{"title":"从属二的模方程计算同基因","authors":"Jean Kieffer , Aurel Page , Damien Robert","doi":"10.1016/j.jalgebra.2024.11.029","DOIUrl":null,"url":null,"abstract":"<div><div>Consider two genus 2 curves over a field whose Jacobians are linked by an isogeny of known type: either an <em>ℓ</em>-isogeny or, in the real multiplication case, an isogeny with cyclic kernel. We present a completely algebraic algorithm to compute this isogeny using modular equations of either Siegel or Hilbert type. An essential step of independent interest is to construct an explicit Kodaira–Spencer isomorphism for principally polarized abelian surfaces.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 331-386"},"PeriodicalIF":0.8000,"publicationDate":"2025-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Computing isogenies from modular equations in genus two\",\"authors\":\"Jean Kieffer , Aurel Page , Damien Robert\",\"doi\":\"10.1016/j.jalgebra.2024.11.029\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Consider two genus 2 curves over a field whose Jacobians are linked by an isogeny of known type: either an <em>ℓ</em>-isogeny or, in the real multiplication case, an isogeny with cyclic kernel. We present a completely algebraic algorithm to compute this isogeny using modular equations of either Siegel or Hilbert type. An essential step of independent interest is to construct an explicit Kodaira–Spencer isomorphism for principally polarized abelian surfaces.</div></div>\",\"PeriodicalId\":14888,\"journal\":{\"name\":\"Journal of Algebra\",\"volume\":\"666 \",\"pages\":\"Pages 331-386\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2025-03-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Algebra\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021869324006537\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2024/12/4 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebra","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021869324006537","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2024/12/4 0:00:00","PubModel":"Epub","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Computing isogenies from modular equations in genus two
Consider two genus 2 curves over a field whose Jacobians are linked by an isogeny of known type: either an ℓ-isogeny or, in the real multiplication case, an isogeny with cyclic kernel. We present a completely algebraic algorithm to compute this isogeny using modular equations of either Siegel or Hilbert type. An essential step of independent interest is to construct an explicit Kodaira–Spencer isomorphism for principally polarized abelian surfaces.
期刊介绍：
The Journal of Algebra is a leading international journal and publishes papers that demonstrate high quality research results in algebra and related computational aspects. Only the very best and most interesting papers are to be considered for publication in the journal. With this in mind, it is important that the contribution offer a substantial result that will have a lasting effect upon the field. The journal also seeks work that presents innovative techniques that offer promising results for future research.
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