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Let $E$ be an elliptic curve over $\mathbb{Q}$, then the group $E(\mathbb{Q})$ satisfies \begin{equation} E(\mathbb{Q}) \cong \mathbb{Z}^r \oplus \mathbf T \end{equation}

for some torsion group $\mathbf T$, and $r$ is the rank of $E$.

I know about the importance of elliptic curves w.r.t. public key cryptography, and I know that bounding the rank of an elliptic curve from above is a Millennium Prize Problem, but I don't know what is the practical use of bounding it from below. Any suggestions?

Yulan van Oppen
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  • Since "half" of the elliptic curves over $\mathbb Q$ has rank 0, so your "bound from below" is just 0. – Hw Chu May 11 '18 at 18:26
  • What does mean "bound from below" that don't be contrary to the Hw Chu's comment? You ask maybe for "bound from above" (any way my English is weak) that the rank is (simply) bounded or not is a great problem of our time. – Piquito May 11 '18 at 19:41
  • Sorry, I wasn't very specific. I meant to ask about the importance of bounding the rank of a given elliptic curve from below. – Yulan van Oppen May 12 '18 at 11:24

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A bound from below for a given elliptic curve could tell you that the equation has infinitely many solutions.

Consider the congruent number problem: for a given positive integer $N$, does there exist a right-angled triangle with rational sides with area $N$? In other words, does there exist $a,b,c \in \mathbb{Q}$ such that $a^2+b^2=c^2$ and $\frac{1}{2}ab=N$?

This turns out to being equivalent to asking whether the elliptic curve $$y^2=x^3-N^2x,$$ has positive rank; a bound from below would help here. In fact, one can actually use the parity conjecture to say this should be true (in particular the rank is odd) whenever $N \equiv 5,6,7 \bmod{8}$.

As a side note, the Birch and Swinnerton-Dyer conjecture (the Millennium problem you mention) doesn't talk about bounding the rank but actually says it should be equal to something coming from analysis (roughly speaking, it comes from counting points $\bmod{p}$ for all $p$).

It is an open question whether the rank is bounded or not for elliptic curves over $\mathbb{Q}$.

Matt B
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  • Thank you! I would like to include this example in a paper, do you know a good reference to the problem you just described? – Yulan van Oppen May 12 '18 at 13:00
  • @User2935032946 I don't know if there is a standard reference but you could use the introduction of Tunnell's paper here: https://link.springer.com/article/10.1007%2FBF01389327 – Matt B May 13 '18 at 13:57
  • Actually Tunnell's theorem gives another (conjectural) characterisation of congruent numbers; one direction is known but the other assumes the Birch and Swinnerton-Dyer conjecture - see https://en.wikipedia.org/wiki/Tunnell%27s_theorem – Matt B May 13 '18 at 14:00