By Michel Courtieu

ISBN-10: 3540407294

ISBN-13: 9783540407294

This ebook, now in its 2d version, is dedicated to the arithmetical concept of Siegel modular varieties and their L-functions. The critical object are L-functions of classical Siegel modular varieties whose targeted values are studied utilizing the Rankin-Selberg process and the motion of definite differential operators on modular kinds that have great arithmetical homes. a brand new approach to p-adic interpolation of those serious values is gifted. a major type of p-adic L-functions treated within the current booklet are p-adic L-functions of Siegel modular types having logarithmic progress. The given development of those p-adic L-functions makes use of special algebraic houses of the arithmetical Shimura differential operator. The booklet should be very worthwhile for postgraduate scholars and for non-experts looking for a fast method of a quickly constructing area of algebraic quantity thought. This re-creation is considerably revised to account for the recent motives that experience emerged long ago 10 years of the most formulation for distinctive L-values when it comes to arithmetical conception of approximately holomorphic modular varieties.

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

**Example text**

Let {fi } be a system of continuous functions fi ∈ C(Y, Op ) in the ring C(Y, Op ) of all continuous functions on the compact totally disconnected group Y with values in the ring of integers Op of Cp such that Cp -linear span of {fi } is dense in C(Y, Cp ). Let also {ai } be any system of elements ai ∈ Op . Then the existence of an Op -valued measure µ on Y with the property fi dµ = ai Y is equivalent to the following congruences, for an arbitrary choice of elements bi ∈ Cp almost all of which vanish bi fi (y) ∈ pn Op for all y ∈ Y implies bi ai ∈ pn Op .

M−1 a=0 f (a)teat . 27) Now let us consider the proﬁnite ring Y = ZS = lim (Z/M Z) ←− M (S(M ) ⊂ S), the projective limit being taken over the set of all positive integers M with support S(M ) in a ﬁxed ﬁnite set S of prime numbers. Then the periodic function f : Z/M Z → C with S(M ) ⊂ S may be viewed as an element of Step(Y, C). We claim that there exists a distribution Ek : Step(Y, C) → C which is uniquely determined by the condition Ek (f ) = Bk,f for all f ∈ Step(Y, C). e. Bk,f ) does not depend on the choice of a period M of the function f .

We suppose that α(q) = 0 for every q ∈ S, and let us extend the deﬁnition of numbers α(n) to all positive integers whose support is contained in S (by multiplicativity): α(q)ordq (n) α(n) = (S(q) ⊂ S). 73). 75), namely, B S (n) = (S(n) ⊂ S), B S (q ordq (n) ) q∈S with Bq,i , for i < mq 0, otherwise. Now we state the main result of the section. 12. 77). 79) with G(χ) = χ(a)e amodCχ a Cχ being the Gauss sum, Cχ the conductor, and S(χ) the support of the conductor of χ. 13. The distribution µs can be obtained as the Fourier transform of the standard zeta-distribution attached to the Diriclet series Hq (χ(q)q −s ) = D(s, χ) = q∈S χ(n)a0 (n)n−s , n≥1 where a0 (qn) = α(q)a0 (n) for each n ∈ N and q ∈ S (see [Co-PeRi]).

### Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms by Michel Courtieu

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