By Michel Courtieu

This publication, now in its 2d version, is dedicated to the arithmetical idea of Siegel modular varieties and their L-functions. The significant object are L-functions of classical Siegel modular kinds whose particular values are studied utilizing the Rankin-Selberg procedure and the motion of convinced differential operators on modular varieties that have great arithmetical houses. a brand new approach to p-adic interpolation of those serious values is gifted. an incredible classification of p-adic L-functions treated within the current ebook are p-adic L-functions of Siegel modular kinds having logarithmic progress. The given development of those p-adic L-functions makes use of particular algebraic houses of the arithmetical Shimura differential operator. The publication should be very necessary for postgraduate scholars and for non-experts looking for a fast method of a speedily constructing area of algebraic quantity thought. This re-creation is considerably revised to account for the recent factors that experience emerged some time past 10 years of the most formulation for specified L-values by way of arithmetical idea of approximately holomorphic modular types.

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

**Example text**

Amice, J. M. Viˇsik (see [Am-Ve], [Vi1]). Let C(Z× S , Cp ) denote the space of Cp -valued functions that can be locally represented by polynomials of degree less than a natural number h of the variable xp ∈ XS introduced above. 11. A Cp -linear form µ : C h (Z× S , Cp ) −→ Cp is called an hadmissible measure if for all r = 0, 1, · · · , h − 1 the following growth condition is satisﬁed (xp − ap )r dµ sup a∈Z× S a+(M) . M. Viˇsik) p ). We know (essentially due to Y. Amice, J. V´ that each h-admissible measure can be uniquely extended to a linear form on the Cp -space of all locally analytic functions so that one can associate to its Mellin transform Lµ : XS −→ Cp χ −→ Lµ (χ) = Z× S χdµ which is a Cp -analytic function on XS of the type o(log(xhp )).

N. Andrianov (see [An1] in the case m = 2. For m = 1 we have that α0 (q) + α0 (q)α1 (q) = a(q), α20 (q)α1 (q) = ψ(q)q k−1 , where f (z) = n≥0 a(n)e(nz) is the Fourier expansion of a normalized elliptic cusp Hecke eigenform, so that the zeta function 1 − a(q)q −s + ψ(q)q k−1−2s Z(s, f ) = −1 ∞ a(n)n−s = qN n=1 (n,N )=1 coincides essentially with the Mellin transform of f ∞ L(s, f ) = a(n)n−s . 33) r=1 1≤i1 >···>ir ≤m λf (q) = λf T (q), T (q) = (Γ gΓ ) ν(g)=q are the Hecke operator for the group Γ = Γ0m (N ), tgJm g = ν(g)Jm .

The group Km is a maximal compact subgroup of the Lie group Spm (R) and it can be identiﬁed with the group U(m) of all unitary m × m-matrices via the ab map γ = −→ a + ib. 4) i≤j d× z = det(y)−κ dz, where z = x + iy, x = (xij ) = tx, y = (yij ) = ty > 0. Then d× z is a diﬀerential × on Hm invariant under the action of the group G+ ∞ , and the measure d y is invariant under the action of elements a ∈ GLm (R) on Y = {y ∈ Mm (R) | t y = y > 0} deﬁned by the rule y −→ taya. 2 Siegel modular forms Let us consider the Siegel modular group denoted Γ m = Spm (Z) and let Γ ⊂ G+ Q be an arbitrary congruence subgroup.