We extend and refine the theory of 'organising modules' of Mazur and Rubin to construct a canonical class of matrices that encodes a range of information about natural families of complexes in arithmetic. We then describe several concrete applications of this theory including the proof of new results on the explicit structures of Galois groups, ideal class groups and wild kernels in higher algebraic K-theory and the formulation of a range of explicit conjectures concerning both the ranks and Galois structures of Selmer groups of abelian varieties over finite (non-abelian) Galois extensions of number fields.

Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the p-component of the equivariant Tamagawa number of the pair h 1 (A /F )(1), Z[Gal(F/k)] . By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of the p-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mordell-Weil group has strictly positive rank and the relevant field extension is both non-abelian and of degree divisible by p. More generally, our approach leads us to the formulation of certain precise families of conjectural p-adic congruences between the values at s = 1 of derivatives of the Hasse-Weil L-functions associated to twists of A, normalised by a product of explicit equivariant regulators and periods, and to explicit predictions concerning the Galois structure of Tate-Shafarevich groups. In several interesting cases we provide theoretical and numerical evidence in support of these more general predictions.

The main aim of the work is to assess physical parameters of forest woodchips and their impact on the prices achieved by the supplier in transactions with a power plant. During fragmentation of logging residue, high content of green matter and contaminants negatively impacts the quality parameters that serve as basis for settlements. The analysis concerns data on the main parameters -water content, fuel value, sulphur and ash content -from 252 days of deliveries of forest chips to a power plant. The deliveries were realised from forested areas on an average about 340 km from the plant. Average water content and the resultant fuel value of forest chips was within 27-47% and 8.7-12.9 GJ×Mg −1 (appropriately), respectively. They depend on the month in which they are delivered to the power plant. The threshold values for the above-mentioned parameters are set by the plant at a real level and the suppliers have no problems with meeting them. The parameter that is most frequently exceeded is ash content (11.5% of cases). The settlement system does not differentiate on the basis of the transport distance but gives possibility to lower the settlement price when the quality parameters are not met but provides no reward for deliveries with parameters better than the average ones. On the basis of results obtained, it was calculated that average annual settlement price is lower than the contract price by about 0.20 PLN×GJ −1 , which in case of the analysed company may translate into an average daily loss of about 700 PLN.

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We establish several close links between the Galois structures of a range of arithmetic modules including certain natural families of ray class groups, the values at strictly positive integers of p-adic Artin L-series, the Shafarevich–Weil Theorem and the conjectural surjectivity of certain norm maps in cyclotomic {\mathbb{Z}_{p}} -extensions. Non-commutative Iwasawa theory and the theory of organising matrices play a key role in our approach.