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Non-Commutative Iwasawa Theory With (φ,Γ)-Local Conditions Over Distribution Algebras

Student thesis: Doctoral ThesisDoctor of Philosophy

In this thesis we formulate a natural non-commutative Iwasawa Main Conjecture for motives which fulfil the Dabrowski-Panchishkin condition on the level of (', 􀀀)-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-(', 􀀀)-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 (', 􀀀)-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.
Original languageEnglish
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Award date2017

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