By Daniel Delbourgo

The mathematics homes of modular types and elliptic curves lie on the middle of contemporary quantity concept. This booklet develops a generalisation of the tactic of Euler platforms to a two-variable deformation ring. The ensuing conception is then used to review the mathematics of elliptic curves, particularly 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 tested. eventually, at weight , the mathematics invariants of those Selmer teams let the keep an eye on of knowledge from the BSD conjecture. because the first e-book at the topic, the fabric is brought from scratch; either graduate scholars quantity theorists will locate this a fantastic advent. fabric on the very vanguard of present study is incorporated, and numerical examples inspire the reader to interpret summary theorems in concrete situations.

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**Example text**

4 Norm relations in K-theory In the ﬁrst section we deﬁned the p-adic L-function, and recalled some well-known conjectures about its behaviour at critical points. The explicit local machinery of Perrin-Riou then allowed the conversion of norm-compatible cocycles into power series, convergent on the open unit disk. Question. What is the correct input into this machine, such that the output is the p-adic L-function attached to a Hecke eigenform? The answer was provided by Kato [Ka1], who proved the norm-compatibility of zeta-elements lying in K2 of modular curves.

E. the ordinary case, then in fact Lp (f, ψ, s) is an Iwasawa function (a continuous function on the unit disk with bounded sup-norm). A special case For the time being, we concern ourselves exclusively with modular elliptic curves. Let E be an elliptic curve over Q, and fE its associated newform of weight two. In this situation there is only one critical point. Consequently, the p-adic L-function should interpolate Dirichlet twists of the algebraic part of the L-value at s = 1. Put Lp (E, ψ, s) = Lp,αp ,Ω± (fE , ψ, s), so E pn 1 × Lp (E, ψ, 1) = αpn whilst at trivial ψ = 1, 1− Lp (E, 1) = 1− ψ −1 (m)e2πim/p m=1 1 αp 2 1 ap (E) × L(E,1) Ω+ E × L(E,1) Ω+ E n × L(E, ψ, j + 1) sign(ψ) ΩE if p does not divide NE if p exactly divides NE .

N−s L(f, s) := which converges for Re(s) > n=1 k+2 . 2 Provided f is a simultaneous eigenform for the whole Hecke algebra, its L-function further admits an Euler product expansion in the same right half-plane. 28. (Hecke) If f is a primitive form of level N and weight k ≥ 2, then the completed L-function Λ(f, s) := N s/2 (2π)−s Γ(s) L(f, s) has an analytic continuation to all of C, and satisﬁes the functional equation Λ f, s = ik Λ f WN , k − s . Note that f WN = i−k w∞ f ∗ where the dual cusp form f ∗ has the q-expansion ∞ f ∗ (z) = n=1 an (f )q n , and the complex sign w∞ ∈ {±1}.