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 capabilities, 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 neighborhood holomorphic services outlined at the area of n+3 tuples of precise issues at the projective line P modulo, the diagonal portion of automobile P=m. For n=1, the characterization might be considered as a generalization of Riemann's classical theorem characterizing hypergeometric services via their exponents at 3 singular points.

This characterization allows the authors to check monodromy teams such as varied 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 job within the facts fantastic variety of lattices in PU(1,2) developed 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 through their exponents on the divisors "at infinity" allows one to turn out generalizations in n-variables of the Kummer identities for n-1 regarding quadratic and cubic adjustments of the variable.

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

Example text

Let Xl, X2 E dom f. Then (Xl, f(Xl)) E epi f and (X2, f(X2)) E epi f. 2 (8), find Ql(Xl, f(xI) + Q2(X2, f(X2)) E epi f. e. tl 2:: f(xd and t2 2:: f(X2) (if dom f = 0 then f(x) = += (x E X) and epi f = 0). 2 = 1. 6. DEFINITION. A mapping p : X ~ JR.. is a sublinear functional provided that epi p is a cone. 7. If dom p i= then the following statements are equivalent: (1) p is a sublinear functional; (2) p is a convex function that is positively homogeneous: p(Qx) = Qp(x) for all 0. + and Xl, X2 E X, then P(QlXl + Q2 X2) :::; QlP(Xt} + Q2P(X2); (4) p is a positively homogeneous functional that is subadditive: P(XI + X2) :::; p(xt} + p(X2) (Xl, X2 EX).

9. Every equicontinuous set consists of uniformly continuous mappings. Every finite set of uniformly continuous mappings is equicontinuous. 3. 1. ¥:= Xl x X 2 • Given x:= (Xl, X2) and y:= (Yl, Y2), put Tben d is a metric on :Z". Moreover, for every X:= (Xl, X2) in :Z" tbe presentation bolds: TX(X) = fil{Ul X U2 : Ul E TXl(Xl), U2 E TX 2 (X2)}. 2. DEFINITION. The topology TX is called the product of TX l and TX 2 or the product topology of Xl x X 2. 3. DEFINITION. A function f : X -+ lR" is lower semicontinuous if its epigraph epi f is closed in the product topology of X x JR.

9. Are there any analogs for a Hamel basis in general modules? 10. When does a sum of projections present a projection itself? 11. Let T be an endomorphism of some vector space which satisfies the conditions Tn-l I- 0 and Tn 0 for a natural n. Prove that the operators TO, T, ... ,Tn-l are linearly independent. 12. Describe the structure of a linear operator defined on the direct sum of spaces and acting into the product of spaces. 13. Find conditions for unique solvability of the following equations in operators ~A B and A~= B (here the operator ~ is unknown).

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