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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Functional analysis
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The Ball and Some Hilbert Problems (Paperback, 1995 ed.)
Loot Price: R1,282
Discovery Miles 12 820
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The Ball and Some Hilbert Problems (Paperback, 1995 ed.)
Series: Lectures in Mathematics. ETH Zurich
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As an interesting object of arithmetic, algebraic and analytic
geometry the complex ball was born in a paper of the French
Mathematician E. PICARD in 1883. In recent developments the ball
finds great interest again in the framework of SHIMURA varieties
but also in the theory of diophantine equations (asymptotic FERMAT
Problem, see ch. VI). At first glance the original ideas and the
advanced theories seem to be rather disconnected. With these
lectures I try to build a bridge from the analytic origins to the
actual research on effective problems of arithmetic algebraic
geometry. The best motivation is HILBERT'S far-reaching program
consisting of 23 prob lems (Paris 1900) " . . . one should succeed
in finding and discussing those functions which play the part for
any algebraic number field corresponding to that of the exponential
function in the field of rational numbers and of the elliptic
modular functions in the imaginary quadratic number field." This
message can be found in the 12-th problem "Extension of KRONECKER'S
Theorem on Abelian Fields to Any Algebraic Realm of Rationality"
standing in the middle of HILBERTS'S pro gram. It is dedicated to
the construction of number fields by means of special value of
transcendental functions of several variables. The close connection
with three other HILBERT problems will be explained together with
corresponding advanced theories, which are necessary to find
special effective solutions, namely: 7. Irrationality and
Transcendence of Certain Numbers; 21."
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