The Arithmetic of Elliptic Curves

Silverman, Joseph H.

doi:10.1007/978-1-4757-1920-8

Cited by 2,352 publications

(914 citation statements)

References 121 publications

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“…This can be done by mimicking the calculations in Speyer's original argument, where the lifts of the points z i lie in the fibers of the reduction map E (R) → E(k). By [30,Theorem 6.4], since we work in residue characteristic 0, the fibers of this map are isomorphic to the additive group R × . The calculations are now identical to those in [31,Lemma 8.3] and the result follows.…”

Section: Remark 33 Davidmentioning

confidence: 99%

Skeletons of stable maps II: superabundant geometries

Ranganathan

2017

Res Math Sci

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We implement new techniques involving Artin fans to study the realizability of tropical stable maps in superabundant combinatorial types. Our approach is to understand the skeleton of a fundamental object in logarithmic Gromov-Witten theory-the stack of prestable maps to the Artin fan. This is used to examine the structure of the locus of realizable tropical curves and derive three principal consequences. First, we prove a realizability theorem for limits of families of tropical stable maps. Second, we extend the sufficiency of Speyer's well-spacedness condition to the case of curves with good reduction. Finally, we demonstrate the existence of liftable genus 1 superabundant tropical curves that violate the well-spacedness condition. BackgroundCentral to the application of tropical techniques to questions in algebraic geometry are so-called lifting theorems. Given a "synthetic" tropical object, such as a weighted balanced polyhedral complex, one must understand whether this object is the tropicalization of an algebraic variety. We deal in this paper with the case of curves. The tropical lifting question in this setting asks, when does an embedded tropical curve in R n arise as the tropicalization of an algebraic curve in a torus over a nonarchimedean field? This question becomes highly nontrivial in the so-called superabundant case and has been the primary obstacle to the application of tropical curve counting techniques in high genus settings. A tropical curve in R n encodes the combinatorial data in a degenerate logarithmic stable map to a toric variety. If the tropical curve is superabundant, i.e., if the tropical deformation space is larger than expected, the obstruction group of this degenerate logarithmic map is nonzero. As a result, such a map may not deform and the tropical curve may fail to be realizable. See Sect. 2.2 for a precise definition of superabundance. The earliest realization theorems for superabundant curves are due to Speyer, who observed a subtle combinatorial condition guaranteeing the realizability of superabundant genus 1 tropical curves [31]. While there has been substantial additional work in the intervening years, the general question remains mysterious [7,15,19,[23][24][25]29,34].In this note, we use recent technical breakthroughs in nonarchimedean geometry and the theory of logarithmic maps to provide a conceptual framework in which the realizability question may be approached. To demonstrate the efficacy of this framework, we use it to give simple proofs of three new results for lifting tropical curves. The same frame- 0123456789().,-: vol

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Section: Remark 33 Davidmentioning

confidence: 99%

Skeletons of stable maps II: superabundant geometries

Ranganathan

2017

Res Math Sci

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“…For definitions and background information about elliptic curves and elliptic surfaces, see Silverman's books [27,28].…”

Section: Resonant Triadsmentioning

confidence: 99%

“…For more information on division polynomials (including a recursive definition), see [26,16], as well as Exercise 3.7 in [27].…”

Section: Rational Parametrizationmentioning

confidence: 99%

The Arithmetic Geometry of Resonant Rossby Wave Triads

Kopp¹

2017

SIAM J. Appl. Algebra Geometry

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Abstract. Linear wave solutions to the Charney-Hasegawa-Mima equation with periodic boundary conditions have two physical interpretations: Rossby (atmospheric) waves, and drift (plasma) waves in a tokamak. These waves display resonance in triads. In the case of infinite Rossby deformation radius, the set of resonant triads may be described as the set of integer solutions to a particular homogeneous Diophantine equation, or as the set of rational points on a projective surface X. The set of all resonant triads was found by Bustamante and Hayat (2013) via mapping to quadratic forms. Our work independently finds all resonant triads via a rational parametrization of X. We provide a fiberwise description of X as a rational singular elliptic surface, yielding many new results about the set of wavevectors belonging to resonant triads. In particular, we show there is an infinite number of resonant triads (with relatively prime wavevectors) containing a wavevector (a, b) with a/b = r, where r is any given rational, and we provide a method to find these triads. This is applied to find all resonant Rossby wave packets that are stationary in the east-west direction.

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“…We show in this paper that there is a planar description of the neutral component of Pic 0 C . This planar description is a direct generalization of the classical planar description of the group law on the neutral component of the set of real points of a real elliptic curve [10]. Such a generalization has already been constructed [7], but used an embedding of C into P 2g .…”

Section: Introductionmentioning

confidence: 99%

A geometric description of the neutral component of the Jacobian of a real plane curve having many pseudo–lines

Fichou

Huisman²

2003

Mathematische Nachrichten

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A pseudo-line of a real plane curve is a real branch that is not homologically trivial in P 2 (R). A real plane curve C of degree d is said to have many pseudo-lines if it has exactly d − 2 pseudo-lines and if the genus of its normalization C is equal to d − 2. Let C be such a curve. We give a planar description of the neutral component of the set of real points of the Jacobian of the normalization C of C.

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The Arithmetic of Elliptic Curves

Cited by 2,352 publications

References 121 publications

Skeletons of stable maps II: superabundant geometries

Skeletons of stable maps II: superabundant geometries

The Arithmetic Geometry of Resonant Rossby Wave Triads

A geometric description of the neutral component of the Jacobian of a real plane curve having many pseudo–lines

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