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 prove an interpolation formula for the values of certain $p$ -adic Rankin-Selberg $L$ -functions associated with non-ordinary modular forms.

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 show that the image of the adelic Galois representation attached to a non-CM modular form is open in the adelic points of a suitable algebraic subgroup of GL2 (defined by F. Momose). We also show a similar result for the adelic Galois representation attached to a finite set of modular forms.

Four new species of cyathaspidid extend the range of the genus Poraspis back into the Late Silurian and indicate that the Canadian arctic was its center of origin. Poraspis heintzae n. sp., P. cracens n. sp., P. thules n. sp. and P. parmula n. sp. also increase the known geographic range of the genus which had previously been reported only from the District of Mackenzie (NWT, Canada), Spitsbergen, western and eastern Europe. The identification of P. sericea from the upper member of the Peel Sound Formation substantiates correlation of this interval with the crouchi zone of the Anglo-Welsh Borders, and provides another rare example of a species common to the Canadian arctic and European successions.

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 give a computationally effective criterion for determining whether a finite-index subgroup of $\mathrm{SL}_2(\mathbf{Z})$ is a congruence subgroup, extending earlier work of Hsu for subgroups of $\mathrm{PSL}_2(\mathbf{Z})$ .

I give an algorithm for computing the full space of automor-phic forms for definite unitary groups over ℚ, and apply this to calculate the automorphic forms of level G(hat{Z}) and various small weights for an example of a rank 3 unitary group. This leads to some examples of various types of endoscopic lifting from automorphic forms for U1 × U1 × U1 and U1 × U2, and to an example of a non-endoscopic form of weight (3, 3) corresponding to a family of 3-dimensional irreducible ℓ-adic Galois representations. I also compute the 2-adic slopes of some automorphic forms with level structure at 2, giving evidence for the local constancy of the slopes.