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On Selmer groups and refined Stark conjectures
Author(s) -
Burns David,
Livingstone Boomla Alice
Publication year - 2018
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12190
Subject(s) - mathematics , abelian group , conjecture , multiplicative function , pure mathematics , multiplicative group , order (exchange) , elementary abelian group , discrete mathematics , mathematical analysis , finance , economics
We prove that the higher Fitting ideals of Selmer groups of the multiplicative group over abelian extensions of global fields admit a natural direct sum decomposition. Using this result, we then formulate an explicit higher order abelian Stark conjecture that significantly refines and extends the existing theory of such conjectures. We show that our conjecture is best possible, establish a direct link between it and the Tamagawa number conjecture of Bloch and Kato and provide strong supporting evidence, including giving a full proof of the conjecture for all abelian extensions of Q and for all abelian extensions of global function fields.