[Paper Review] Deligne's conjecture for automorphic motives over CM-fields, Part I: factorization

Harald Grobner, Michael Harris+1 more|arXiv (Cornell University)|Feb 8, 2018

Advanced Algebra and Geometry43 references3 citations

TL;DR

This paper establishes relations between automorphic periods and critical values of Rankin-Selberg and Asai L-functions over CM fields by combining the Ichino-Ikeda-Nekovar-Harris (IINH) formula with cup product structures on coherent cohomological automorphic forms. It shows that automorphic periods of holomorphic forms factor into products of coherent cohomological forms, compatibly with the motivic factorization predicted by the Tate conjecture, conditional on the IINH formula and a non-vanishing conjecture for twisted L-functions.

ABSTRACT

This is the first of two papers devoted to the relations between Deligne's conjecture on critical values of motivic $L$-functions and the multiplicative relations between periods of arithmetically normalized automorphic forms on unitary groups. The present paper combines the Ichino-Ikeda-Neal Harris (IINH) formula with an analysis of cup products of coherent cohomological automorphic forms on Shimura varieties to establish relations between certain automorphic periods and critical values of Rankin-Selberg and Asai $L$-functions of $GL(n) imes GL(m)$ over CM fields. The second paper reinterprets these critical values in terms of automorphic periods of holomorphic automorphic forms on unitary groups. As a consequence, we show that the automorphic periods of holomorphic forms can be factored as products of coherent cohomological forms, compatibly with a motivic factorization predicted by the Tate conjecture. All of these results are conditional on the IINH formula (which is still partly conjectural), as well as a conjecture on non-vanishing of twists of automorphic $L$-functions of $GL(n)$ by anticyclotomic characters of finite order.

Motivation & Objective

• To explore the connection between Deligne's conjecture on critical values of motivic L-functions and multiplicative relations among automorphic periods on unitary groups.
• To analyze how cup products of coherent cohomological automorphic forms on Shimura varieties relate to critical values of Rankin-Selberg and Asai L-functions over CM fields.
• To establish a factorization of automorphic periods of holomorphic forms into products of coherent cohomological forms, compatible with motivic factorization predicted by the Tate conjecture.
• To lay the foundation for the second paper, which reinterprets critical values in terms of automorphic periods on unitary groups.

Proposed method

• Utilizes the Ichino-Ikeda-Nekovar-Harris (IINH) formula to relate automorphic periods to special values of L-functions.
• Applies cup product structures on coherent cohomological automorphic forms over Shimura varieties to derive period relations.
• Analyzes Rankin-Selberg and Asai L-functions of GL(n) × GL(m) over CM fields to connect them with automorphic periods.
• Combines cohomological techniques with automorphic representation theory to study period factorization.
• Relies on the IINH formula, which remains partly conjectural, and a non-vanishing conjecture for twists of GL(n) L-functions by finite-order anticyclotomic characters.
• Establishes compatibility of period factorizations with the motivic factorization predicted by the Tate conjecture.

Experimental results

Research questions

• RQ1 How do automorphic periods of holomorphic forms on unitary groups factor in terms of coherent cohomological forms over CM fields?
• RQ2 What is the precise relation between critical values of Rankin-Selberg and Asai L-functions and automorphic periods on Shimura varieties?
• RQ3 To what extent do the IINH formula and cup product structures on cohomological forms explain period relations in the context of Deligne's conjecture?
• RQ4 How do these period relations align with the motivic factorization predicted by the Tate conjecture?
• RQ5 What conditions are required to ensure the non-vanishing of twisted L-functions for GL(n) over CM fields?

Key findings

• The paper establishes that automorphic periods of holomorphic forms on unitary groups can be factored as products of coherent cohomological automorphic forms over CM fields.
• Critical values of Rankin-Selberg and Asai L-functions for GL(n) × GL(m) are shown to be related to automorphic periods via cup product constructions on Shimura varieties.
• The factorization of automorphic periods is compatible with the motivic factorization predicted by the Tate conjecture.
• The results are conditional on the Ichino-Ikeda-Nekovar-Harris formula and a non-vanishing conjecture for twists of GL(n) L-functions by finite-order anticyclotomic characters.
• The framework provides a pathway to reinterpret critical L-values in terms of automorphic periods on unitary groups, as developed in the second paper.
• The analysis reveals a deep interplay between cohomological automorphic forms, L-function special values, and motivic arithmetic structures over CM fields.