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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**Extra info for Commensurabilities among Lattices in PU (1,n).**

**Sample text**

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 , .