Algebraic geometry and its applications : dedicated to Gilles Lachaud on his 60th birthday : proceedings of the First SAGA Conference, Papeete, France, 7-11 May 2007 / editors, Jean Chaumine, James Hirschfeld, Robert Rolland.

Includes bibliographical references and index.

Print version record.

""CONTENTS""; ""To Gilles Lachaud on the occasion of his 60th birthday R. Rolland, M. Tsfasman""; ""Preface J.-M. Goursaud""; ""Organizing Committees""; ""Fast addition on non-hyperelliptic genus 3 curves S. Flon, R. Oyono, C. Ritzenthaler""; ""Introduction""; ""1. Geometric description of the algorithm""; ""2. Rationality of the points on a canonical divisor""; ""2.1. Structure of the canonical divisor""; ""2.2. The general and tangent case""; ""2.3. The flex case""; ""2.4. The hyperflex case""; ""3. Algebraic description""; ""3.1. Mumford representation and typical divisors""

""3.2. The tangent case""""3.3. Flex case""; ""3.4. Comments on implementation""; ""4. Examples""; ""4.1. AGM-method""; ""4.2. 3-dimensional factors of Jnew(X0(N))""; ""5. Conclusion""; ""6. Appendix""; ""Acknowledgment""; ""References""; ""Computing endomorphism rings of Jacobians of genus 2 curves over finite fields D. Freeman, K.Lauter""; ""1. Introduction""; ""2. Computing zeta functions and the Frobenius element""; ""3. Constructing a generating set for OK""; ""4. Determining fields of definition""; ""4.1. The brute force method""; ""4.2. The Gaudry-Harley-Schost method""

""4.3. A probabilistic method""""5. Computing the action of Frobenius""; ""5.1. The brute force method""; ""5.2. A probabilistic method""; ""5.3. The Couveignes method""; ""6. Bounding the field of definition of the .d-torsion points""; ""7. Computing Igusa class polynomials""; ""8. Implementation notes""; ""9. Examples""; ""References""; ""Complex multiplication and canonical lifts D. Kohel""; ""1. Introduction""; ""2. Complex multiplication""; ""2.1. Complex multiplication in genus 1""; ""2.2. Complex multiplication in higher dimension""; ""3. Canonical lifts""

""3.1. Canonical lifting conditions""""3.2. Canonical lifts as CM constructions""; ""4. Constructive CM algorithms""; ""4.1. Canonical 2-adic AGM algorithm""; ""4.2. Canonical .-adic Richelot lifting algorithm""; ""5. Conclusion""; ""References""; ""Two letters to Jaap Top J.-P. Serre""; ""On some questions of Serre on abelian threefolds G. Lachaud, C. Ritzenthaler""; ""1. Introduction""; ""1.1. Geometric Torelli�s theorem""; ""1.2. Arithmetic Torelli�s theorem""; ""1.3. Serre�s questions""; ""2. Ciani Quartics""; ""2.1. Definition of Ciani quartics""

""2.2. Discriminant of a ternary form""""2.3. Product of elliptic curves""; ""2.4. The theory of Howe, Leprevost and Poonen revisited""; ""2.5. Relation with Serreâ€?s assertion""; ""3. Complex abelian varieties""; ""3.1. The symplectic group""; ""3.2. Abelian varieties""; ""3.3. Isotropy and quotients""; ""3.4. Theta functions""; ""3.5. The modular function Ï?""; ""4. Comparison of analytic and algebraic discriminants""; ""4.1. Expression of the algebraic discriminant""; ""4.2. The subgroup W""; ""4.3. Expression of Ï? ( ′) as a discriminant""; ""Appendix A.""; ""References""

This volume covers many topics, including number theory, Boolean functions, combinatorial geometry, and algorithms over finite fields. It contains many new, theoretical and applicable results, as well as surveys that were presented by the top specialists in these areas. New results include an answer to one of Serre's questions, posted in a letter to Top; cryptographic applications of the discrete logarithm problem related to elliptic curves and hyperelliptic curves; construction of function field towers; construction of new classes of Boolean cryptographic functions; and algorithmic applicati.

English.

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