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bound for the rank of an elliptic curve
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(Theorem)
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Example 1
As an application of the theorem above, we can prove that $E_1\colon y^2=x^3-x$ has only finitely many rational solutions. Indeed, the discriminant of $E_1$ , $\Delta=64$ , is only divisible by $p=2$ , which is a prime of (bad) multiplicative reduction. Therefore $R_{E_1}=0$ . Moreover, the Nagell-Lutz theorem implies that the only torsion points on $E_1$ are those of order $2$ . Hence, the only rational points on $E_1$ are: $$\{ \mathcal{O}, (0,0),(1,0),(-1,0)\}.$$
- 1
- James Milne, Elliptic Curves, online course notes.
http://www.jmilne.org/math/CourseNotes/math679.html
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"bound for the rank of an elliptic curve" is owned by alozano.
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Cross-references: order, points, torsion, implies, Nagell-Lutz theorem, multiplicative reduction, divisible, discriminant, solutions, rational, theorem, application, rank, additive reduction, multiplicative, number, bad reduction, primes, equation, elliptic curve
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This is version 3 of bound for the rank of an elliptic curve, born on 2004-06-10, modified 2005-01-31.
Object id is 5908, canonical name is BoundForTheRankOfAnEllipticCurve.
Accessed 2211 times total.
Classification:
| AMS MSC: | 14H52 (Algebraic geometry :: Curves :: Elliptic curves) |
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Pending Errata and Addenda
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