Download Commensurabilities among Lattices in PU (1,n). by Pierre Deligne PDF

By Pierre Deligne

The first a part of this monograph is dedicated to a characterization of hypergeometric-like services, that's, twists of hypergeometric services in n-variables. those are taken care of as an (n+1) dimensional vector area of multivalued in the community holomorphic services outlined at the house of n+3 tuples of specific issues at the projective line P modulo, the diagonal element of vehicle P=m. For n=1, the characterization should be considered as a generalization of Riemann's classical theorem characterizing hypergeometric services by way of their exponents at 3 singular points.

This characterization allows the authors to check monodromy teams reminiscent of varied parameters and to turn out commensurability modulo internal automorphisms of PU(1,n).

The ebook comprises an research of elliptic and parabolic monodromy teams, in addition to hyperbolic monodromy teams. the previous play a task within the evidence magnificent variety of lattices in PU(1,2) built because the primary teams of compact complicated surfaces with consistent holomorphic curvature are in truth conjugate to projective monodromy teams of hypergeometric features. The characterization of hypergeometric-like services by means of their exponents on the divisors "at infinity" allows one to end up generalizations in n-variables of the Kummer identities for n-1 concerning quadratic and cubic adjustments of the variable.

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Commensurabilities among Lattices in PU (1,n).

The 1st a part of this monograph is dedicated to a characterization of hypergeometric-like features, that's, twists of hypergeometric services in n-variables. those are taken care of as an (n+1) dimensional vector house of multivalued in the neighborhood holomorphic services outlined at the house of n+3 tuples of exact issues at the projective line P modulo, the diagonal part of automobile P=m.

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For x0 ∈ S, the set {rm : m ∈ Zn } is linearly independent. 2, we have |bm | < ∞, m∈Zn where 1 T →∞ T bm = lim T 1 n−1 DR2 (x0 )eirm R dR = C , n rm 22 C1. An introduction to multiple Fourier series for C = 0. However, it is evident that m∈Zn 1 = n rm m∈Zn 1 = +∞. 13) holds. 6 (M. Riesz convexity theorem) Let x0 ∈ Rn and t > 0. We define 0 α Aα (t) = tα D√ (x0 ) = (t − |m|2 )α eim·x . t √ |m|< t Suppose that V (t), W (t) are both positive increasing functions on (0, +∞) and let Uθ (t) = (V (t))1−θ (W (t))θ , for 0 ≤ θ ≤ 1.

Bochner-Riesz means of multiple Fourier integral Proof. Assume lim R→∞ α BR (f ) − f p =0 α (f ) for every f ∈ Lp (Rn ). Thus, we obtain that BR p has bound with respect to R. It follows from the uniform boundedness theorem in functional analysis that α (f ) p ≤ Cp f p . BR Conversely, we suppose that there exists a constant Cp such that the inequality α BR (f ) p ≤ Cp f p holds for any f ∈ Lp (Rn ). Notice that the space D(Rn ), the class of infinitely differential functions which have compact support, is dense in Lp (Rn ).

2 1 −β 2 dt 1 t fx ( ) − s tβ− 2 Jn− 1 −β (t)dt 2 R ∞ 1 t fx ( ) − s tβ− 2 Jn− 1 −β (t)dt. 9) 2 R 1 as t → 0, we immediately have 1 1 − s tβ− 2 Jn− 1 −β (t)dt = O 2 0 fx t R − s tn−1 dt . 4), we easily have 1 fx 0 t R 1 − s tβ− 2 Jn− 1 −β (t)dt = o(1), 2 as R → ∞. From the equation Jν (t) = 2 νπ π cos t − − + O t−3/2 , πt 2 4 as t → +∞, we have ∞ t R fx 1 ∞ = 1 fx 1 − s tβ− 2 Jn− 1 −β (t)dt 2 t R ∞ +O 1 fx n−β 2 cos t − 2 π dt −s π t1−β dt t − s 2−β , R t where the second term can be estimated as ∞ 1 fx t R −s dt t2−β t = t−(2−β+n−1) 0 ∞ +O 1 fx t−(n+2−β) τ ∞ − s τ n−1 dτ 1 R t τ fx − s τ n−1 dτ dt .

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