Elliptic Curves and Big Galois Representations

Abstract

The arithmetic properties of modular forms and elliptic curves lie at the heart of modern number theory. This book develops a generalisation of the method of Euler systems to a two-variable deformation ring. The resulting theory is then used to study the arithmetic of elliptic curves, in particular the Birch and Swinnerton-Dyer (BSD) formula. Three main steps are outlined: the first is to parametrise 'big' cohomology groups using (deformations of) modular symbols. Finiteness results for big Selmer groups are then established. Finally, at weight two, the arithmetic invariants of these Selmer groups allow the control of data from the BSD conjecture. As the first book on the subject, the material is introduced from scratch; both graduate students and professional number theorists will find this an ideal introduction. Material at the very forefront of current research is included, and numerical examples encourage the reader to interpret abstract theorems in concrete cases.

Rankin–Eisenstein classes and explicit reciprocity laws

“…While this can be proved directly, by purely geometric methods, we give an alternative argument based on Kato's explicit reciprocity law for his GL 2 Euler system, as this is more in keeping with the flavour of the present paper and involves less translation between different normalisations. This argument is closely based on work of Daniel Delbourgo [Del08], in particular Theorem 6.4 of op.cit.. 9.1. Kato's GL 2 Euler system.…”

Equivariant Iwasawa theory for elliptic curves

“…There is also a work by Delbourgo [6, Appendix A] which generalize [18, (12.5)] to K = Q(µ m ). The method of [6] is similar to (the first half of) the following proof, but the integrality is not asserted in [6].…”

“…γ is independent of the choices of α 1 , α 2 , c, d. We consider the integrality of z (p) γ . The key tool is [18, (12.6)], generalized in [6,p.255,Key Claim]. The former treats K = Q and the latter treats K = Q(µ m ), but the following assertion for general K can be deduced from the latter.…”

P-adic weight pairings on pro-Jacobians

“…In a forthcoming work [2], we consider "refined" Selmer groups ord and H 1 f Q, e * ord · Ta p ( t J r ) which have slightly stricter local conditions than their Bloch-Kato cousins. Skipping the precise definitions, we now describe a p-adic pairing linking these two objects.…”

“…Taking Theorems 4 and 5 in tandem, if L(E, 1) = 0 then ord p III E [p ∞ ] ord p C(p, T ∞ ) + δ p,T ∞ + ord p #E(Q) 2 Tam Q (E) × L(E, 1) Ω +…”