Shadow line distributions.
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| Title: | Shadow line distributions. |
|---|---|
| Authors: | Balakrishnan, Jennifer S.1 (AUTHOR), Çiperiani, Mirela2 (AUTHOR), Mazur, Barry3 (AUTHOR), Rubin, Karl4 (AUTHOR) |
| Source: | Mathematics of Computation. Sep2026, Vol. 95 Issue 361, p2539-2557. 19p. |
| Subjects: | Elliptic curves, Quadratic fields, Hypothesis, Abelian groups |
| Abstract: | Let E be an elliptic curve over \mathbb {Q} with Mordell–Weil rank 2 and p be an odd prime of good ordinary reduction. For every imaginary quadratic field K satisfying the Heegner hypothesis, there is (subject to the Shafarevich–Tate conjecture) a line, i.e., a free \mathbb {Z}_p-submodule of rank 1, in E(K)\otimes \mathbb {Z}_p given by universal norms coming from the Mordell–Weil groups of subfields of the anticyclotomic \mathbb {Z}_p-extension of K; we call it the shadow line. When the twist of E by K has analytic rank 1, the shadow line is conjectured to lie in E(\mathbb {Q})\otimes \mathbb {Z}_p; we verify this computationally in all our examples. We study the distribution of shadow lines in E(\mathbb {Q})\otimes \mathbb {Z}_p as K varies, framing conjectures based on the computations we have made. [ABSTRACT FROM AUTHOR] |
| Copyright of Mathematics of Computation is the property of American Mathematical Society and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 194543851 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Shadow line distributions. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Balakrishnan%2C+Jennifer+S%2E%22">Balakrishnan, Jennifer S.</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Çiperiani%2C+Mirela%22">Çiperiani, Mirela</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Mazur%2C+Barry%22">Mazur, Barry</searchLink><relatesTo>3</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Rubin%2C+Karl%22">Rubin, Karl</searchLink><relatesTo>4</relatesTo> (AUTHOR) – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Mathematics+of+Computation%22">Mathematics of Computation</searchLink>. Sep2026, Vol. 95 Issue 361, p2539-2557. 19p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Elliptic+curves%22">Elliptic curves</searchLink><br /><searchLink fieldCode="DE" term="%22Quadratic+fields%22">Quadratic fields</searchLink><br /><searchLink fieldCode="DE" term="%22Hypothesis%22">Hypothesis</searchLink><br /><searchLink fieldCode="DE" term="%22Abelian+groups%22">Abelian groups</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Let E be an elliptic curve over \mathbb {Q} with Mordell–Weil rank 2 and p be an odd prime of good ordinary reduction. For every imaginary quadratic field K satisfying the Heegner hypothesis, there is (subject to the Shafarevich–Tate conjecture) a line, i.e., a free \mathbb {Z}_p-submodule of rank 1, in E(K)\otimes \mathbb {Z}_p given by universal norms coming from the Mordell–Weil groups of subfields of the anticyclotomic \mathbb {Z}_p-extension of K; we call it the shadow line. When the twist of E by K has analytic rank 1, the shadow line is conjectured to lie in E(\mathbb {Q})\otimes \mathbb {Z}_p; we verify this computationally in all our examples. We study the distribution of shadow lines in E(\mathbb {Q})\otimes \mathbb {Z}_p as K varies, framing conjectures based on the computations we have made. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Mathematics of Computation is the property of American Mathematical Society and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1090/mcom/4110 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 19 StartPage: 2539 Subjects: – SubjectFull: Elliptic curves Type: general – SubjectFull: Quadratic fields Type: general – SubjectFull: Hypothesis Type: general – SubjectFull: Abelian groups Type: general Titles: – TitleFull: Shadow line distributions. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Balakrishnan, Jennifer S. – PersonEntity: Name: NameFull: Çiperiani, Mirela – PersonEntity: Name: NameFull: Mazur, Barry – PersonEntity: Name: NameFull: Rubin, Karl IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 09 Text: Sep2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 00255718 Numbering: – Type: volume Value: 95 – Type: issue Value: 361 Titles: – TitleFull: Mathematics of Computation Type: main |
| ResultId | 1 |
