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This question is not very precise; I hope it is suitable for the site.

I have come to a situation where I have to study rational points on an elliptic curve defined over $\mathbb{Q}$. I don't know much about the curve, let alone its equation. I already have one rational point, which sits on a bounded real connected component. What I want to avoid is that this is the only rational point (other than the marked point).

I am not sure what to use about my curve that will help me get there, so I turn the questio the other way round:

What is known about elliptic curves $E$ over $\mathbb{Q}$ such that $E(\mathbb{Q}) \cong \mathbb{Z}/2 \mathbb{Z}$?

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Torsion points are integral or half-integral in suitable natural coordinates (Lutz-Nagell thm. see, e.g., Silverman) so if you can prove that your point is not integral, then it's of infinite order. Similarly, if you can show it has non-zero canonical height. Your question is too vague. –  Felipe Voloch May 13 '11 at 10:29
    
Yes, I know my question is vague. That is because I do not know much about this point. I was hoping that having Mordell-Weil group of order 2 would imply some restrictions, and then work backwards. –  Andrea Ferretti May 13 '11 at 10:35
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You will have to tell us more about your $E$ if you hope that we can help. Stated this way, your question is too vague. –  Chris Wuthrich May 13 '11 at 11:08
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The fact is, I don't know much more than stated in the question. I can probably extract more information, if I know what to look for. So, in a sense, any theorem starting like "Let $E/\mathbb{Q}$ be a an elliptic curve with $E(\mathbb{Q}) = \mathbb{Z}/(2)$; then..." would help me. –  Andrea Ferretti May 13 '11 at 11:59

2 Answers 2

up vote 14 down vote accepted

Mazur's theorem ensures that there are exactly 15 possible cases for the torsion part of the Mordell-Weil group of an elliptic curve: the cyclic groups $\mathbb{Z}_n$ (with $1\leq n\leq 10$ or $n=12$) and the groups $\mathbb{Z}_2\times\mathbb{Z}_n$ for $n=2,4,6,8$.

In his paper Universal Bounds on The Torsion of Elliptic Curves, Proc. London. Math. Soc.(1976) 33, 193-237 , Daniel Sion Kubert (who was a student of Mazur) presents in table 3 (page 217) a list of parametrizations for the different possible cases.

In particular, curves with a $\mathbb{Z}_2$ torsion are parametrized by the following family: $$ \mathbb{Z}_2\ \text{torsion}:\quad y^2=x(x^2+a x+ b), \quad b^2(a^2-4b)\neq 0. $$

The example given in Francesco's answer is a special case with $a=0$. As another example, the case with torsion $\mathbb{Z}_2\times \mathbb{Z}_2$ is parametrized by the Legendre family:

$$ \mathbb{Z}_2\times\mathbb{Z}_2 \ \text{torsion}:\quad y^2=x(x+r)(x+s), \quad r\neq 0 \neq s \neq r. $$

A slight generalization of the Hesse family parametrizes the curves with torsion $\mathbb{Z}_3$:

$$ \mathbb{Z}_3 \ \text{torsion}:\quad y^2+a_1 x y +a_3 y =x^3, \quad a_3^3( a_1^3-27 a_3)\neq 0. $$ For the other groups you might have to use Tate's normal form $$ E(b,c): \quad y^2+(1-c)x y - b y =x^3- b x^2 $$ and the condition for a given torsion is expressed as an algebraic condition on $b$ and $c$.

For example for $\mathbb{Z}_4$, we have $c=0$ and $b^4(1+16b)\neq 0$, which gives: $$ \mathbb{Z}_4 \ \text{torsion}:\quad E(b,c=0): \quad y^2+x y - b y =x^3- b x^2, \quad b^4(1+16b)\neq 0. $$

For a review, you can read chapter 4 of the book of Husemoller . A friendly short review is also available in section 2 of this string theory paper by Aspinwall and Morrison ( they don't present all the 15 cases but for those they analyze, they express everything in Weierstrass form).

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I think not so much is known.

Conjecturally, elliptic curves (defined over $\mathbb{Q}$) of rank $0$ have density $1/2$, and since at any rate only a finite number of cases for the torsion part of $E(\mathbb{Q})$ are possible (by Mazur's celebrated result), I guess that elliptic curves with $E(\mathbb{Q}) \cong \mathbb{Z}/2 \mathbb{Z}$ should have density $1/2$ too.

For this reason, it seems to me that a complete classification is out of reach.

If you want to see an infinite family of cubic curves with this property, take

$y^2=x^3 + px$

where $p$ is a prime number such that $p \equiv 7$ (mod $16$), see [Silverman-Tate, Rational points on elliptic curves, p. 105].

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4  
I don't think that the statement that curves with torsion groups of order 2 have density 1/2 is correct; it's probably, to paraphrase Pauli, not even false: these density statements only make sense if you think of elliptic curves as being ordered, and this can be done in several ways: with respect to the size of its coefficients, its discriminant, its conductor, etc. –  Franz Lemmermeyer May 13 '11 at 10:38
    
In addition, torsion of order 2 means that the defining polynomial on the right hand side splits into a linear and a quadratic factor over the rationals, plus the rather complicated condition of the absence of further torsion points of odd order. –  Franz Lemmermeyer May 13 '11 at 10:38
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That's right. If we consider elliptic curves as polynomials $x^3+Ax+B$ with integral $A,B$ ordered by size, you should get density zero for those with roots. From this point of view (and I think most other reasonable ones), the density of elliptic curves with 2-torsion should be zero. –  Minhyong Kim May 13 '11 at 10:45
    
Dear Franz and Minhyong, thank you for your comments. I'm actually not an expert in the field, so probably my idea of density is too vague and intuitive. What I really want to say in this answer in that elliptic curves with $E(\mathbb{Q})=\mathbb{Z}/2 \mathbb{Z}$ are probably "too many" to admith an exhaustive description. –  Francesco Polizzi May 13 '11 at 11:00
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@Francesco: it's not really a matter of size. Families of elliptic curves whose torsion parts have subgroups of certain orders have been classified. –  Franz Lemmermeyer May 13 '11 at 17:51

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