When do Heegner points lead to a basis points?

Question

For a rank 1 elliptic curve, a rational point can be obtained from Heegner points. When is this rational point a basis point?

If sometimes additional work is required to obtain a basis point, is there an easy calculation to determine when this is the case?

There are also "mock" heegner points. https://www.uni-math.gwdg.de/tschinkel/cmi/mockheegner-clay-nopause.pdf — PluckyBird, Jan 31 '19 at 22:38

score 2 · Accepted Answer · answered Jan 18 '19 at 20:42

2

For reference, the index of the Heegner point modulo torsion is referred to as the Heegner index. You can use the Gross-Zagier formula to numerically estimate the height of the Heegner point and thus reduce it to a finite computation by bounding the height of possible generators. This is, for example, what Sage does in its heegner_index computation.

answered Jan 18 '19 at 20:42

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So this just bounds a search for the generators? There is no way to use the now known rational point to somehow work backwards to the basis points? – PluckyBird Jan 26 '19 at 00:06

When do Heegner points lead to a basis points?

1 Answers1