On the "Section Conjecture" in anabelian geometry

A main problem in Galois theory is to characterize the fields with a given absolute Galois group. We apply a K-theoretic method for constructing valuations to study this problem in various situations. As a first application we obtain an algebraic proof of the 0-dimensional case of Grothendieck's anabelian conjecture (proven by Pop), which says that finitely generated infinite fields are determined up to purely inseparable extensions by their absolute Galois groups. As a secon...

We prove a generalization of Shafarevich's Conjecture for fields of Laurent series in two variables over an arbitrary field. While not projective, the absolute Galois group of such a field is shown to be semi-free. We also show that the function field of a smooth projective curve over a large field has semi-free absolute Galois group. In the first edition of this paper it was shown that these groups are quasi-free, which is weaker.

Let $X$ be a proper, smooth, and geometrically connected curve of genus $g(X)\ge 1$ over a $p$-adic local field. We prove that there exists an effectively computable open affine subscheme $U\subset X$ with the property that $period (X)=1$, and $index (X)$ equals $1$ or $2$ (resp. $period(X)=index (X)=1$, assuming $period (X)=index (X)$), if (resp. if and only if) the exact sequence of the geometrically abelian fundamental group of $U$ splits. We compute the torsor of splittin...

The main purpose of this paper is to describe the abelian part $\mathcal G^{ab}_{K}$ of the absolute Galois group of a global function field $K$ as pro-finite group. We will show that the characteristic $p$ of $K$ and the non $p$-part of the class group of $K$ are determined by $\mathcal G^{ab}_{K}$. The converse is almost true: isomorphism type of $\mathcal G_K^{ab}$ as pro-finite group is determined by the invariant $d_K$ of the constant field $\mathbb F_q$ introduced in fi...

The paper has three main applications. The first one is this Hilbert-Grunwald statement. If $f:X\rightarrow \Pp^1$ is a degree $n$ $\Qq$-cover with monodromy group $S_n$ over $\bar\Qq$, and finitely many suitably big primes $p$ are given with partitions $\{d_{p,1},..., d_{p,s_p}\}$ of $n$, there exist infinitely many specializations of $f$ at points $t_0\in \Qq$ that are degree $n$ field extensions with residue degrees $d_{p,1},..., d_{p,s_p}$ at each prescribed prime $p$. Th...

Let $C$ and $D$ be smooth, proper and geometrically integral curves over a finite field $F$. Any morphism from $D$ to $C$ induces a morphism of their \'etale fundamental groups. The anabelian philosophy proposed by Grothendieck suggests that, when $C$ has genus at least $2$, all open homomorphisms between the \'etale fundamental groups should arise in this way from a nonconstant morphism of curves. We relate this expectation to the arithmetic of the curve $C_K$ over the globa...

In this paper we prove a version of Grothendieck's section conjecture for the restriction of the universal complete curve over M_{g,n}, g > 4, to the function field k(M_{g,n}) where k is, for example, a number field. In this version, the fundamental group of the closed fiber is replaced by its ell-adic unipotent completion when n > 1.

In 1981, Uchida proved a conditional version of the Hom-form of the Grothendieck birational anabelian conjecture for number fields. In this paper we prove an m-step solvable conditional version of the Grothendieck birational anabelian conjecture for number fields whereby our conditions are slightly weaker than the ones in Uchida's theorem. Furthermore, as in Uchida's work, we show that our result is unconditional in the case where the number field relating to the domain of th...

In the first part of this paper we try to explain to a general mathematical audience some of the remarkable web of conjectures linking representations of Galois groups with algebraic geometry, complex analysis and discrete subgroups of Lie groups. In the second part we briefly review some limited recent progress on these conjectures.

In 1922, Mordell conjectured the striking statement that for a polynomial equation $f(x,y)=0$, if the topology of the set of complex number solutions is complicated enough, then the set of rational number solutions is finite. This was proved by Faltings in 1983, and again by a different method by Vojta in 1991, but neither proof provided a way to provably find all the rational solutions, so the search for other proofs has continued. Recently, Lawrence and Venkatesh found a th...