By Daniel Delbourgo
The mathematics homes of modular types and elliptic curves lie on the middle of recent quantity concept. This booklet develops a generalisation of the tactic of Euler structures to a two-variable deformation ring. The ensuing thought is then used to check the mathematics of elliptic curves, specifically the Birch and Swinnerton-Dyer (BSD) formulation. 3 major steps are defined: the 1st is to parametrise 'big' cohomology teams utilizing (deformations of) modular symbols. Finiteness effects for giant Selmer teams are then validated. ultimately, at weight , the mathematics invariants of those Selmer teams enable the keep an eye on of information from the BSD conjecture. because the first ebook at the topic, the fabric is brought from scratch; either graduate scholars quantity theorists will locate this a great advent. fabric on the very leading edge of present study is integrated, and numerical examples inspire the reader to interpret summary theorems in concrete situations.
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