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

By Pierre Deligne

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

This characterization allows the authors to match monodromy teams such as diversified parameters and to end up commensurability modulo internal automorphisms of PU(1,n).

The publication contains an research of elliptic and parabolic monodromy teams, in addition to hyperbolic monodromy teams. the previous play a task within the evidence stunning variety of lattices in PU(1,2) developed because the primary teams of compact advanced surfaces with consistent holomorphic curvature are in reality conjugate to projective monodromy teams of hypergeometric services. The characterization of hypergeometric-like services via their exponents on the divisors "at infinity" allows one to end up generalizations in n-variables of the Kummer identities for n-1 regarding quadratic and cubic adjustments of the variable.

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Proof. Let x ∈ X with x = 1. Find x∗ ∈ X ∗ such that x∗ = x∗ (x) = 1. For every u ∈ X we choose a norm one functional yu∗ in X ∗ such that yu∗ (x+u) = x+u . Then yu∗ + x∗ ≥ yu∗ (x) + x∗ (x) = x + u − yu∗ (u) + 1. Hence yu∗ + x∗ → 2 as u → 0. By the local uniform convexity of X ∗ we deduce that yu∗ − x∗ → 0 as u → 0. It follows that ∗ 0 ≤ x + u + x − u − 2 = yu∗ (x + u) + y−u (x − u) − 2 ∗ ∗ = yu∗ (x) + y−u (x) − 2 + (yu∗ − y−u )(u) ∗ ≤ yu∗ − y−u u = o( u ). 1. Since both X and X ∗ are separable there are a dense sequence (xn )∞ n=1 in the unit sphere of X and an increasing sequence of finite dimensional sub∗ spaces (Fn )∞ n=1 of X such that X∗ = Fn .

This is enough for our purposes, but let us note that we will actually use only the following consequence of uniform equicontinuity. For every ε > 0 there is δ > 0 such that whenever F ∈ F(U ), u, v ∈ U p , y ∈ Y p , and u − v < δ, then |F (u, y) − F (v, y)| < ε. 3. If ε and δ are as above and W is a subspace of X, then for every x, x ˜ ∈ W with x − x ˜ < δ, β(f, x, W, F) ≤ β(f, x ˜, W, F) + ε. Proof. Fix any 0 < c < β(f, x, W, F). By definition of β(f, x, W, F) there is a separable subspace U ⊂ W such that β(f, x, U, F ) > c for every F ∈ F(U ).

F ) + C 2 + Cy ∗ (f (x)(e) − w) > C 2 + C − 2, implying that y ∗ (f (x)(e) − w)) > − 2 (1 + Lip C · (f )), which, multiplied by b − a and for C large enough, gives the statement. So far we have considered what could be termed one-dimensional mean value estimates: although the range could be even infinite dimensional, the estimate involved only derivative in a single direction. 3). The natural formulation may then sound somewhat awkward, but we may follow the same reasoning as above. If, instead of a line segment, we imagine an n-dimensional (say C 1 ) surface in X, then for any Lipschitz f : X −→ Y and u∗1 , .

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