Coverings of Curves of Genus 2

Flynn, E. V.

doi:10.1007/10722028_3

Cited by 6 publications

(2 citation statements)

References 17 publications

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“…The problem whether the Jacobian variety of a hyperelliptic curve of genus 2 is isogenous to the direct product of elliptic curves is considered in many papers of number theory (e.g., [12,13,14,21,27,28,39,43,46,47,53]). The problems discussed in our article are related to the well-known open problem about rational points on the Jacobian variety of a curve of genus 2 [31,43].…”

Section: Introductionmentioning

confidence: 99%

Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions

Ayano¹,

Buchstaber²

2022

SIGMA

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The article is devoted to the classical problems about the relationships between elliptic functions and hyperelliptic functions of genus 2. It contains new results, as well as a derivation from them of well-known results on these issues. Our research was motivated by applications to the theory of equations and dynamical systems integrable in hyperelliptic functions of genus 2. We consider a hyperelliptic curve V of genus 2 which admits a morphism of degree 2 to an elliptic curve. Then there exist two elliptic curves E i , i = 1, 2, and morphisms of degree 2 from V to E i . We construct hyperelliptic functions associated with V from the Weierstrass elliptic functions associated with E i and describe them in terms of the fundamental hyperelliptic functions defined by the logarithmic derivatives of the two-dimensional sigma functions. We show that the restrictions of hyperelliptic functions associated with V to the appropriate subspaces in C 2 are elliptic functions and describe them in terms of the Weierstrass elliptic functions associated with E i . Further, we express the hyperelliptic functions associated with V on C 2 in terms of the Weierstrass elliptic functions associated with E i . We derive these results by describing the homomorphisms between the Jacobian varieties of the curves V and E i induced by the morphisms from V to E i explicitly.

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Section: Introductionmentioning

confidence: 99%

Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions

Ayano¹,

Buchstaber²

2022

SIGMA

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show abstract

“…These techniques take their name from one of the two methods available to compute these sets explicitly in the number field case: the method of Chabauty [7] (made 'effective' by Coleman [8]) applies to genus-g curves whose Jacobian has rank less than g over K; the other method, due to Dem'janenko [12] and generalised by Manin [26], applies to curves having m independent morphisms to an elliptic curve of rank less than m over K. This last method also works in the function field case, provided that the sets of points of bounded height are finite. While Chabauty's method has been successfully applied, and has given rise to new techniques of investigation (see, for instance, [4,5,8,15,16, 17] among others), there are only a few examples of computations using the Dem'janenko-Manin method [6,23,33]. One reason could be that the conditions for the application of this method are rather restrictive geometrically, and some particular curves of interest thus fail to satisfy the criterion.…”

Section: Introductionmentioning

confidence: 99%