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[parent] bound for the rank of an elliptic curve (Theorem)
Theorem 1   Let $ E/\mathbb{Q}$ be an elliptic curve given by the equation:
$\displaystyle E\colon y^2=x(x-t)(x-s),$    with $\displaystyle t,s\in \mathbb{Z}$
and suppose that $ E$ has $ s=m+a$ primes of bad reduction, with $ m$ and $ a$ being the number of primes with multiplicative and additive reduction respectively. Then the rank of $ E$, denoted by $ R_E$, satisfies:
$\displaystyle R_E\leq m+2a-1$
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:

$\displaystyle \{ \mathcal{O}, (0,0),(1,0),(-1,0)\}.$

Bibliography

1
James Milne, Elliptic Curves, online course notes.
http://www.jmilne.org/math/CourseNotes/math679.html



"bound for the rank of an elliptic curve" is owned by alozano.
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See Also: the arithmetic of elliptic curves


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Cross-references: order, points, torsion, implies, Nagell-Lutz theorem, multiplicative reduction, divisible, discriminant, solutions, rational, rank, additive reduction, multiplicative, 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 1577 times total.

Classification:
AMS MSC: 14H52 (Algebraic geometry :: Curves :: Elliptic curves)
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