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January 20, 2017 | Discrete Mathematics | By admin | 0 Comments

2 Triangle teams: An creation 279 three simple Shimura Curves 281 four Examples of Shimura Curves 282 five Congruence Zeta services 283 6 Diophantine houses of Shimura Curves 284 7 Klein Quartic 285 eight Supersingular issues 289 Towers of Elkies nine 289 7. CRYPTOGRAPHY AND functions 291 1 creation 291 Discrete Logarithm challenge 2 291 Curves for Public-Key Cryptosystems three 295 Hyperelliptic Curve Cryptosystems four 297 CM-Method five 299 6 Cryptographic Exponent three hundred 7 positive Descent 302 eight Gaudry and Harley set of rules 306 nine Picard Jacobians 307 Drinfeld Module established Public Key Cryptosystems 10 308 eleven Drinfeld Modules and a method capabilities 308 12 Shimura's Map 309 thirteen Modular Jacobians of Genus 2 Curves 310 Modular Jacobian Surfaces 14 312 15 Modular Curves of Genus 313 sixteen Hecke Operators 314 eight. REFERENCES 317 345 Index Xll Preface The background of counting issues on curves over finite fields is particularly ex- tensive, beginning with the paintings of Gauss in 1801 and carrying on with with the paintings of Artin, Schmidt, Hasse and Weil of their examine of curves and the similar zeta features Zx(t), the place m Zx(t) = exp (2: N t ) m m 2': 1 m with N = #X(F qm). If X is a curve of genus g, Weil's conjectures m nation that L(t) Zx(t) = (1 - t)(l - qt) the place L(t) = rr~!l (1 - O'.

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