On period relations for automorphic 𝐿-functions I

Abstract

We study Hecke algebras for pairs (g, K) over arbitrary fields E of characteristic 0, define the Bernstein functor and give another definition of the Zuckerman functor over E. Building on this and the author's previous work on rational structures on automorphic representations, we show that hard duality remains valid over E and apply this result to the study of rationality properties of Sun's cohomologically induced functionals. Our main application are period relations for the special values of standard L-functions of automorphic representations of GL(2n) admitting Shalika models.

Rational structures on automorphic representations

“…Such a rational classification could reveal fundamental arithmetic patterns in the representation theory of reductive groups, local and global. As already hinted in the previous sections certain rationality patterns reflect motivic arithmetic structure as demonstrated in [30,31].…”

“…Since h is non-degenerate, so is b. A direct calculation, exploiting relations (27), (31) and (32), yields…”

The nonvanishing hypothesis at infinity for Rankin-Selberg convolutions

“…See Kazhdan-Mazur-Schmidt [KMS], Mahnkopf [Mah], Raghuram [Rag1], Kasten-Schmidt [KS], Januszewski [Jan1], RS2], and Schmidt [Sch1,Sch2]. For more recent works using Theorem A (and Theorem C of Section 6), see Grobner-Harris [GH], Raghuram [Rag2], and Januszewski [Jan2,Jan3,Jan4].…”

$\mathrm{SO}(3)$-homogeneous decomposition of the flag scheme of $\mathrm{SL}_3$ over $\mathbb{Z}\left[1/2\right]$

“…Motivated by applications to special values of automorphic L-functions, Michael Harris, Günter Harder, and Fabian Januszewski started to work on (g, K)modules over number fields and localization of the rings of their integers in the 2010s ( [9,10,8,18,19,17]). For general theory of (g, K)-modules over commutative rings, see [13] and [12].…”