# How did height in algeb. number theory/elliptic curves started?

Maybe this is obvious but it isn't to me yet. What is the history of heights used in say points of the project plane over a number field or of elliptic curve over a number field? I would guess people working with valuations (like Ostrowski, Krull, Dedekind, Hensel et al) would have started this idea. Just an idea.. there is a nice article on the history of valuation written by Peter Roquette, maybe I can find this there?

Edit: Maybe Keith is right. In fact I took a look at the text by Roquette and as far as I could see (I didnt read the whole text though) height was not mentioned.

• I believe heights go back to Weil in his work on the Mordell-Weil theorem. – KConrad Sep 11 '14 at 12:01
• so it really started as a neccessity in elliptic curves. I really thought having a rich involvement of valuation theory, it had come out of valuation theory. I personally would never have thought of collection of valuations and absolute values to define height unless I was very involved in valuation theory. – Jose Capco Sep 11 '14 at 12:38
• Well, Weil was "very involved" in valuation theory. – KConrad Sep 11 '14 at 13:13

I think that Keith Conrad is correct and heights via maxs of valuations are in Weil's "Mordell-Weil" paper. But note that Weil's paper generalized Mordell's work in two ways. First, he extended from $\mathbb{Q}$ to number fields, and second from elliptic curves to abelian varieties (or at least Jacobians). So he would have needed some sort of way of measuring the arithmetic complexity of points on varieties, which he presumably viewed as embedded in projective space.

Note that there's also the more general theory of heights that associates to each divisor $D$ on a variety $X$ (everything defined over $\overline{\mathbb{Q}}$) a height function $h_{X,D}$, or more properly, an equivalence class of functions modulo bounded functions. I believe that this, too, is due to Weil, although this belief comes primarily from the fact that everyone calls them Weil height functions. (If I had to guess, the name "Weil height function" is due to Lang.)

Weil would have fixed a number field $K$ and defined a height function $h_K$ on $\mathbb{P}^n(K)$, and used that to define a height on $X(K)$ for any given embedding $X\hookrightarrow\mathbb{P}^n(K)$. I believe that it was Northcott (Annals paper in 1950) who observed that one can define an absolute height on $\mathbb{P}^n(\overline{\mathbb{Q}})$ by taking $$h(x) = \frac{1}{[K:\mathbb{Q}]} h_{K}(x)$$ for any number field $K$ such that $x\in\mathbb{P}^n(K)$. Then $h(x)$ is independent of the choice of $K$. So now people often prove theorems saying that "such and such a subset of $\mathbb{P}^n(\overline{\mathbb{Q}})$ is a set of bounded height", which implies in particular that it has only finite intersection with $\mathbb{P}^n(K)$ for any number field $K$, but even more, that it contains only finitely many points defined over all number fields of a given bounded degree. For example, the set of all torsion points on an abelian variety is a set of bounded height.

Note that one can also define height on projective space mimicking the "largest coordinate after clearing denominators" approach, which doesn't involve valuations per se. So, let $a=[a_0,\ldots,a_n]\in\mathbb{P}^n(K)$ for some number field $K$. WLOG, we may assume that all of the $a_i$ are in $O_K$. Then we more or less want to set $$h(a) = \sum_{\sigma:K\hookrightarrow\mathbb{C}} \log\max_{0\le i\le n} |\sigma(a_i)|,$$ but there's a problem if the $a_i$ have a nontrivial common factor. Since $O_K$ isn't a PID, we don't have gcds, but we do if we consider ideals. So let $$\mathfrak{A} = a_1O_K+a_2O_K+\cdots+a_nO_K$$ be the ideal generated by the coordinates of $a$. Then the correction factor is more-or-less to divide by $\mathfrak{A}$. Precisely, an alternative formula for what we now call the Weil height is $$h(a) = \log\left(\frac{1}{N_{K/\mathbb{Q}}\mathfrak{A}} \prod_{\sigma:K\hookrightarrow\mathbb{C}} \max_{0\le i\le n} |\sigma(a_i)|\right).$$ Now, if you write the norm as a product of prime powers and write the log of the product as a sum, you'll get to usual definition as a sum over all absolute values.

• Thanks Joe. I like your detailed answer, especially about what your wrote about the observation of Northcott. I marked this as the answer. I hope I havent discredited Keith, I also upvoted his comment. So to put it mildly, it was in the 50's when all these things about height was developed. It is interesting to note that it came about because of a problem in elliptic curve then made its way into algebraic number theory and older maths. – Jose Capco Sep 11 '14 at 17:25
• @JoseCapco it was not "all" in the 50's. The early papers from Weil, especially the Mordell--Weil mentioned in the answer, are from around 1927 to 1929, and there were some contributions in between. – user9072 Sep 11 '14 at 17:34

The relevance of Weil was already mentioned, and the collected works of Weil contain some commentary due to himself at the end of each volume. Reading that commentary (of the first volume) gives some overview of the developpment of early ideas (by him and others) related to heights.

Some of the papers could serve as further sources; not only or even mainly the very first ones from the 20's but e.g. "Arithmetic on algebraic varieties" Ann. of Math. 1951.