Let E be an elliptic curve defined over $\mathbb {Q}$ with good ordinary reduction at a prime $p\geq 5$ and let F be an imaginary quadratic field. Under appropriate assumptions, we show that the Pontryagin dual of the fine Mordell–Weil group of E over the $\mathbb {Z}_{p}^2$-extension of F is pseudo-null as a module over the Iwasawa algebra of the group $\mathbb {Z}_{p}^2$.

Let p be a fixed prime number, and let F be a global function field with characteristic not equal to p. In this article, we shall study the variation properties of the Sylow p-subgroups of the even K-groups in a p-adic Lie extension of F. When the p-adic Lie extension is assumed to contain the cyclotomic $\mathbb {Z}_p$-extension of F, we obtain growth estimate of these groups. We also establish a duality between the direct limit and inverse limit of the even K-groups.

This paper is concerned with the study of the fine Selmer group of an abelian variety over a $\mathbb{Z}_{p}$-extension which is not necessarily cyclotomic. It has been conjectured that these fine Selmer groups are always torsion over $\mathbb{Z}_{p}[[ \Gamma ]]$, where $\Gamma$ is the Galois group of the $\mathbb{Z}_{p}$-extension in question. In this paper, we shall provide several strong evidences towards this conjecture. Namely, we show that the conjectural torsionness is consistent with the pseudo-nullity conjecture of Coates–Sujatha. We also show that if the conjecture is known for the cyclotomic $\mathbb{Z}_{p}$-extension, then it holds for almost all $\mathbb{Z}_{p}$-extensions. We then carry out a similar study for the fine Selmer group of an elliptic modular form. When the modular forms are ordinary and come from a Hida family, we relate the torsionness of the fine Selmer groups of the specialization. This latter result allows us to show that the conjectural torsionness in certain cases is consistent with the growth number conjecture of Mazur. Finally, we end with some speculations on the torsionness of fine Selmer groups over an arbitrary p-adic Lie extension.

Let f be an elliptic modular form and p an odd prime that is coprime to the level of f. We study the link between divisors of the characteristic ideal of the p-primary fine Selmer group of f over the cyclotomic $\mathbb {Z}_p$ extension of $\mathbb {Q}$ and the greatest common divisor of signed Selmer groups attached to f defined using the theory of Wach modules. One of the key ingredients of our proof is a generalisation of a result of Wingberg on the structure of fine Selmer groups of abelian varieties with supersingular reduction at p to the context of modular forms.

The plus and minus norm groups are constructed by Kobayashi as subgroups of the formal group of an elliptic curve with supersingular reduction, and they play an important role in Kobayashi’s definition of the signed Selmer groups. In this paper, we study the cohomology of these plus and minus norm groups. In particular, we show that these plus and minus norm groups are cohomologically trivial. As an application of our analysis, we establish certain (quasi-)projectivity properties of the non-primitive mixed signed Selmer groups of an elliptic curve with good reduction at all primes above p. We then build on these projectivity results to derive a Kida formula for the signed Selmer groups under a slight weakening of the usual µ = 0 assumption, and study the integrality property of the characteristic element attached to the signed Selmer groups.

Let $p$ be an odd prime number and $E$ an elliptic curve defined over a number field $F$ with good reduction at every prime of $F$ above $p$. We compute the Euler characteristics of the signed Selmer groups of $E$ over the cyclotomic $\mathbb{Z}_{p}$-extension. The novelty of our result is that we allow the elliptic curve to have mixed reduction types for primes above $p$ and mixed signs in the definition of the signed Selmer groups.

Let $p$ be an odd prime. We study the variation of the $p$-rank of the Selmer groups of Abelian varieties with complex multiplication in certain towers of number fields.