Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms.
- 1st ed.
- 1 online resource (258 pages)
- Contemporary Mathematics ; v.587 .
- Contemporary Mathematics .
Intro -- In Memoriam -- Contents -- Preface -- Boris Venkov's Theory of Lattices and Spherical Designs -- 1. Introduction -- 2. Lattices, designs and modular forms -- 3. Lattices and spherical designs -- 4. Unimodular lattices -- 5. Tight spherical designs -- 6. Hecke operators -- References -- Generalized Theta Series and Spherical Designs -- 1. Introduction -- 2. The generalized theta series Θ_ -- 3. Characterizations of -designs -- 4. Root lattices | ADE -- 5. The Leech lattice Λ₂₄ -- References -- Representations of integral quadratic polynomials -- 1. Introduction -- 2. Universal Ternary Quadratic Polynomials -- 3. Regular Ternary Triangular Forms -- 4. Representations of Cosets -- References -- Dense lattices as Hermitian tensor products -- 1. Introduction -- 2. Tensor products over ℤ -- 3. Preliminaries on Hermitian lattices -- 4. Hermitian ℤ[(1+√-11)/2]-lattices. -- 5. Hermitian ℤ[(1+√-7)/2]-lattices. -- References -- Small zeros of homogeneous cubic congruences -- 1. Introduction -- Acknowledgement -- 2. Quaternary cubic forms having no small zeros -- 3. Preparations -- 4. An application of Bertini's Theorem -- 5. Proofs of Theorems 1.2 and 1.3 -- References -- Strictly Regular Diagonal Positive Definite Quaternary Integral Quadratic Forms -- 1. Introduction -- 2. Infinite families of diagonal regular quaternary lattices -- 3. Candidates for strict regularity -- References -- Heights and quadratic forms: Cassels' theorem and its generalizations -- 1. Introduction: Cassels' theorem -- 2. Notation and heights -- 3. Extensions over global fields -- 4. Multiple zeros and isotropic subspaces -- 5. Effective structure theorems -- 6. Effective results with additional conditions -- 7. Open problems -- Acknowledgment -- References -- On the positive integers satisfying the equation _= ²+ ² -- 1. Introduction. 2. Preliminary results -- 3. The proof of Theorem 1.1 -- 4. Comments -- Acknowledgements. -- References -- Algorithms for computing maximal lattices in bilinear (and quadratic) spaces over number fields -- 1. Introduction and Notation -- 2. Duality for Bilinear Lattices -- 3. Maximal Bilinear Lattices -- 4. Maximal Quadratic Lattices -- 5. Neighbors and Genera -- References -- -adic Zeros of Systems of Quadratic Forms -- References -- The Number of Function Fields with Given Genus -- Introduction -- Notation and Auxiliary Results -- Proof of the Theorem -- References -- Unique Factorization in the Theory of Quadratic Forms -- 1. Theorem Statement -- 2. Proof of Key Case -- 3. Proof of Remaining Cases -- References -- Golden lattices -- 1. Introduction -- 2. Hilbert theta series of golden lattices -- 3. Examples -- References -- The extremal lattice of dimension 14, level 7 and its genus -- 1. Background on extremal lattices and modular forms -- 2. The extremal 7-modular lattice -- 3. Properties of the genus \ ₁₄(7⁺⁷) -- 4. Weakening the condition of extremality -- Acknowledgment -- References -- Strict Periodic Extreme Lattices -- 1. Introduction -- 2. A parameter space for periodic sets -- 3. Characterizing strict periodic extreme sets -- References -- Exceptional units and cyclic resultants, II -- 1. Introduction -- 2. Preliminary lemmas -- 3. Proof of Theorem 1.1 -- 4. Proof of Theorem 1.2 -- 5. Proof of Theorem 1.3 -- 6. Computations for small degrees -- References -- A note on generators of number fields -- 1. Introduction -- 2. Preliminaries -- 3. Proof of Theorem 1.2 -- 4. A general strategy -- 5. Application of Chebotarev's density theorem and GRH -- 6. Proof of Theorem 1.3 -- 7. The field ℚ(√-163) -- References -- Voronoï's reduction theory of _ over a totally real number field -- Introduction -- 1. Preliminaries. 2. Voronoï's algorithm of Λ₀-perfect forms -- 3. Polyhedral reduction of _( _)/ (Λ₀)* -- 4. Ryshkov polyhedra of real quadratic fields -- References -- Some comments about Indefinite LLL -- 1. Review of LLL -- Acknowledgments -- 2. Quadratic equations -- 3. Back to indefinite LLL -- References.
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Linear algebraic groups -- Congresses. Forms, Quadratic -- Congresses. Number theory -- Congresses.