How could we show that the abelian group has $\text{ rank}=0$?

Question

Let $E/\mathbb{Q}$ the elliptic curve $Y^2=X^3+p^2X$ with $p \equiv 5 \pmod 8$. Show that the abelian group $E(\mathbb{Q})$ has $\text{rank}=0$.

Could you give me a hint how we could do this?

It is known that if $E|_{\mathbb{Q}} Y^2=X^3+aX+b \ \ (a,b \in \mathbb{Z}, D(f) \neq 0)$ then:

$$E(\mathbb{Q})=\{ (\alpha, \beta) \in \mathbb{Q} \times \mathbb{Q} | \beta^2=\alpha^3+a \alpha+ \beta \} \cup \{ [0,1,0]\}$$

EDIT: $p$ is a prime.

We would like to help you, but we first should have a correct statement. What exactly is assumed on $p$ ? — Dietrich Burde, Dec 28 '14 at 21:43
It is obviously wrong, unfortunately. Take $p=69,77$ etc. Where did you find this "exact statement" ? — Dietrich Burde, Dec 28 '14 at 21:50

score 4 · Accepted Answer · edited Apr 13 '17 at 12:58

4

This is not true in general. For $p=-3$ we have $p\equiv 5\bmod 8$, but the elliptic curve $y^2=x^3+9x$ has rank $1$ over $\mathbb{Q}$. Also, for positive $p$ this need not be true. Take $p=69$. Then $p\equiv 5 \bmod 8$ and the elliptic curve $y^2=x^3+4761x$ has rank $2$. So we need an additional assumption on $p$. A possible reference for such assumptions is Silverman's "The Arithmetic of Elliptic Curves", Chapter X, Section 6, "The curve $y^2=x^3+Dx$".

Edit: For primes $p$ the question has been asked already on MO and on this site in $2012$, see here and here. However, there seems to be no obvious condition, except the ones in Silverman's book. Perhaps we should ask your professor about the solution.

edited Apr 13 '17 at 12:58

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answered Dec 23 '14 at 14:52

Dietrich Burde

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Did you look at section $6$ in Silverman ? Where did you find this statement ? – Dietrich Burde Dec 23 '14 at 15:45
Dietrich We also assume that $p$ is a prime. How could we show it then? – evinda Dec 30 '14 at 17:55
Indeed the professor gave it to us.. What is the rank of the curve for example for $p=5$ ? – evinda Dec 30 '14 at 20:07
The curve $y^2=x^3+25x$ has rank zero. – Dietrich Burde Dec 30 '14 at 20:11
Could you maybe take a look at this? http://math.stackexchange.com/questions/1131121/determine-if-e-mathbbq-is-finite-or-infinite – evinda Feb 03 '15 at 15:10

How could we show that the abelian group has $\text{ rank}=0$?

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