We investigate a question of Burns and Sano concerning the structure of the module of Euler systems for a general p-adic representation. Assuming the weak Leopoldt conjecture, and the vanishing of μ-invariants of natural Iwasawa modules, we obtain an Iwasawa-theoretic classification criterion for Euler systems which can be used to study this module. This criterion, taken together with Coleman's conjecture on circular distributions, leads us to pose a refinement of the aforementioned question for which we provide strong, and unconditional, evidence. We furthermore answer this question in the affirmative in many interesting cases in the setting of the multiplicative group over number fields. As a consequence of these results, we derive explicit descriptions of the structure of the full collection of Euler systems for the situations in consideration.

It is a pleasure for the presently named author to thank David Burns for suggesting this project to him as well as for numerous stimulating conversations and comments concerning the present manuscript. He also wishes to thank the anonymous referee for their comments and suggestions which have improved the exposition of the article.

The author would additionally like to extend his gratitude to Dominik Bullach for his careful reading of earlier versions of this article, as well as Andrew Graham, Daniel Macias Castillo, and Takamichi Sano for their helpful comments and discussions.

The present article is based on material contained in the author's King's College London Ph.D. thesis [17], the preparation and writing of which was financially supported by the Engineering and Physical Sciences Research Council [EP/N509498/1].