# bound for the rank of an elliptic curve

###### Theorem.

Let $E/\mathbb{Q}$ be an elliptic curve  given by the equation:

 $E\colon y^{2}=x(x-t)(x-s),\text{ with }t,s\in\mathbb{Z}$

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:

 $R_{E}\leq m+2a-1$
###### Example.

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:

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

## References

• 1 James Milne, Elliptic Curves, online course notes.
http://www.jmilne.org/math/CourseNotes/math679.htmlhttp://www.jmilne.org/math/CourseNotes/math679.html
Title bound for the rank of an elliptic curve BoundForTheRankOfAnEllipticCurve 2013-03-22 14:24:25 2013-03-22 14:24:25 alozano (2414) alozano (2414) 6 alozano (2414) Theorem msc 14H52 ArithmeticOfEllipticCurves