{"title":"A semi-numerical algorithm for the homology lattice and periods of complex elliptic surfaces over P1","authors":"Eric Pichon-Pharabod","doi":"10.1016/j.jsc.2024.102357","DOIUrl":null,"url":null,"abstract":"<div><p>We provide an algorithm for computing a basis of homology of elliptic surfaces over <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> that is sufficiently explicit for integration of periods to be carried out. This allows the heuristic recovery of several algebraic invariants of the surface, notably the Néron–Severi lattice, the transcendental lattice, the Mordell–Weil group and the Mordell–Weil lattice. This algorithm comes with a SageMath implementation.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Symbolic Computation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0747717124000610","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
We provide an algorithm for computing a basis of homology of elliptic surfaces over that is sufficiently explicit for integration of periods to be carried out. This allows the heuristic recovery of several algebraic invariants of the surface, notably the Néron–Severi lattice, the transcendental lattice, the Mordell–Weil group and the Mordell–Weil lattice. This algorithm comes with a SageMath implementation.
期刊介绍：
An international journal, the Journal of Symbolic Computation, founded by Bruno Buchberger in 1985, is directed to mathematicians and computer scientists who have a particular interest in symbolic computation. The journal provides a forum for research in the algorithmic treatment of all types of symbolic objects: objects in formal languages (terms, formulas, programs); algebraic objects (elements in basic number domains, polynomials, residue classes, etc.); and geometrical objects.
It is the explicit goal of the journal to promote the integration of symbolic computation by establishing one common avenue of communication for researchers working in the different subareas. It is also important that the algorithmic achievements of these areas should be made available to the human problem-solver in integrated software systems for symbolic computation. To help this integration, the journal publishes invited tutorial surveys as well as Applications Letters and System Descriptions.
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