A motivic approach to Shimura's zeta functions and attached $p$-adic $L$-functions via admissible measures (Automorphic forms, automorphic representations and related topics)

Abstract

A motivic approach is presented to Shimura's zeta functions Z.(s, f) [47] attched to holomorphic automorphic forms f on unitary groups UK (n, n) over an imaginary quadratic field K=Q(√-Dk). A motivically normalized L-function D(s, f) attached to Z(s, f) is defined in accordance with Deligne's conjectures [14]. An explicit description of Shimura's Г-factors is used. The attached p-adic L-functions of D(s, f) satisfies conjecture of Coates-Perrin-Riou [11] and it is constructed via admissible measures of Amiee-Vélu, see also [31]. The p-ordinary case was treated in [17] via algebraic geometry (method of Katz). The main result is stated in terms of the Hodge polygon PH(t): [0, d] → R and the Newton polygon PN(t) = PN, v(t) : [O, d] → R of the zeta function D(s, f) of degree d = 4n. Main theorem gives a p-adic analytic interpolation of the L values in the form of certain integrals with respect to Mazur-type measures, saisfying Coates-Perrin-Riou conjectures. Both Rankin-Selberg and doubling methods are used.