Topics in Arithmetic Geometry

The theory of 'Euler systems' was introduced by Kolyvagin in [1] and was later systematically developed by Rubin in [2] and by Mazur and Rubin in [3].
It has since played an indispensable role in the proof of many of the most spectacular, and most famous, results in arithmetic geometry concerning relations between the special values of L-series and the structure of the Selmer groups of associated p-adic representations over number fields. With a view towards extending the range of such applications to important new classes of examples and, in particular, to attack key problems that arise in deformation theory, the theory of Euler systems has also recently been expanded by Mazur and Rubin in [4] to a natural 'higher rank' setting. This aspect of the theory is currently undergoing rapid development and is attracting the interest of many leading researchers.
However, in order to apply the general theory in any given arithmetic setting, one must first supply an explicit example of an Euler system (of the relevant rank) that is related to the values of L-series. Unfortunately, the search for such examples has so far proven extremely difficult!
The most classical example of an Euler system is provided by the so-called 'cyclotomic elements' that arise in the multiplicative group of abelian extensions of the field of rational numbers. In this context, earlier work of Robert Coleman gave a beautiful reinterpretation of the relevant properties of cyclotomic elements in terms of so-called 'circular distributions' and this led him to conjecture that every Euler system that could arise in the aforementioned setting must arise in a straightforward fashion from the Euler system of cyclotomic elements (see [5] and [6]). The validity of this striking conjecture would therefore offer a precise explanation for the mysterious scarcity of Euler systems. It would also seem reasonable to believe that any methods leading to a proof of the conjecture could also shed light on the difficulty of obtaining Euler systems in other significant settings.
During the course of my PhD I will be investigating whether or not one can formulate a natural ana- logue of Coleman's conjecture in the very general setting of higher rank Euler systems that arise in the multiplicative group of abelian extensions of arbitrary number fields. In this setting I will first aim to construct a module of so-called 'basic' Euler systems, elements of which can be seen as a natural generalisation of cyclotomic elements. This module of basic Euler systems will be constructed as a 'global' analogue of its namesake that is constructed in the setting of p-adic representations by Burns and Sano in [7]. I will then seek to precisely formulate the aforementioned generalisation of Coleman's conjecture in this setting. Such a conjecture should, modulo minor technical details, in effect state that every higher rank Euler system in this setting is necessarily obtained from a basic Euler system in a straightforward way.
I will then aim to provide evidence for this conjecture by building upon the techniques developed in the setting of Coleman's original conjecture by Seo in [9] and [10].
I expect a key role to be played by the recent proof of Burns, Sakamato and Sano in [8] of the main conjecture of Mazur and Rubin concerning the theory of higher rank Euler systems.