The converse of the Nagell-Lutz theorem

The Nagell-Lutz tells us that rational points of finite order have integer coordinates, but it doesn't tell us that points with integer coordinates have finite order. As a reminder, here's the statement of the Nagell-Lutz theorem.

Let y2 = x3 + ax + b be an elliptic curve over the rationals with integer coefficients and let D = 4 a3 + 27 b2. Then if P = (xP,yP) is a rational point of finite order then P has integer coordinates and either yP = 0 or yP2|D.

Here are some examples of points with integer coordinates that don't have finite order.

The point P = (1,2) is on the elliptic curve

y2 = x3 + 3

but (1,2) isn't a point of finite order. We have that

2P = (-23/16,-11/64)

for example. Since 2P doesn't have integer coordinates, it's not a point of finite order, so P isn't either.

For another example, consider the elliptic curve

y2 = x3 + 17

There are 16 points with integer coordinates on this curve. These are the following

(-2,±3)

(-1,±4)

(2,±5)

(4,±9)

(8,±23)

(43,±282)

(52,±375)

(5234,±378661)

Although we can find a few cases where adding these points gives another point with integer coordinates, like

(-2,3) + (-1,4) = (4,9)

most cases don't. We have that

(-1,4) + (-1,4) = (137/64, -2651/512)

for example.

Even worse, we have that

(5234,378661)  + (5234,378661)  = (187618163896928/143384152921, -1/4)

None of these points actually have finite order although they have integer coordinates. So points of finite order have to have integer coordinates, but not all points with integer coordinates have finite order.

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