A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory

By O. Venjakob. 2002

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In this paper and a forthcoming joint one with Y. Hachimori we study Iwasawa modules over an infinite Galois extension K of a number field k whose Galois group G=G(K/k) is isomorphic to the semidirect product of two copies of the p-adic numbers. After first analyzing some general algebraic properties of the corresponding Iwasawa algebra, we apply these results to the Galois group of the p-Hilbert class field over K. As a main tool we prove a Weierstrass preparation theorem for certain skew power series rings. One striking result in our work is the discovery of the abundance of faithful torsion modules, i.e. non-trivial torsion modules whose global annihilator ideal is zero. Finally we show that the completed group algebra with coefficients in the finite field of p elements is a unique factorization domain in the sense of Chatters.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8e89a73b259c4d3767a03ad4f12b3d6b0f0715aa

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Citations in the corpus

Equivariant Weierstrass Preparation and Values of L-functions at Negative Integers (2003, background): “…Mathematica · Extra Volume Kato (2003) 157–185 In the remainder of the manuscript our main aim is to show that, if G is abelian, then the above considerations can be combined with the constructions of Deligne and Ribet and the theorem of Wiles to shed light on a number of interesting questions.”
Ring-Theoretic Properties of Iwasawa Algebras: A Survey 1 (2005, background): “Theorem. [ 37, Theorem 7.1 ] Let G = X ⋊ Y be a nonabelian semidirect product of two copies of Zp. Then the only prime ideals of G are 0, wX and J(G), and each one is completely prime.”
On descent theory and main conjectures in non-commutative Iwasawa theory (2007, background): “These include proving a Weierstrass Preparation Theorem for Whitehead groups of Iwasawa algebras (this result is related to, but different from, the Weierstrass Preparation Theorem discussed by the second named author in [ 33 ]), defining a canonical ‘characteristic series’ for torsion modules over ”
Fine Selmer groups of elliptic curves over p-adic Lie extensions (2005, background): “Moreover, the modules E(Wi) are pseudo-null for j ≥ 2 (see [29]), and hence have μ-invariant zero.”
Torsion p-adic Galois representations and a conjecture of Fontaine (2007, background): “There are several versions of such a theorem available; the version we use is from [23].”