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Heights on squares of modular curves

hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorPARENT, Pierre
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorAUTISSIER, Pascal
dc.date2018-12
dc.date.accessioned2024-04-04T03:02:23Z
dc.date.available2024-04-04T03:02:23Z
dc.date.issued2018-12
dc.identifier.issn1937-0652
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/192977
dc.description.abstractEnWe develop a strategy for bounding from above the height of rational points of modular curves with values in number fields, by functions which are polynomial in the curve's level. Our main technical tools come from effective Arakelov descriptions of modular curves and jacobians. We then fulfill this program in the following particular case: If $p$ is a not-too-small prime number, let $X_0 (p )$ be the classical modular curve of level $p$ over $\bf Q$. Assume Brumer's conjecture on the dimension of winding quotients of $J_0 (p)$. We prove that there is a function $b(p)=O(p^{5} \log p )$ (depending only on $p$) such that, for any quadratic number field $K$, the $j$-height of points in $X_0 (p ) (K)$ which are not lifts of elements of $X_0^+ (p) ({\bf Q})$, is less or equal to $b(p)$.
dc.language.isoen
dc.publisherMathematical Sciences Publishers
dc.subject.enArakelov geometry and Diophantine applications
dc.subject.enModular curves
dc.title.enHeights on squares of modular curves
dc.typeArticle de revue
dc.identifier.doi10.2140/ant.2018.12.2065
dc.subject.halMathématiques [math]/Théorie des nombres [math.NT]
dc.subject.halMathématiques [math]/Géométrie algébrique [math.AG]
dc.identifier.arxiv1606.09553v3
bordeaux.journalAlgebra & Number Theory
bordeaux.page2065--2122
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue12:9
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-01926076
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-01926076v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Algebra%20&%20Number%20Theory&rft.date=2018-12&rft.issue=12:9&rft.spage=2065--2122&rft.epage=2065--2122&rft.eissn=1937-0652&rft.issn=1937-0652&rft.au=PARENT,%20Pierre&AUTISSIER,%20Pascal&rft.genre=article


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