By Michel Courtieu

This booklet, now in its second version, is dedicated to the arithmetical thought of Siegel modular varieties and their L-functions. The important object are L-functions of classical Siegel modular types whose particular values are studied utilizing the Rankin-Selberg procedure and the motion of yes differential operators on modular varieties that have great arithmetical houses. a brand new approach to p-adic interpolation of those severe values is gifted. an incredible category of p-adic L-functions treated within the current e-book are p-adic L-functions of Siegel modular kinds having logarithmic development. The given building of those p-adic L-functions makes use of designated algebraic houses of the arithmetical Shimura differential operator. The publication could be very important for postgraduate scholars and for non-experts looking for a quickly method of a swiftly constructing area of algebraic quantity concept. This new version is considerably revised to account for the recent motives that experience emerged long ago 10 years of the most formulation for designated L-values when it comes to arithmetical thought of approximately holomorphic modular types.

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Extra resources for Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms

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In the construction of non-Archimedean convolutions of Hilbert modular forms given in chapter 4, we give another approach to local distributions, which differs from the given above and is applicable only to certain Dirichlet series (namely, to convolutions of Rankin type). However, there is an interesting link between these two approaches, which is based on a general construction of p-adic distributions attached to motives. It turns out that both types of distributions can be obtained using the Fourier transform in the distribution space (see [Co-PeRi], [Co-Schn]).

C Springer-Verlag Berlin Heidelberg 2004 46 2 Siegel modular forms and the holomorphic projection operator is transpose to z ∈ Mr,s (A), and write ξ[η] for tηξη. For a non-degenerate square matrix ξ we put ξ = tξ −1 . If ξ is a hermitian matrix then we write ξ ≥ 0 or ξ > 0 according as ξ is non negative or positive definite. Let Hm denote the Siegel upper half plane on the degree m, Hm = {z ∈ Mm (C) | t z = z = x + iy, y > 0}, so that Hm is a complex analytic variety whose dimension is denoted by m = m(m + 1)/2.

2 Measures Let R be a topological ring, and C(Y, R) be the topological module of all R-valued functions on a profinite set Y . 6. A measure on Y with values in the topological R-module A is a continuous homomorphism of R-modules µ : C(Y, R) −→ A. The restriction of µ to the R-submodule Step(Y, R) ⊂ C(Y, R) defines a distribution which we denote by the same letter µ, and the measure µ is uniquely determined by the corresponding distribution since the R-submodule Step(Y, R) is dense in C(Y, R). The last statement expresses the well known fact about the uniform continuity of a continuous function over a compact topological space.

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