2012, ISBN 9780691151199, cm.

.... The key to the conjecture lies in elliptic curves, which may appear simple, but arise from some very deep--and often very mystifying--mathematical ideas...

Mathematics | Elliptic functions | History & Philosophy | Abstract | PDZ | PBX | Calculus | PBKA | Geometry | Algebra | PBF | Algebraic | PBH | PBMW | Number Theory | Curves, Elliptic | Number theory

Mathematics | Elliptic functions | History & Philosophy | Abstract | PDZ | PBX | Calculus | PBKA | Geometry | Algebra | PBF | Algebraic | PBH | PBMW | Number Theory | Curves, Elliptic | Number theory

Book

2006, ISBN 9781584885184, xxxiv, 808

Book

Journal of Pure and Applied Algebra, ISSN 0022-4049, 01/2020, Volume 224, Issue 1, pp. 272 - 299

We determine which of the modular curves XΔ(N), that is, curves lying between X0...

Bielliptic | Modular curve | Hyperelliptic | Infinitely many quadratic points | COMPACT RIEMANN SURFACES | MATHEMATICS | AUTOMORPHISM-GROUPS | ELLIPTIC-CURVES | MATHEMATICS, APPLIED | TORSION POINTS | GENUS

Bielliptic | Modular curve | Hyperelliptic | Infinitely many quadratic points | COMPACT RIEMANN SURFACES | MATHEMATICS | AUTOMORPHISM-GROUPS | ELLIPTIC-CURVES | MATHEMATICS, APPLIED | TORSION POINTS | GENUS

Journal Article

Journal of algebra, ISSN 0021-8693, 2019, Volume 519, pp. 474 - 490

We give an efficient algorithm to compute equations of twists of hyperelliptic curves C of arbitrary genus over any perfect field k...

Hyperelliptic curves | Twists | Conics | Galois cohomology | MATHEMATICS | Hyperclliptic curves | Algorithms | Mathematics - Number Theory | Mathematics | Number Theory

Hyperelliptic curves | Twists | Conics | Galois cohomology | MATHEMATICS | Hyperclliptic curves | Algorithms | Mathematics - Number Theory | Mathematics | Number Theory

Journal Article

Journal of Algebra, ISSN 0021-8693, 09/2019, Volume 533, pp. 44 - 89

In this paper, we will give an algebraic proof for determining the sections for the universal pointed hyperelliptic curves CHg,n/k→Hg,n/k, when g...

Hyperelliptic curves | Arithmetic geometry | Unipotent completions | Lie algebras | Moduli space of hyperelliptic curves | Hyperelliptic mapping class groups | Algebraic geometry | Fundamental groups | MATHEMATICS | FUNDAMENTAL GROUP | MAPPING CLASS GROUP | GALOIS ACTIONS | Algebra

Hyperelliptic curves | Arithmetic geometry | Unipotent completions | Lie algebras | Moduli space of hyperelliptic curves | Hyperelliptic mapping class groups | Algebraic geometry | Fundamental groups | MATHEMATICS | FUNDAMENTAL GROUP | MAPPING CLASS GROUP | GALOIS ACTIONS | Algebra

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 07/2019, Volume 88, Issue 318, pp. 1913 - 1927

We give an algorithm to compute the conductor for curves of genus 2. It is based on the analysis of 3-torsion of the Jacobian for genus 2 curves over 2-adic...

Hyperelliptic curves | Local fields | 3-torsion | Conductor | FIELDS | MATHEMATICS, APPLIED | hyperelliptic curves | local fields | JACOBIANS

Hyperelliptic curves | Local fields | 3-torsion | Conductor | FIELDS | MATHEMATICS, APPLIED | hyperelliptic curves | local fields | JACOBIANS

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 01/2019, Volume 88, Issue 315, pp. 421 - 438

Given a smooth curve defined over a field k that admits a non-singular plane model over \overline {k...

Automorphism groups | Twist | Non-singular plane curves | HYPERELLIPTIC CURVES | MATHEMATICS, APPLIED | automorphism groups | twist | Algebraic Geometry | Mathematics | Number Theory

Automorphism groups | Twist | Non-singular plane curves | HYPERELLIPTIC CURVES | MATHEMATICS, APPLIED | automorphism groups | twist | Algebraic Geometry | Mathematics | Number Theory

Journal Article

Journal de mathématiques pures et appliquées, ISSN 0021-7824, 03/2019, Volume 123, pp. 41 - 77

Oort has conjectured that there do not exist Shimura varieties of dimension >0 contained generically in the Torelli locus of genus-g curves when g is sufficiently large...

