How important is Weil's decomposition theorem today?

Question

Andre Weil's Apprenticeship of a Mathematician (p. 46) tells how he as a student realized that all of Fermat's uses of descent are unified in one principle: "If $P(x,y)$ and $Q(x,y)$ are homogeneous polynomials algebraically prime to each other, with integer coefficients, and $x,y$ are integers prime to each other, then $P(x,y)$ and $Q(x,y)$ are `almost"prime to each other." Serre agrees with Weil's estimate of the result and says that Weil put an end to the time when "a different little miracle" seemed to happen in each use of descent (André Weil. 6 May 1906-6 August 1998. Biographical Memoirs of Fellows of the Royal Society, 45 (1999), 520--529).

One special case of this is omnipresent in undergraduate number theory: when $x$ and $y$ are relatively prime integers then the GCD of $x+y$ and $x-y$ is either 1 or 2.

The theorem is widely praised but it is not widely given. Books that mention it often go on to say they will not use the full strength.

Lang's Fundamentals of Diophantine Geometry (p. 263) gives a "reformulation of the existence of Weil functions in the words of Weil's decomposition theorem" and I believe the two formulas he gives after that actually make up a form of Weil's theorem but am not sure of that. Anyone who does not already know these formulas would need considerable explanation of notation to read them, so I will not type them here. The passage can be found by searching "Weil's decomposition theorem" in Google books .

Is Lang's Theorem 3.7 a modern form of Weil's theorem?

Are there other modern expositions of the theorem today? Weil's own dissertation is not too hard to read but he was inventing a lot of ideas and the terminology is not what we use today. And anyway I wonder how important the theorem is today. Has some other insight tended to replace it?

Vesselin Dimitrov's answer led me to see that the suggestive statement Weil gives in his autobiography is precisely proven in many books today in the following form. Define the height function $H$ for rational numbers so $H(m/n)$ is $\mathrm{Max}(|m|,|n|)$ for relatively prime $m,n$. Then for relatively prime polynomials $f(T),g(T)$ and $d$ the maximum of their degrees there is some real $C>0$ with $$H(x)^d \leq C\cdot H(g(x)/f(x))$$ whenever $f(x)\neq 0$. This seems to suffice for Fermat's uses of descent. It is never called Weil's decomposition theorem to my knowledge.

Weil's dissertation proves the theorem for algebraic points on curves rather than for rational numbers. That is called his decomposition theorem and it has various modern forms which are only given in rather advanced texts as Vesselin Dimitrov describes.

Answer 1

Yes, this is the modern statement of Weil's theorem of decomposition. It is a basic component of the theory of heights. For a more recent exposition see 2.7.15 in Bombieri and Gubler's Heights in Diophantine Geometry.

If you look for a specific application of the theorem and its point of view, you should be aware of Bombieri's paper [On Weil's "Theoreme de decomposition," Amer. J. Math., 1983]. There, Bombieri employs the theory of heights (Weil's theorem of decomposition and a theorem of Neron, cf. 9.3.10 in Bombieri-Gubler), to extend work of Sprindzhuk on the Hilbert irreducibility theorem and deduce a generalization of an old theorem of Runge stating the finiteness of solutions $(x,y) \in \mathbb{Z} \times \mathbb{Q}$ to $G(x,y) = 0$ for an irreducible $G \in \mathbb{Z}[x,y]$ whose leading homogeneous part is not proportional to a power of an irreducible polynomial.

Bombieri's main result in that paper is roughly that if $f : C \to \mathbb{P}^1$ is a morphism from a curve over a number field $K$, then for $P \in C(\bar{K})$, the contribution from every given pole of $f$ to the height of $f(P)$ can be approximately read from the factorization of the function $f \in K(C)$. This exactly captures the spirit of the theorem of decomposition.

To get at the precise statement, let $\iota : C \hookrightarrow \mathbb{P}_K^N$ a projective embedding of $C$ and $d_v(\cdot,\cdot)$ the $v$-adic chordal distance on $\mathbb{P}^N(\mathbb{C}_v)$. Then, for all $P \in C(K) \setminus f^{-1}(\infty)$, and for each pole $Q$ of $f$, it holds $$ \sum_{\substack{v \\ d_v(\iota(P),\iota(Q)) < 1}} \log^+{|f(P)|_v} = \frac{\mathrm{ord}_Q (1/f)}{\deg{f}} \sum_v \log^+{|f(P)|_v} + O \Big( \sqrt{\sum_v \log^+{|f(P)|_v}} \Big), $$ with the sum being over the places $v$ of $K$ at which $\iota(P)$ belongs to the open unit disk centered at $\iota(Q)$ (which for almost all $v$ means that $P$ is closer to $Q$ than to any other pole), and with an implied constant depending only on $f$ and the embedding $\iota$. With some care in the notation, the statement moreover extends to all $P \in C(\bar{K})$.

Here is the intuitive meaning of this. Note that the local height of $f(P)$ at $v$ is large if and only if $P$ is $v$-adically close to some pole of $f$. Then as $v$ varies, each pole is approached with frequency proportional to the order of the pole.

A full treatment of this result of Bombieri's and its consequent generalization of Runge's theorem is presented in the chapter 9 on Neron-Tate heights in Bombieri and Gubler's book. There, it is used to give an essentially algebro-geometric proof of the Hilbert irreducibility theorem.