We construct an Euler system—a compatible family of global cohomology classes—for the Galois representations appearing in the geometry of Hilbert modular surfaces. If a conjecture of Bloch and Kato on injectivity of regulator maps holds, this Euler system is nontrivial, and we deduce bounds towards the Iwasawa main conjecture for these Galois representations.

We study the asymptotic behaviour of the Bloch–Kato–Shafarevich–Tate group of a modular form $f$ over the cyclotomic ${{\mathbb{Z}}_{p}}$ -extension of $\mathbb{Q}$ under the assumption that $f$ is non-ordinary at $p$ . In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using $p$ -adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara, and Sprung for supersingular elliptic curves.

We construct an Euler system attached to a weight 2 modular form twisted by a Grössencharacter of an imaginary quadratic field $K$, and apply this to bounding Selmer groups.

We study the Selmer group of an elliptic curve over an admissible p-adic Lie extension of a number field F. We give a formula for the Akashi series attached to this module, in terms of the corresponding objects for the cyclotomic ℤp-extension and certain correction terms. This extends our earlier work [16], in particular since it applies to elliptic curves having split multiplicative reduction at some primes above p, in which case the Akashi series can have additional zeros.

Let $E$ be an elliptic curve defined over a number field $F$. The paper concerns the structure of the $p^{\backslash}\infty$-Selmer group of $E$ over $p$-adic Lie extensions $F_{\backslash}\infty$ of $F$ which are obtained by adjoining to $F$ the $p$-division points of an abelian variety $A$ defined over $F$. The main focus of the paper is the calculation of the Gal$(F_{\backslash}\infty/F)$-Euler characteristic of the $p^{\backslash}\infty$-Selmer group of $E$. The main theory is illustrated with the example of an elliptic curve of conductor 294.