zbMATH — the first resource for mathematics

Admissible non-Archimedean standard zeta functions associated with Siegel modular forms. (English) Zbl 0837.11029
Jannsen, Uwe (ed.) et al., Motives. Proceedings of the summer research conference on motives, held at the University of Washington, Seattle, WA, USA, July 20-August 2, 1991. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 55, Pt. 2, 251-292 (1994).
Let \(p\) be a prime number. Consider a Siegel cusp form \(f\) of even degree \(m\) and of weight \(k > 2m + 2\). The author describes \(p\)-adic properties of the special critical values of the standard zeta function \(D(s,f,\chi)\) with varying Dirichlet character \(\chi\). The work extends previous investigations of the author, where he treated the case with \(p\)-ordinary \(f\). Here the \(p\)-supersingular case is included, and the existence of the corresponding \(h\)-admissible measure is proved. The author also gives an interpretation of \(D (s,f,\chi)\) in terms of a conjectural motive \(M(f)\) over \(Q\) (provided \(f\) does not belong to the generalized Maass subspace). He also checks that the number \(h\) coincides with \(h(p,M(f))\) (attached to an arbitrary motive over \(Q\) by A. Dabrowski).
It should be noticed that recently S. Böcherer and C.-G. Schmidt (“\(p\)-adic measures attached to Siegel modular forms”, preprint 1995) removed the condition that \(m\) be even; the method avoids holomorphic projection by using holomorphic differential operators.
For the entire collection see [Zbl 0788.00054].

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F85 \(p\)-adic theory, local fields
11S40 Zeta functions and \(L\)-functions