L-functions and periods of adjoint motives

Harris, Michael

doi:10.2140/ant.2013.7.117

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$L$-functions and periods of adjoint motives

Michael Harris

Vol. 7 (2013), No. 1, 117–155

DOI: 10.2140/ant.2013.7.117

Abstract

The article studies the compatibility of the refined Gross–Prasad (or Ichino–Ikeda) conjecture for unitary groups, due to Neal Harris, with Deligne’s conjecture on critical values of $L$ -functions. When the automorphic representations are of motivic type, it is shown that the $L$ -values that arise in the formula are critical in Deligne’s sense, and their Deligne periods can be written explicitly as products of Petersson norms of arithmetically normalized coherent cohomology classes. In some cases this can be used to verify Deligne’s conjecture for critical values of adjoint type (Asai) $L$ -functions.

Keywords

adjoint $L$-functions, automorphic forms, motives, Ichino–Ikeda conjecture, periods

Mathematical Subject Classification 2010

Primary: 11F67

Secondary: 11F70, 14G35, 11G09

Milestones

Received: 10 July 2011

Revised: 12 October 2011

Accepted: 20 February 2012

Published: 28 March 2013

Authors

Michael Harris
Tour 15-25, 4ème étage, bureau 420 Institut de Mathématiques de Jussieu 4, place Jussieu 75252 Paris CEDEX 05 France
http://people.math.jussieu.fr/~harris/

Vol. 7, No. 1, 2013