Shimura curves | Hyperelliptic curves | Torelli locus | Complex multiplication | MATHEMATICS | MATHEMATICS, APPLIED | ABELIAN-VARIETIES | FAMILIES | MODULI SPACE | SURFACES

Shimura curves | Hyperelliptic curves | Torelli locus | Complex multiplication | MATHEMATICS | MATHEMATICS, APPLIED | ABELIAN-VARIETIES | FAMILIES | MODULI SPACE | SURFACES

Journal Article

Mathematische Nachrichten, ISSN 0025-584X, 12/2017, Volume 290, Issue 17-18, pp. 2890 - 2900

The main result of this paper states that if C is a hyperelliptic curve of even genus over an arbitrary field K, then there is a natural bijection between the set of equivalence classes of elliptic...

hyperelliptic curves | rationality questions | Curve covers | 14H30 | 14H52 | 14H40 | elliptic curves | Jacobian varieties | 11G30 | Hyperelliptic curves | Rationality questions | Elliptic curves | MATHEMATICS | NUMBER | THEOREM | GENUS 2

hyperelliptic curves | rationality questions | Curve covers | 14H30 | 14H52 | 14H40 | elliptic curves | Jacobian varieties | 11G30 | Hyperelliptic curves | Rationality questions | Elliptic curves | MATHEMATICS | NUMBER | THEOREM | GENUS 2

Journal Article

Revista Matematica Iberoamericana, ISSN 0213-2230, 2017, Volume 33, Issue 1, pp. 169 - 182

In this paper we present a method for computing the set of twists of a non-singular projective curve defined over an arbitrary (perfect) field k...

Non-hyperelliptic curves | Galois embedding problems | Twists | MATHEMATICS | GENUS-2 | SATO-TATE DISTRIBUTIONS

Non-hyperelliptic curves | Galois embedding problems | Twists | MATHEMATICS | GENUS-2 | SATO-TATE DISTRIBUTIONS

Journal Article

Journal of Algebra, ISSN 0021-8693, 01/2019, Volume 517, pp. 457 - 512

We exploit three classical characterizations of smooth genus two curves to study their tropical and analytic counterparts...

Tropical geometry | Tropical modifications | Hyperelliptic covers | Faithful tropicalizations | Berkovich spaces | Igusa invariants | FANS | MATHEMATICS | SKELETONS | MODULI SPACES | PLANE-CURVES | Algorithms

Tropical geometry | Tropical modifications | Hyperelliptic covers | Faithful tropicalizations | Berkovich spaces | Igusa invariants | FANS | MATHEMATICS | SKELETONS | MODULI SPACES | PLANE-CURVES | Algorithms

Journal Article

Journal of algebraic combinatorics, ISSN 1572-9192, 2012, Volume 37, Issue 2, pp. 331 - 359

We study the locus of tropical hyperelliptic curves inside the moduli space of tropical curves of genus g...

Hyperelliptic curves | Metric graphs | Convex and Discrete Geometry | Tropical geometry | Tropical curves | Mathematics | Order, Lattices, Ordered Algebraic Structures | Group Theory and Generalizations | Harmonic morphisms | Combinatorics | Computer Science, general | Metric graphs | LINEAR-SYSTEMS | MATHEMATICS | RIEMANN-ROCH | GEOMETRY

Hyperelliptic curves | Metric graphs | Convex and Discrete Geometry | Tropical geometry | Tropical curves | Mathematics | Order, Lattices, Ordered Algebraic Structures | Group Theory and Generalizations | Harmonic morphisms | Combinatorics | Computer Science, general | Metric graphs | LINEAR-SYSTEMS | MATHEMATICS | RIEMANN-ROCH | GEOMETRY

Journal Article

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), ISSN 0302-9743, 2014, Volume 8873, pp. 338 - 357

Journal Article

The Quarterly Journal of Mathematics, ISSN 0033-5606, 06/2018, Volume 69, Issue 2, pp. 747 - 768

Abstract We study hyperelliptic curves C with an action of an affine group of automorphisms G...

MATHEMATICS

MATHEMATICS

Journal Article

IEEE transactions on computers, ISSN 0018-9340, 2006, Volume 55, Issue 10, pp. 1306 - 1311

Digit serial multipliers are used extensively in hardware implementations of elliptic and hyperelliptic curve cryptography...

elliptic/hyperelliptic curve cryptography | digit serial multiplier | Bit serial multiplier | public key cryptography | least significant digit multiplier | Public key cryptography | Elliptic/hyperelliptic curve cryptography | Least significant digit multiplier | Digit serial multiplier | elliptic/hyperel liptic curve cryptography | COMPUTER SCIENCE, HARDWARE & ARCHITECTURE | bit serial multiplier | ENGINEERING, ELECTRICAL & ELECTRONIC | Circuit design | Analysis | Methods | Public key encryption | Multipliers | Digits | Serials | Hardware | Cryptography | LSD | Time measurements | Optimization

elliptic/hyperelliptic curve cryptography | digit serial multiplier | Bit serial multiplier | public key cryptography | least significant digit multiplier | Public key cryptography | Elliptic/hyperelliptic curve cryptography | Least significant digit multiplier | Digit serial multiplier | elliptic/hyperel liptic curve cryptography | COMPUTER SCIENCE, HARDWARE & ARCHITECTURE | bit serial multiplier | ENGINEERING, ELECTRICAL & ELECTRONIC | Circuit design | Analysis | Methods | Public key encryption | Multipliers | Digits | Serials | Hardware | Cryptography | LSD | Time measurements | Optimization

