The motivic fundamental group of \(\mathbf P^1\setminus\{0,1,\infty\}\) and the theorem of Siegel. (English) Zbl 1090.14006
The aim of the paper is to show how results in Diophantine geometry can be achieved by studying the arithmetic fundamental group of an algebraic variety \(X\) over a number field \(F\). Here the author takes \(F =\mathbb Q\) and \(X =\mathbb P^1_{\mathbb Q}-\{0,1,\infty\}\) and considers the unipotent motivic fundamental group in the sense of P. Deligne [in: Galois groups over \(\mathbb Q\), Publ. Math. Sci. Res. Inst. 16, 79–297 (1989; Zbl 0742.14022)]. Then he proves Siegel’s theorem on the finiteness of integral points for the thrice-punctured projective line. The following is an outline of his proof.
Let \(S\) be a finite set of primes, fix \(p\not\in S\) and an integral point \(x\in{\mathcal X} = \mathbb P^1_{\mathbb Z}-\{0,1,\infty\}\). Put \(T = S\cup \{p\}\) and let \(Y\) be the reduction of \(\mathcal X\bmod p\), \(y\in Y\) the reduction mod\(p\) of the point \(x\). The \(p\)-adic unipotent Albanese map is then the following map: \[ U\text{\,Alb}_x:X(\mathbb Q_p)\cap \overline Y\to \pi_{1, \text{DR}}(X,x)(\mathbb Q_p) \] where \(\overline Y\) are the points that reduce mod\(\,p\) to points in \(Y\). The image of the integral points \({\mathcal X}(\mathbb Z_S)\) under the Albanese map is essentially contained inside the image of another map \[ H^1_f (\Gamma_T,\pi_{1,\text{ét}} (X,x))\to \pi_{1, \text{DR}}(X\otimes \mathbb Q_p,x) \] from a suitable continuous global cohomology set to the De Rham fundamental group.
\(H^1_f(\Gamma_T,\pi_{1,\text{ét}}(X,x))\) has the natural structure of a proalgebraic variety. By looking at various quotients \([\pi_{1,\text{DR}}]_n\) and \([\pi_{1,\text{ét}}]_n\) the author shows that for large \(n\) and \(p\) the image of \(H^1_f(\Gamma_T, [\pi_{1,\text{ét}}(X,x)]_n)\) under the map to \([\pi_{1,\text{DR}}(X\otimes \mathbb Q_p,x)]_n\) lies in some proper subvariety. Thus fact, together with the identity principle for Colemann’s functions and compactness yields the finiteness of Siegel’s theorem.
The results of this paper – as noted by the author – may be viewed as a non abelian lift of C. Chabauty’s proof [C. R. Acad. Sci., Paris 212, 882–885 (1941; Zbl 0025.24902)].
Let \(S\) be a finite set of primes, fix \(p\not\in S\) and an integral point \(x\in{\mathcal X} = \mathbb P^1_{\mathbb Z}-\{0,1,\infty\}\). Put \(T = S\cup \{p\}\) and let \(Y\) be the reduction of \(\mathcal X\bmod p\), \(y\in Y\) the reduction mod\(p\) of the point \(x\). The \(p\)-adic unipotent Albanese map is then the following map: \[ U\text{\,Alb}_x:X(\mathbb Q_p)\cap \overline Y\to \pi_{1, \text{DR}}(X,x)(\mathbb Q_p) \] where \(\overline Y\) are the points that reduce mod\(\,p\) to points in \(Y\). The image of the integral points \({\mathcal X}(\mathbb Z_S)\) under the Albanese map is essentially contained inside the image of another map \[ H^1_f (\Gamma_T,\pi_{1,\text{ét}} (X,x))\to \pi_{1, \text{DR}}(X\otimes \mathbb Q_p,x) \] from a suitable continuous global cohomology set to the De Rham fundamental group.
\(H^1_f(\Gamma_T,\pi_{1,\text{ét}}(X,x))\) has the natural structure of a proalgebraic variety. By looking at various quotients \([\pi_{1,\text{DR}}]_n\) and \([\pi_{1,\text{ét}}]_n\) the author shows that for large \(n\) and \(p\) the image of \(H^1_f(\Gamma_T, [\pi_{1,\text{ét}}(X,x)]_n)\) under the map to \([\pi_{1,\text{DR}}(X\otimes \mathbb Q_p,x)]_n\) lies in some proper subvariety. Thus fact, together with the identity principle for Colemann’s functions and compactness yields the finiteness of Siegel’s theorem.
The results of this paper – as noted by the author – may be viewed as a non abelian lift of C. Chabauty’s proof [C. R. Acad. Sci., Paris 212, 882–885 (1941; Zbl 0025.24902)].
Reviewer: Claudio Pedrini (Genova)
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