## Local measures related with parabolic Jacquet-Langlands forms over CM-fields.(Russian)Zbl 0417.12004

This paper is closely connected with the paper written by Yu. I. Manin in [Usp. Mat. Nauk 31, No. 1(187), 5–54 (1976; Zbl 0336.12007) and devoted to the construction of $$p$$-adic analogy of $$L$$-functions of modular forms. These functions were connected with representations of $$\mathrm{GL}(2)$$ over totally real fields. In the present paper Manin’s construction is generalized on CM-fields. The author investigates all over again Manin’s formula for a local measure and then he interprets the main arithmetical functions of the measure formula as periods of a closed differential form on a variety with unique singularity. The main theorem is devoted to establish estimates for some measures.

### MSC:

 11S40 Zeta functions and $$L$$-functions 11S85 Other nonanalytic theory 20G05 Representation theory for linear algebraic groups 11G15 Complex multiplication and moduli of abelian varieties

Zbl 0336.12007
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