Journal Article

Compositio mathematica, ISSN 0010-437X, 12/2017, Volume 153, Issue 12, pp. 2534 - 2576

We show that a genus $2$ curve over a number field whose jacobian has complex multiplication will usually have stable bad reduction at some prime...

complex multiplication | curves | hyperelliptic jacobians | MATHEMATICS | EQUIDISTRIBUTION | ABELIAN-VARIETIES | DISCRIMINANT | CONDUCTOR | FORMULA | POINTS | HEIGHT | MODULI | Mathematics - Number Theory

complex multiplication | curves | hyperelliptic jacobians | MATHEMATICS | EQUIDISTRIBUTION | ABELIAN-VARIETIES | DISCRIMINANT | CONDUCTOR | FORMULA | POINTS | HEIGHT | MODULI | Mathematics - Number Theory

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 01/2014, Volume 83, Issue 285, pp. 365 - 409

Let p be a prime and C a genus one curve over a number field k representing an element of order dividing p in the Shafarevich-Tate group of its Jacobian...

Homomorphisms | Algebra | Mathematical theorems | Maps | Hyperplanes | Rational functions | Coordinate systems | Jacobians | Curves | Arithmetic | EXPLICIT N-DESCENT | HYPERELLIPTIC CURVES | MATHEMATICS, APPLIED | BIRCH | JACOBIANS | SELMER GROUP | CONJECTURES | ALGEBRA | SWINNERTON-DYER | PERIOD | INDEX | Mathematics - Number Theory

Homomorphisms | Algebra | Mathematical theorems | Maps | Hyperplanes | Rational functions | Coordinate systems | Jacobians | Curves | Arithmetic | EXPLICIT N-DESCENT | HYPERELLIPTIC CURVES | MATHEMATICS, APPLIED | BIRCH | JACOBIANS | SELMER GROUP | CONJECTURES | ALGEBRA | SWINNERTON-DYER | PERIOD | INDEX | Mathematics - Number Theory

Journal Article

PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES, ISSN 0386-2194, 07/2019, Volume 95, Issue 7, pp. 65 - 69

In this paper, we find all Weierstrass points on the hyperelliptic modular curves X-0(N...

MATHEMATICS | modular curve | Weierstrass points | CUSPS | hyperelliptic curve

MATHEMATICS | modular curve | Weierstrass points | CUSPS | hyperelliptic curve

Journal Article

Journal of Cryptology, ISSN 0933-2790, 7/2011, Volume 24, Issue 3, pp. 446 - 469

Efficiently computable homomorphisms allow elliptic curve point multiplication to be accelerated using the Gallant–Lambert–Vanstone (GLV) method...

GLV method | Computational Mathematics and Numerical Analysis | Multiexponentiation | Elliptic curves | Computer Science | Isogenies | Coding and Information Theory | Probability Theory and Stochastic Processes | Communications Engineering, Networks | Combinatorics | Point multiplication | HYPERELLIPTIC CURVES | PRECOMPUTATION | MATHEMATICS, APPLIED | PRIME FIELDS | POLLARD | ENGINEERING, ELECTRICAL & ELECTRONIC | SCALAR MULTIPLICATION | EXPONENTIATION | CRYPTOSYSTEMS | COMPUTER SCIENCE, THEORY & METHODS | Cryptography | Universities and colleges

GLV method | Computational Mathematics and Numerical Analysis | Multiexponentiation | Elliptic curves | Computer Science | Isogenies | Coding and Information Theory | Probability Theory and Stochastic Processes | Communications Engineering, Networks | Combinatorics | Point multiplication | HYPERELLIPTIC CURVES | PRECOMPUTATION | MATHEMATICS, APPLIED | PRIME FIELDS | POLLARD | ENGINEERING, ELECTRICAL & ELECTRONIC | SCALAR MULTIPLICATION | EXPONENTIATION | CRYPTOSYSTEMS | COMPUTER SCIENCE, THEORY & METHODS | Cryptography | Universities and colleges

Journal Article

Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8121, 10/2010, Volume 43, Issue 43, p. 434009

.... By giving a homology basis well adapted to the symmetries of Klein's curve, presented as a plane curve, we derive a new expression for its period matrix...

HYPERELLIPTIC CURVES | PERIOD MATRICES | PHYSICS, MULTIDISCIPLINARY | PHYSICS, MATHEMATICAL | SURFACES

HYPERELLIPTIC CURVES | PERIOD MATRICES | PHYSICS, MULTIDISCIPLINARY | PHYSICS, MATHEMATICAL | SURFACES

Journal Article

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