Vol. 7, No. 1, 2013

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$L$-functions and periods of adjoint motives

Michael Harris

Vol. 7 (2013), No. 1, 117–155
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/