In this thesis we formulate a natural non-commutative Iwasawa Main Conjecture for motives which fulfil the Dabrowski-Panchishkin condition on the level of (φ,Γ)-modules. The basic framework we employ is still Fukaya-Kato’s but we work systematically over Schneider-Teitelbaum’s distribution algebras of compact p-adic Lie groups instead of Iwasawa algebras. This allows us to consider as local conditions not just subrepresentations of the p-adic realisation which fulfil the Dabrowski-Panchishkin conditions but also sub-(φ,Γ)-modules which fulfil the analogous Dabrowski-Panchishkin conditions. We then combine this with Pottharst’s Selmer complexes and a generalisation of Nakamura’s Local Epsilon Conjecture for (φ,Γ)-modules to conjecturally define p-adic L-functions. We prove that the validity of our main conjecture for these p-adic L-functions follows from the validity of Fukaya-Kato’s Equivariant Tamagawa Number Conjecture and our generalisation of Nakamura’s Local Epsilon Conjecture. Moreover we are also able to compute the values of these p-adic L-functions at motivic points. Our formalism allows us, for example, to unify the GL2-main conjecture of elliptic curves which have either ordinary or supersingular reduction at p. In addition, we can use our formalism to give a new, and very natural, interpretation of Pollack’s ±-construction in the context of supersingular elliptic curves and we are hopeful that this new interpretation will in the future lead to the construction of natural non-commutative generalizations.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:721690 |
| Date | January 2017 |
| Creators | Zähringer, Yasin Hisam Julian |
| Contributors | Kakde, Mahesh Ramesh ; Burns, David John |
| Publisher | King's College London (University of London) |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://kclpure.kcl.ac.uk/portal/en/theses/noncommutative-iwasawa-theory-with-local-conditions-over-distribution-algebras(77477392-e3b4-4eb1-8acc-e59789517360).html |
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