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| = John Cremona (Warwick University): Tables of elliptic curves = | = John Cremona (University of Warwick): Tables of elliptic curves = |
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| 1. Tabulating modular elliptic curves over Q (based on the first chapter of my book) | 1. Tabulating modular elliptic curves over Q (based on Chapter II of my book Algorithms for Modular Elliptic Curves, which is available free online here:http://www.warwick.ac.uk/staff/J.E.Cremona/book/fulltext/index.html). The project would be to extend the existing tables from the current conductor limit of 130,000 to something larger, using my C++ code (distributed with Sage) and Sage itself, and some combination of the two. |
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| 2. Finding all elliptic curves with good reduction outside a finite set S of prime, over number fields. I could lecture on the theory, based on my paper on the subject which reduces it to finding all S-integral points on e.c.s over number fields, and then the project could be to implement it in Sage. Over Q, I already did that, using the S-integral points functions implemented two summers ago. But over number fields there will be other challenges, and this is likely to spill over into more general computation of Mordell-Weil groups. | 2. Finding all elliptic curves with good reduction outside a finite set S of primes, over number fields. The theory, is based on my paper on the subject (see Cremona and Lingham, Experimental Mathematics 16 No.3 (2007), 303-312, also available here: http://www.warwick.ac.uk/staff/J.E.Cremona/papers/egros.pdf ) which reduces the problem to finding all S-integral points on e.c.s over number fields, together with some algebraic number theory. The project would be to implement this in Sage, over number fields. Over Q, I have already done this, using the S-integral points functions implemented two summers ago. I also have a partial implementation over number fields in Magma. Over number fields there will be other challenges, and this is likely to spill over into more general computation of Mordell-Weil groups of elliptic curves over number fields. |
John Cremona (University of Warwick): Tables of elliptic curves
Description
Tabulating modular elliptic curves over Q (based on Chapter II of my book Algorithms for Modular Elliptic Curves, which is available free online here:http://www.warwick.ac.uk/staff/J.E.Cremona/book/fulltext/index.html). The project would be to extend the existing tables from the current conductor limit of 130,000 to something larger, using my C++ code (distributed with Sage) and Sage itself, and some combination of the two.
Finding all elliptic curves with good reduction outside a finite set S of primes, over number fields. The theory, is based on my paper on the subject (see Cremona and Lingham, Experimental Mathematics 16 No.3 (2007), 303-312, also available here: http://www.warwick.ac.uk/staff/J.E.Cremona/papers/egros.pdf ) which reduces the problem to finding all S-integral points on e.c.s over number fields, together with some algebraic number theory. The project would be to implement this in Sage, over number fields. Over Q, I have already done this, using the S-integral points functions implemented two summers ago. I also have a partial implementation over number fields in Magma. Over number fields there will be other challenges, and this is likely to spill over into more general computation of Mordell-Weil groups of elliptic curves over number fields.