Galois theory and Diophantine geometry (Q2900376)

The paper under review is a survey paper concerning a modification of some sort of Albanese map in order to study the set \(X(K)\) of the \(K\)-points of a arithmetic variety \(X\) defined over \(K\), where \(K\) is a number field. As usual \(\overline{K}\) the algebraic closure of \(K\) and \(b\in X(K)\), \(G_K\) the galois group of \(\overline{K}\) over \(K\) and \(H^1(G_K,*)\) denotes the continuous cohomology group.NEWLINENEWLINEFor example take \(X\) a smooth projective curve of genus \(\geq 2\) the section conjecture of \textit{A. Grothendieck} [in: Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 242, 49--58; English translation 285--293 (1997; Zbl 0901.14002)] proposes a non-abelian sort of Albanese map NEWLINE\[NEWLINE\kappa:X(K)\rightarrow H^1(G_K,\pi_1^{\mathrm{et}}(X\times_{\mathrm{Spec}(K)}\mathrm{Spec}(\overline{K}),b))NEWLINE\]NEWLINE and claims its surjectivity, where \(\pi_1^{\mathrm{et}}(X\times_{\mathrm{Spec}(K)}\mathrm{Spec}(\overline{K}),b)\) the profinite étale fundamental group of \(X\).NEWLINENEWLINEThe paper under review would like emphasize the above map but replacing the above étale fundamental group by the \(\mathbb{Q}_p\)-pro-unipotent étale fundamental group \(U\), and more concretely by use a filtration NEWLINE\[NEWLINEU=U^1\supset U^2\supset\ldotsNEWLINE\]NEWLINE and its finite-dimensional algebraic quotient \(U_n:=U/U_{n+1}\), (for example \(U_1\) is the Tate module of the Jacobian of \(X\) tensored by the \(p\)-adic numbers \(\mathbb{Q}_p\)). Then one obtains a collection of mapsNEWLINENEWLINENEWLINE\[NEWLINE\kappa_n: X(K)\rightarrow H^1_f(G_K,U_n),NEWLINE\]NEWLINENEWLINENEWLINEwhere the subindex \(f\) is a subspace of \(H^1(G_K,U_n)\) defined by local type conditions, named finite part, which for example impose crystalline condition at the primes dividing \(p\). For any place \(v\) of \(K\) we have a natural localization map \(\mathrm{loc}_v:H^1_f(G_K,U_n)\rightarrow H^1_f(G_v,U_n)\) where \(G_v=Gal(\overline{K_v}/K_v)\), as usual \(K_v\) denotes the completion of \(K\) at \(v\) and \(\overline{K_v}\) its algebraic closure. For the places \(v|p\), there is a natural isomorphism \(D:H^1_f(G_v,U_n)\rightarrow U_n^{\mathrm{dR}}/F^0\) where \(U_n^{\mathrm{dR}}\) follows from the De Rham fundamental group of \(X\times \mathrm{Spec}(K_v)\) in the setting. Under the assumption that \(K_v=\mathbb{Q}_p\) the author proves that when \(D\circ \mathrm{loc}_p(H^1_f(G_K,U_n))\subset U_n^{\mathrm{dR}}/F^0\) is not Zariski dense for some \(n\), then \(X(K)\) is finite. After, the author refers different works where this idea was applied, see [\textit{M. Kim}, Invent. Math. 161, No. 3, 629--656 (2005; Zbl 1090.14006), Ann. Math. (2) 172, No. 1, 751--759 (2010; Zbl 1223.11080)] or [\textit{J. Coates} and \textit{M. Kim}, Kyoto J. Math. 50, No. 4, 827--852 (2010; Zbl 1283.11092].NEWLINENEWLINEThe author of the paper under review is interested in making as explicit as possible the map \(D\circ \mathrm{loc}_p\) when \(v=p\) and \(K=\mathbb{Q}\). The approach is by having a function \(\psi:H^1_f(G_p,U_n)\rightarrow \mathbb{Q}_p\), which should encode some sort of reciprocity law map. This speculation is followed in the survey paper under review taking \(X=E\setminus\{e\}\) with \(E\) an elliptic curve satisfying that the rank of the \(\mathbb{Q}\)-points of \(E\) is one, the \(p\)-torsion of the Sha-group of \(E\) is finite, and the local Tamagawa numbers of \(E\) are one for each prime \(\ell\) of \(\mathbb{Q}\).NEWLINENEWLINEFor the entire collection see [Zbl 1237.11001].