These are a very preliminary (and incomplete!) version of the author’s lectures for the 2018 Arizona Winter School on Iwasawa Theory.

Let f be a modular eigenform of weight at least two and let F be a finite abelian extension of Q. Fix an odd prime p at which f is ordinary in the sense that the p Fourier coefficient of f is not divisible by p. In Iwasawa theory, one associates two objects to f over the cyclotomic Zp-extension F∞ of F : a Selmer group Sel(F∞, Af ) (whereAf denotes the divisible version of the two-dimensional G...

Let E be an elliptic curve over Q. We assume that E has good supersingular reduction at a prime p, and for simplicity, assume p is odd and ap = p+ 1− #E(Fp) is zero. Then, as the second author showed, the p-adic L-function Lp,α(E) of E corresponding to α = ±√−p (by Amice-Vélu and Vishik) can be written as Lp,α(E) = f log+p +g logp α by using two Iwasawa functions f and g ∈ Zp[[Gal(Q∞/Q)]] ([20]...