What is the canonical height of an elliptic curve?

Question

Aside from the math involved, curious if there is a good layman's explanation for the notion of canonical height for an elliptic curve? I.e. if there is a geometric intepretation? Or perhaps if anyone can help link to a brief history of why it was developed?

Answer 1

COMMENT.- According to Cassels the descent method is particularly appropriate for elliptic curves. To understand the full meaning of these words, various apparently dissimilar definitions are needed around the notion of "descent" and its inseparable correlative, that of "height", since both notions, even within the simple context of elliptic curves, have evolved quite a lot since their foundation by Fermat, with whom the idea of "height" was still implicit and unformulated but was nonetheless essentially used. At present, these two notions of fundamental importance present such a variety of nuances that it is difficult, at least at a non-advanced level, to give them a single precise and encompassing meaning (for example, the famous Faltings theorem on Mordell's conjecture, begins with a vast generalization of "height" and a "bounded-height principle" followed by a formula of "height variation" by action of isogenies on abelian manifolds).

It has been said that in elliptic curves the method of descent is a kind of inverse of the method of chords and tangents that turns the curve into a group and it is indeed so, but clearly seeing why is a matter of familiarity with the subject. In its most elementary application, the method rests on the property of good order in $\mathbb N$ and the difficulty begins when wanting to use it in $\mathbb Q$ and in $\mathbb R$ because between two different elements there are always an infinity of others. However many heights are defined to values in $\mathbb R_{+}$, in particular the much used canonical Neron-Tate height on an elliptic curve is nothing less than a limit of a convergent sequence of real numbers.