Kay Wingberg

Ein Analogon zur Fundamentalgruppe einer Riemann'schen Fläche im Zahlkörperfall

Nach einem Satz von Grothendieck ist die Struktur der maximalen zu f primen Faktorgruppe der Fundamentalgruppe einer glatten projektiven Kurve )7 fiber einem algebraisch abgeschlossenen K6rer F der Charakteristik E > 0 explizit gegeben. Ffir die maximale pro-p-Faktorgruppe fg, p 4= f , hat dies Ergebnis zur Folge, dal3 ff eine Demugkingruppe vom Rang 2g ist, wenn mit g das Geschlecht von X bezeichnet wird. Genauer gibt es 2g Erzeugende von ~ mit der einzigen definierenden Relation g

Galois groups of number fields generated by torsion points of elliptic curves

Coates and Wiles [1] and B. Perrin-Riou (see [2]) study the arithmetic of an elliptic curve E defined over a number field F with complex multiplication by an imaginary quadratic field K by using p-adic techniques, which combine the classical descent of Mordell and Weil with ideas of Iwasawa’s theory of Z p -extensions of number fields. In a special case they consider a non-cyclotomic Z p -extension F ∞ defined via torsion points of E and a certain Iwasawa module attached to E/F , which can be interpreted as an abelian Galois group of an extension of F ∞ . We are interested in the corresponding non-abelian Galois group and we want to show that the whole situation is quite analogous to the case of the cyclotomic Z p -extension (which is generated by torsion points of G m ).

On the Fontaine–Mazur Conjecture for CM-Fields

Fontaine and Mazur conjecture that a number field k has no infinite unramified Galois extension such that its Galois group is a p-adic analytic pro-p-group. We consider this conjecture for the maximal unramified p-extension of a CM-field k.

Cohomology of Local Fields

We now begin the development of cohomology in number theory. As a ground field we take a nonarchimedean local field k, i.e. a field which is complete with respect to a discrete valuation and has a finite residue field. This covers two cases, namely p-adic local fields, i.e. finite extensions of \(\mathbb{Q}_{p}\) for some prime number p, and fields of formal Laurent series \(\mathbb{F}((t))\) in one variable over a finite field. For the basic properties of local fields we refer to [160], chapters II and V. As always, \(\bar{k}|k\) denotes a separable closure of k and K|k the subextensions of \(\bar{k}|k\). v k denotes the valuation of k, normalized by \(v_{k}(k^{\times}) = \mathbb{Z}\), and κ the residue field. For every Galois extension K|k we set

Duality Properties of Profinite Groups

H^i(K|k) := H^i(G(K|k),K^{\times}),\quad i \geq 0.

Some Homological Algebra

If K|k is finite, we also set \(\hat{H}^{i}(K|k) = \hat{H}^{i}(G(K|k),K^{\times})\) for \(i \in \mathbb{Z}\). The basis for the results in this chapter is the following theorem.

Cohomology of Global Fields

Profinite groups are topological groups which naturally occur in algebraic number theory as Galois groups of infinite field extensions or more generally as etale fundamental groups of schemes. Their cohomology groups often contain important arithmetic information.

Iwasawa Theory of Number Fields

Let G be a finite group. If A and B are two G-modules, the cup-product associated with the canonical pairing