An interesting family of curves of genus 1
We study the family of elliptic curves y2=x3−t2x+1, both over ℚ(t) and over ℚ. In the former case, all integral solutions are determined; in the latter case, computation in the range 1≤t≤999 shows large ranks are common, giving a particularly simple example of curves which (admittedly over a small r...
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| Format: | Article |
| Language: | English |
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Wiley
2000-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171200002210 |
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| _version_ | 1832560329518743552 |
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| author | Andrew Bremner |
| author_facet | Andrew Bremner |
| author_sort | Andrew Bremner |
| collection | DOAJ |
| description | We study the family of elliptic curves y2=x3−t2x+1, both over
ℚ(t) and over ℚ. In the former case, all
integral solutions are determined; in the latter case, computation
in the range 1≤t≤999 shows large ranks are common, giving
a particularly simple example of curves which (admittedly over a
small range) apparently contradict the once held belief that the
rank under specialization will tend to have minimal rank consistent
with the parity predicted by the Selmer conjecture. |
| format | Article |
| id | doaj-art-9827d960c46747cdb62c2e48892a7464 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2000-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-9827d960c46747cdb62c2e48892a74642025-02-03T01:27:48ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252000-01-0123643143410.1155/S0161171200002210An interesting family of curves of genus 1Andrew Bremner0Department of Mathematics, Arizona State University, Tempe 85287-1804, AZ, USAWe study the family of elliptic curves y2=x3−t2x+1, both over ℚ(t) and over ℚ. In the former case, all integral solutions are determined; in the latter case, computation in the range 1≤t≤999 shows large ranks are common, giving a particularly simple example of curves which (admittedly over a small range) apparently contradict the once held belief that the rank under specialization will tend to have minimal rank consistent with the parity predicted by the Selmer conjecture.http://dx.doi.org/10.1155/S0161171200002210Elliptic curveintegral solutionrank.. |
| spellingShingle | Andrew Bremner An interesting family of curves of genus 1 International Journal of Mathematics and Mathematical Sciences Elliptic curve integral solution rank.. |
| title | An interesting family of curves of genus 1 |
| title_full | An interesting family of curves of genus 1 |
| title_fullStr | An interesting family of curves of genus 1 |
| title_full_unstemmed | An interesting family of curves of genus 1 |
| title_short | An interesting family of curves of genus 1 |
| title_sort | interesting family of curves of genus 1 |
| topic | Elliptic curve integral solution rank.. |
| url | http://dx.doi.org/10.1155/S0161171200002210 |
| work_keys_str_mv | AT andrewbremner aninterestingfamilyofcurvesofgenus1 AT andrewbremner interestingfamilyofcurvesofgenus1 |