Jacobian Nullwerte, periods and symmetric equations for hyperelliptic curves

Guàrdia, Jordi

doi:10.5802/aif.2293

Cited by 5 publications

(6 citation statements)

References 25 publications

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“…These symmetric roots were extensively studied by J. Guàrdia [4] in the context of an effective Torelli theorem for hyperelliptic period matrices. They are defined as follows.…”

Section: ⎠ Holds Where ( ) a Is Zhang's Admissible Pairing On Div(xmentioning

confidence: 99%

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Symmetric roots and admissible pairing

Jong

2011

Trans. Amer. Math. Soc.

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Abstract. Using the discriminant modular form and the Noether formula it is possible to write the admissible self-intersection of the relative dualising sheaf of a semistable hyperelliptic curve over a number field or function field as a sum, over all places, of a certain adelic invariant χ. We provide a simple geometric interpretation for this invariant χ, based on the arithmetic of symmetric roots. We propose the conjecture that the invariant χ coincides with the invariant ϕ introduced in a recent paper by S.-W. Zhang. This conjecture is true in the genus 2 case, and we obtain a new proof of the Bogomolov conjecture for curves of genus 2 over number fields.

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“…These symmetric roots were extensively studied by J. Guàrdia [4] in the context of an effective Torelli theorem for hyperelliptic period matrices. They are defined as follows.…”

Section: ⎠ Holds Where ( ) a Is Zhang's Admissible Pairing On Div(xmentioning

confidence: 99%

“…we are done once we prove that f (a j )/f (a i ) is a unit in R. Let a r be an arbitrary root of f ; we will show that ν(f (a r )) = ν(4), so that ν(f (a r )) is independent of r. (4). According to [7,Proposition 6.3] we have however:…”

Section: Holds Where ν(·) Denotes Order Of Vanishing Along the Closementioning

confidence: 99%

Symmetric roots and admissible pairing

Jong

2011

Trans. Amer. Math. Soc.

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“…In the particular case that the abelian variety is the Jacobian variety J(C) of a hyperelliptic curve C, we have proposed to make this choice in a way that makes present the geometry of the curve (cf. [5]): Theorem 8. Let C /K be a hyperelliptic curve defined over a number field K, and let Z be a period matrix for C, so that J(C) ≃ C g /(1 g | Z).…”

Section: Periods Of Algebraic Differential Forms On Hyperelliptic Curvesmentioning

confidence: 99%

“…We refer to [5] for the detailed presentation of symmetric models for hyperelliptic curves. We only mention here two of the main results about them:…”

Section: Symmetric Models For Hyperelliptic Curvesmentioning

confidence: 99%

“…The article contains an overview of previous results by the author on the application of theta functions to the determination of good equations and their periods for general hyperelliptic curves ( [4], [5]). We have included them to help the reader to have a complete panorama of the ideas that lead to the particular method for elliptic curves.…”

Section: Introductionmentioning

confidence: 99%

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Thetanullwerte: From periods to good equations

Guàrdia¹

2007

Publicacions Matemàtiques

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We will show the utility of the classical Jacobi Thetanullwerte for the description of certain period lattices of elliptic curves, providing equations with good arithmetical properties. These equations will be the starting point for the construction of families of elliptic curves with everywhere good reduction. IntroductionThe determination of an equation for an elliptic curve associated to a given complex torus is a common task in number theory. The common solution follows from the uniformization of the torus by means of the Weierstrass ℘, ℘ ′ functions. Unfortunately, such a direct approach is not always satisfactory, especially if one desires an equation with good arithmetical properties.A very basic example illustrates this phenomenon: let us consider the complex torus C/Z[i]. We may compute numerically the common Weierstrass coefficients and find While the second one seems to vanish clearly, the first one is difficult to recognize, since it turns to be the transcendental number4Γ(3/4) 4 . On the other hand, it is well-known that the equation Y 2 = X 3 − X gives an algebraic model of this torus, with good reduction outside 2.We will explain here a general method to find such nice equations for elliptic curves. The key ingredient will be the classical Jacobi theta functions. The models provided have very good arithmetical properties. For instance, they are not far from having good reduction everywhere.2000 Mathematics Subject Classification. 11G05, 14K25.

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Special values of canonical Green’s functions

Jong

2012

Math. Z.

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Jacobian Nullwerte, periods and symmetric equations for hyperelliptic curves

Cited by 5 publications

References 25 publications

Symmetric roots and admissible pairing

Symmetric roots and admissible pairing

Thetanullwerte: From periods to good equations

Special values of canonical Green’s functions

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