Onp-adicL-Functions for GL (n) × GL (n-1) Over Totally Real Fields

Rational structures on automorphic representations

“…Statement (i) follows from Matsushima's Formula and our Proposition 2.3, a consequence of the Homological Base Change Theorem, and the fact that we may associate to an even finite order character ξ a cohomology class of degree 0, such that cup product with this class realizes the twist with ξ. Fruthermore, if ξ is odd, or if φ is a non-cohomological cusp form in (69) generating an irreducible representation, the form φ ξ (g) := ξ(det(g)) · φ(g) lies again in (69) by Arthur-Clozel [2], is K-finite if and only if φ is K-finite, and thus this correspondence may serve as the bridge to transport any normalization of the rational structure on the subspace of cohomoligcal forms to the space of non-cohomological forms. For an explicit exposition of the Eichler-Shimura map realizing Matsushima's formula in the case of totally real F (yet valid in general), we refer to Section 6.2 in [29]. Finally the uniqueness statement is a consequence of the construction and the uniqueness statement (f).…”

“…Proof. Recall that Y = Res C/R X and Y ′ is the base change of Y to C. In the infinitesimally unitary case the existence of the isomorphism (29) implies that decomposition (21) reads…”

p-adic L-functions for Rankin–Selberg convolutions over number fields

“…We remark that in the ordinary case the p-adic measure is uniquely determined by the evaluation at sufficiently ramified characters (cf. [Jan14]).…”

“…Proof. The proof proceeds as the proof of Theorem 4.5 in [Jan14]. The main ingredient being Theorem 7.1 for the computation of c(χ p , −) in the ramified case.…”

“…The first general construction of a p-adic measure on GL(n+1)×GL(n) over Q is Schmidt's complement [Sch2] to [KMS], which still imposed a trivial central character at infinity and had restrictions on p, depending on n, in particular excluding small primes. The latter restriction was once and for all removed in [Jan11], and the correct formulation for arbitrary class numbers for totally real k was given in [Jan14], which also overcame the restriction of the central character at infinity.…”