Riemann surfaces and abelian varieties with an automorphism of prime order
type: Item
instance of: scholarly article
title: Riemann surfaces and abelian varieties with an automorphism of prime order
zbMATH Open document ID: 0790.14039
DOI: 10.1215/S0012-7094-93-06910-4
author: Gonzalo Riera
author: Rubí E. Rodríguez
published in: Duke Mathematical Journal
publication date: 1994-06-20T00:00:00Z
review text: The authors compute the number of isomorphism classes of principally polarized abelian varieties of dimension \(g=(p-1)/2\) admitting a complex multiplication of prime order \(p\). The number is equal to \(h(p-1)^{- 1}\sum_{d\in S(p)}\varphi(d)2^{(p-1)/2d}\), where \(S(p)\) is the set of odd positive integers \(d\) which divide \(p-1\), \(\varphi\) denotes the Euler function and \(h\) denotes the ideal class number. Some of these varieties (but not all) can be realized as the Jacobian varieties of compact Riemann surfaces admitting a cyclic involution of order \(p\) with the quotient of genus zero. Each such a Riemann surface can be defined by the equation \(y^ p=x^ a(x-1)\) for some \(1\leq a<p\). The number of their isomorphism classes was computed by S. Lefschetz. It is equal to \((p+1)/6\) if \(p\equiv 2\pmod 3\) and to \((p+5)/6\) otherwise. The authors present another proof of this classical result.
Mathematics Subject Classification ID: 14K22
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 179236
zbMATH Keywords: number of isomorphism classes of principally polarized abelian varieties
zbMATH Keywords: complex multiplication
zbMATH Keywords: ideal class number
zbMATH Keywords: Jacobian varieties of compact Riemann surfaces
reviewed by: Igor V. Dolgachev
MaRDI profile type: Publication
cites work: Q5632528
cites work: Riemann surface transformation groups
cites work: Q3268276
cites work: Sur les Périodes des Intégrales Abéliennes
cites work: On generalized Riemann matrices
full work available at URL: s0012-7094-93-06910-4
OpenAlex ID: W1533755086
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description: scientific article; zbMATH DE number 179236