Katz N. M.  Rigid local systems
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Название: 
Rigid local systems 
Автор: 
Katz N. M. 
Категория: 
Математика

Тип: 
Книга 
Дата: 
04.01.2009 12:02:25 
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Описание: 
It is now nearly 140 years since Riemann introduced [RieEDL]
he concept of a "local system" on P^  {a finite set of points). His ea was that one could (and should) study the solutions of an nth order linear differential equation by studying the rank n local system (of its local holomorphic solutions) to which it gave rise.
Riemann knew that a rank n local system on P*  (m points) was "nothing more" than a collection of m invertible matrices Aj in
GL(n, C) which satisfy the matrix equation A^A2Am = (idn)> such collections taken up to simultaneous conjugation by a single element of GL(n, C). He also knew each individual Aj was, up to GL(n, C) conjugacy, just the effect of analytic continuation along a small loop encircling the i'th missing point.
His first application of these then revolutionary ideas was to study the classical Gauss hyper geometric function (RieSGl, which he
did by studying rank two local systems on P*  (three points). His investigation was a stunning success, in large part because any such (irreducible) local system is rigid in the sense that it is determined up to isomorphism as soon as one knows separately the individual conjugacy classes of all its local monodromies. By exploiting this rigidity, Riemann was able to recover Hummer's transformation theory of hypergeometric functions "almost without calculation" [RieAPMl.
It soon became clear that Riemann had been "lucky", in the sense that the most local systems are not rigid. For instance, rank
two irreducible local systems onP^ (m points), all of whose local monodromies are nonscalar, are rigid precisely for m»3. And rank
n irreducible local systems on F*  {three points), each of whose local monodromies has n distinct eigenvalues, are rigid precisely for n=l and n»2.
On the other hand, some of the best known classical functions are solutions of differential equations whose local systems are rigid, including both of the standard generalizations of the hypergeometric function, namely nFn_i, which gives a rank n local system on P* (0,1,eo), and the Pochhammer hypergeometric functions, which give rank n local systems on P*  (n+1 points).
In the classical literature, rigidity or its lack is expressed in 
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