Zahlentheorie-Skripte

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Zahlentheorie	Bücher	Publikationen	Themen	Quellen	Mathematik

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• Vorlesungsskript 2000 von Winfried Bruns, Fachbereich Mathematik der Universität Osnabrück:
  • Zahlentheorie (pdf file)

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• Scripts by James S. Milne:
  • Algebraic Number Theory (1998)
  Contents: 1.Preliminaries From Commutative Algebra   2.Rings of Integers   3.Dedekind Domains; Factorization   4.The Finiteness of the Class Number   5.The Unit Theorem   6.Cyclotomic Extensions; Fermat's Last Theorem   7.Valuations; Local Fields   8.Global Fields
  • Modular Functions and Modular Forms
  Contents: 1.Preliminaries   2.Elliptic modular curves as Riemann surfaces   3.Elliptic functions   4.Modular functions and modular forms   5.Hecke operators   6.The modular equation for Gamma_0(N)   7.The canonical model of X_0(N) over Q   8.Modular curves as moduli varieties    9.Modular forms, Dirichlet series, and functional equations 10.Correspondences on curves; the theorem of Eichler and Shimura   11.Curves and their zeta functions   12.Complex multiplication for elliptic curves  
  • Elliptic Curves
  Contents: 1.Review of Plane Curves   2.Rational Points on Plane Curves   3.The Group Law on a Cubic Curve   4.Functions on Algebraic Curves and the Riemann-Roch Theorem   5.Definition of an Elliptic Curve   6.Reduction of an Elliptic Curve Modulo p   7.Elliptic Curves over Q_p   8.Torsion Points   9.Neron Models   10.Elliptic Curves over the Complex Numbers   11.The Mordell-Weil Theorem: Statement and Strategy   12.Group Cohomology   13.The Selmer and Tate-Shafarevich Groups   14.The Finiteness of the Selmer Group   15.Heights   16.Completion of the Proof of the Mordell-Weil Theorem, and Further Remarks   17.Geometric Interpretation of the Cohomology Groups; Jacobians   18.The Tate-Shafarevich Group; Failure of the Hasse Principle   19.Elliptic Curves over Finite Fields   20.The Conjecture of Birch and Swinnerton-Dyer   21.Elliptic Curves and Sphere Packings   22.Algorithms for Elliptic Curves   23.The Riemann Surfaces X₀(N)   24.X₀ as an Algebraic Curve over Q   25.Modular Forms   26.Modular Forms and the L-series of Elliptic Curves   27.Statements of the Main Theorems   28.How to get an Elliptic Curve from a Cusp Form   29.Why the L-Series of E Agrees with the L-series of f   30.Wiles's Proof   31.Fermat, At Last  
  • Class Field Theory
  Contents: 1.Local Class Field Theory: Lubin-Tate Extensions   2.Cohomology of Groups.   3.Local Class Field Theory Continued.   4.Brauer Groups   5.Global Class Field Theory: Statements   6.L-series and the Density of Primes   7.Global Class Field Theory: Proofs   8.Complements (Power reciprocity laws; quadratic forms; etc.)

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kersten@mathematik.uni-bielefeld.de