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Hazama, Fumio

doi:10.1023/a:1000106427229

Compositio Mathematica

1997

DOI: 10.1023/a:1000106427229

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Fumio Hazama

Abstract: Abstract. The jacobian variety of the Catalan curve y q = x p 1 is shown to be nondegenerate.As an application, a 0-1-matrix whose determinant computes the relative class number of the pq-th cyclotomic field is constructed.Mathematics Subject Classifications (1991): 14C30, 11R29.

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Cited by 9 publications

References 9 publications

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Formulae for the relative class number of an imaginary abelian field in the form of a determinant

Kučera

2001

Nagoya Mathematical Journal

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Abstract. There is in the literature a lot of determinant formulae involving the relative class number of an imaginary abelian field. Usually such a formula contains a factor which is equal to zero for many fields and so it gives no information about the class number of these fields. The aim of this paper is to show a way of obtaining most of these formulae in a unique fashion, namely by means of the Stickelberger ideal. Moreover some new and non-vanishing formulae are derived by a modification of Ramachandra's construction of independent cyclotomic units. §1. Introduction

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Formulae for the relative class number of an imaginary abelian field in the form of a determinant

Kučera

2001

Nagoya Mathematical Journal

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Endomorphism algebras of factors of certain hypergeometric Jacobians

Xue

2015

Trans. Amer. Math. Soc.

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We classify the endomorphism algebras of factors of the Jacobian of certain hypergeometric curves over a field of characteristic zero. Other than a few exceptional cases, the endomorphism algebras turn out to be either a cyclotomic field E = Q(ζq), or a quadratic extension of E, or E⊕E. This result may be viewed as a generalization of the well known results of the classification of endomorphism algebras of elliptic curves over C.

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3-Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number field

Bhargava¹,

Klagsbrun²,

Oliver³

et al. 2019

Duke Math. J.

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For an abelian variety A over a number field F , we prove that the average rank of the quadratic twists of A is bounded, under the assumption that the multiplication-by-3 isogeny on A factors as a composition of 3-isogenies over F . This is the first such boundedness result for an absolutely simple abelian variety A of dimension greater than one. In fact, we exhibit such twist families in arbitrarily large dimension and over any number field.In dimension one, we deduce that if E/F is an elliptic curve admitting a 3-isogeny, then the average rank of its quadratic twists is bounded. If F is totally real, we moreover show that a positive proportion of twists have rank 0 and a positive proportion have 3-Selmer rank 1. These results on bounded average ranks in families of quadratic twists represent new progress towards Goldfeld's conjecture -which states that the average rank in the quadratic twist family of an elliptic curve over Q should be 1/2 -and the first progress towards the analogous conjecture over number fields other than Q.Our results follow from a computation of the average size of the φ-Selmer group in the family of quadratic twists of an abelian variety admitting a 3-isogeny φ.

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Untitled

Abstract: Abstract. The jacobian variety of the Catalan curve y q = x p 1 is shown to be nondegenerate.As an application, a 0-1-matrix whose determinant computes the relative class number of the pq-th cyclotomic field is constructed.Mathematics Subject Classifications (1991): 14C30, 11R29.

Cited by 9 publications

References 9 publications

Formulae for the relative class number of an imaginary abelian field in the form of a determinant

Formulae for the relative class number of an imaginary abelian field in the form of a determinant

Endomorphism algebras of factors of certain hypergeometric Jacobians

3-Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number field

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