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xx Contents

6 Characterizations of Calabi–Yau Manifolds: First Examples . . ..... 366
7 Examples of Calabi–Yau Varieties from Toric Geometry .......... 369
8 Line Bundles and Divisors: Picard and N´ eron–Severi Groups . ..... 371
9 Numerical Invariants of Surfaces . . ............................ 374
10 Enriques Classification for Surfaces ........................... 377
11 Introduction to K3 Surfaces .................................. 378
20 Families of Elliptic Curves ..................................... 383
1 Algebraic and Analytic Geometry . ............................ 384
2 Morphisms Into Projective Spaces Determined by Line Bundles,
Divisors, and Linear Systems ................................. 387
3 Fibrations Especially Surfaces Over Curves..................... 390
4 Generalities on Elliptic Fibrations of Surfaces Over Curves . . ..... 392
5 Elliptic K3 Surfaces ........................................ 395
6 Fibrations of 3 Dimensional Calabi–Yau Varieties ............... 397
7 Three Examples of Three Dimensional Calabi–Yau Hypersurfaces
in Weight Projective Four Space and Their Fibrings .............. 400

Appendix I: Calabi–Yau Manifolds and String Theory ................. 403
Stefan Theisen
Why String Theory? ............................................. 403
Basic Properties . . .............................................. 404
String Theories in Ten Dimensions ................................. 406
Compactification . . .............................................. 407
Duality ........................................................ 409
Summary . . .................................................... 411

Appendix II: Elliptic Curves in Algorithmic Number Theory and
Cryptography ............................................... 413
Otto Forster
1 Applications in Algorithmic Number Theory.................... 413
1.1 Factorization ........................................ 413
1.2 Deterministic Primality Tests .......................... 415
2 Elliptic Curves in Cryptography . . ............................ 417
2.1 The Discrete Logarithm ............................... 417
2.2 Diffie–Hellman Key Exchange ......................... 417
2.3 Digital Signatures . . .................................. 418
2.4 Algorithms for the Discrete Logarithm . ................. 419
2.5 Counting the Number of Points . . ...................... 421
2.6 Schoof’s Algorithm .................................. 421
2.7 Elkies Primes ....................................... 423
References ..................................................... 424