As a natural generalization of the notion of 'higher rank Euler system', we develop a theory of 'higher special elements' in the exterior power biduals of the Galois cohomology of p-adic representations. We show, in particular, that such elements encode detailed information about the structure of Galois cohomology groups and are related by families of congruences involving natural height pairings on cohomology. As a first concrete application of the approach, we use it to refine, and extend, a variety of existing results and conjectures concerning the values of derivatives of Dirichlet L-series.
We extend the work of [LLSTT21] and study the change of µ-invariants, with respect to a finite Galois p-extension K ′ /K, of an ordinary abelian variety A over a Z d p -extension of global fields L/K (whose characteristic is not necessarily positive) that ramifies at a finite number of places at which A has ordinary reductions. We obtain a lower bound for the µ-invariant of A along LK ′ /K ′ and deduce that the µ-invariant of an abelian variety over a global field can be chosen as big as needed. Finally, in the case of elliptic curve over a global function field that has semi-stable reduction everywhere we are able to improve the lower bound in terms of invariants that arise from the supersingular places of A and certain places that split completely over L/K.
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