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The topics that we will discuss have their origin in Mazur's synthesis of the theory of elliptic curves and Iwasawa's theory of Z Z p -extensions in the early 1970s. We first recall some results from Iwasawa's theory. Suppose that F is a finite extension of Q and that F ∞ is a Galois extension of F such that Gal(F ∞ /F ) ∼ = Z Z p , the additive group of p-adic integers, where p is any prime. Equivalently, F ∞ = n≥0 F n , where, for n ≥ 0, F n is a cyclic extension of F of degree p n andthe class number of F n , p en the exact power of p dividing h n . Then Iwasawa proved the following result.Theorem 1.1. There exist integers λ, µ, and ν, which depend only on F ∞ /F , such that e n = λn + µp n + ν for n ≫ 0.

Let p be an odd prime. Suppose that E is a modular elliptic curve/Q with good ordinary reduction at p. Let Q_{oo} denote the cyclotomic Z_p-extension of Q. It is conjectured that Sel_E(Q_{oo}) is a cotorsion Lambda-module and that its characteristic ideal is related to the p-adic L-function associated to E. Under certain hypotheses we prove that the validity of these conjectures is preserved by congruences between the Fourier expansions of the associated modular forms.Comment: Abstract added in migration (taken from Greenberg's web site

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