Ebook: The Ball and Some Hilbert Problems

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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The famous twelfth Hilbert problem calls for holomorphic functions in several variables with properties analogous to the exponential function and the elliptic modular function with a view to the explicit construction of (Hilbert) class fields by means of special values. The lecture notes present those functions living on the two-dimensional complex unit ball. In the course of their construction, the reader is introduced to work with complex multiplication, moduli fields, moduli space of curves, surface uniformizations, Gauss-Manin connection, Jacobian varieties, Torelli's theorem, Picard modular forms, Theta functions, class fields and transcen- dental values in an effective manner.

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The famous twelfth Hilbert problem calls for holomorphic functions in several variables with properties analogous to the exponential function and the elliptic modular function with a view to the explicit construction of (Hilbert) class fields by means of special values. The lecture notes present those functions living on the two-dimensional complex unit ball. In the course of their construction, the reader is introduced to work with complex multiplication, moduli fields, moduli space of curves, surface uniformizations, Gauss-Manin connection, Jacobian varieties, Torelli's theorem, Picard modular forms, Theta functions, class fields and transcen- dental values in an effective manner.
Content:
Front Matter....Pages i-vii
Elliptic Curves, the Finiteness Theorem of Shafarevi?....Pages 1-9
Picard Curves....Pages 11-37
Uniformizations and Differential Equations of Euler-Picard Type....Pages 39-63
Algebraic Values of Picard Modular Theta Functions....Pages 65-100
Transcendental Values of Picard Modular Theta Constants....Pages 101-118
Arithmetic Surfaces of Kodaira-Picard Type and some Diophantine Equations....Pages 119-131
Appendix I A Finiteness Theorem for Picard Curves With Good Reduction (by J. Estrada-Sarlabous)....Pages 133-143
Appendix II The Hilbert Problems 7, 12,21 and 22....Pages 145-148
Back Matter....Pages 149-160

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The famous twelfth Hilbert problem calls for holomorphic functions in several variables with properties analogous to the exponential function and the elliptic modular function with a view to the explicit construction of (Hilbert) class fields by means of special values. The lecture notes present those functions living on the two-dimensional complex unit ball. In the course of their construction, the reader is introduced to work with complex multiplication, moduli fields, moduli space of curves, surface uniformizations, Gauss-Manin connection, Jacobian varieties, Torelli's theorem, Picard modular forms, Theta functions, class fields and transcen- dental values in an effective manner.
Content:
Front Matter....Pages i-vii
Elliptic Curves, the Finiteness Theorem of Shafarevi?....Pages 1-9
Picard Curves....Pages 11-37
Uniformizations and Differential Equations of Euler-Picard Type....Pages 39-63
Algebraic Values of Picard Modular Theta Functions....Pages 65-100
Transcendental Values of Picard Modular Theta Constants....Pages 101-118
Arithmetic Surfaces of Kodaira-Picard Type and some Diophantine Equations....Pages 119-131
Appendix I A Finiteness Theorem for Picard Curves With Good Reduction (by J. Estrada-Sarlabous)....Pages 133-143
Appendix II The Hilbert Problems 7, 12,21 and 22....Pages 145-148
Back Matter....Pages 149-160
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