, regardless of whether $\Pi'$ is conjugate self-dual.
• The conjecture holds for $n>n'$ when $n$ and $n'$ have different parity and the numbers $-b_j$ lie in distinct gaps between the $a_i$.
• The conjecture is verified for $m=1$ when $n$ and $n'$ have the same parity.
• The automorphic periods factorize through infinite places, generalizing a conjecture of Shimura and enabling extension to CM fields.
• The functoriality of automorphic periods under base change is established, with explicit relations between periods of $\Pi$ on $L$ and its base change $\pi$ on $F$.
• The conjectural relation between automorphic periods and $L$-values is shown to match the prediction of the Deligne conjecture when motivic and automorphic periods are identified.