Let p be a prime. In this paper, we use techniques from Iwasawa theory to study questions about rank jump of elliptic curves in cyclic extensions of degree p. We also study growth of the p-primary Selmer group and the Shafarevich–Tate group in cyclic degree-p extensions and improve upon previously known results in this direction.

We study the growth of p-primary Selmer groups of abelian varieties with good ordinary reduction at p in ${{Z}}_p$ -extensions of a fixed number field K. Proving that in many situations the knowledge of the Selmer groups in a sufficiently large number of finite layers of a ${{Z}}_p$ -extension over K suffices for bounding the over-all growth, we relate the Iwasawa invariants of Selmer groups in different ${{Z}}_p$ -extensions of K. As applications, we bound the growth of Mordell–Weil ranks and the growth of Tate-Shafarevich groups. Finally, we derive an analogous result on the growth of fine Selmer groups.

Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex multiplication. Let $p\geq 5$ be a prime in $\mathbb{Q}$ and suppose that $E$ has good ordinary reduction at $p$. We study the dual Selmer group of $E$ over the compositum of the $\text{GL}_{2}$ extension and the anticyclotomic $\mathbb{Z}_{p}$-extension of an imaginary quadratic extension as an Iwasawa module.

We study the growth of $\unicode[STIX]{x0428}$ and $p^{\infty }$-Selmer groups for isogenous abelian varieties in towers of number fields, with an emphasis on elliptic curves. The growth types are usually exponential, as in the ‘positive ${\it\mu}$-invariant’ setting in the Iwasawa theory of elliptic curves. The towers we consider are $p$-adic and $l$-adic Lie extensions for $l\neq p$, in particular cyclotomic and other $\mathbb{Z}_{l}$-extensions.

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.

Let $A$ be an abelian variety over a global field $K$ of characteristic $p\geqslant 0$. If $A$ has nontrivial (respectively full) $K$-rational $l$-torsion for a prime $l\neq p$, we exploit the fppf cohomological interpretation of the $l$-Selmer group $\text{Sel}_{l}\,A$ to bound $\#\text{Sel}_{l}\,A$ from below (respectively above) in terms of the cardinality of the $l$-torsion subgroup of the ideal class group of $K$. Applied over families of finite extensions of $K$, the bounds relate the growth of Selmer groups and class groups. For function fields, this technique proves the unboundedness of $l$-ranks of class groups of quadratic extensions of every $K$ containing a fixed finite field $\mathbb{F}_{p^{n}}$ (depending on $l$). For number fields, it suggests a new approach to the Iwasawa ${\it\mu}=0$ conjecture through inequalities, valid when $A(K)[l]\neq 0$, between Iwasawa invariants governing the growth of Selmer groups and class groups in a $\mathbb{Z}_{l}$-extension.

Extending the idea of Dabrowski [‘On the proportion of rank 0 twists of elliptic curves’, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 483–486] and using the 2-descent method, we provide three general families of elliptic curves over $\mathbb{Q}$ such that a positive proportion of prime-twists of such elliptic curves have rank zero simultaneously.

For an abelian variety $A$ over a number field $k$ we discuss the maximal divisible subgroup of ${\mathrm{H} }^{1} (k, A)$ and its intersection with the subgroup Ш$(A/ k)$. The results are most complete for elliptic curves over $ \mathbb{Q} $.

Let E/ℚ be an elliptic curve and p a prime of supersingular reduction for E. Denote by the anticyclotomic ℤp-extension of an imaginary quadratic field K which satisfies the Heegner hypothesis. Assuming that p splits in K/ℚ, we prove that has trivial Λ-corank and, in the process, also show that and both have Λ-corank two.

Let $H$ be a torsion-free compact $p$-adic analytic group whose Lie algebra is split semisimple. We show that the quotient skewfield of fractions of the Iwasawa algebra $\varLambda_H$ of $H$ has trivial centre and use this result to classify the prime $c$-ideals in the Iwasawa algebra $\varLambda_G$ of $G:=H\times\mathbb{Z}_p$. We also show that a finitely generated torsion $\varLambda_G$-module having no non-zero pseudo-null submodule is completely faithful if and only if it is has no central torsion. This has an application to the study of Selmer groups of elliptic curves.

This paper is a continuation of the author's previous work, where we studied one of the inequalities between the characteristic ideal of the Selmer group and the ideal of the $p$-adic $L$-function predicted by the two-variable Iwasawa main conjecture for a nearly ordinary Hida deformation $\mathcal{T}$. In this paper, we study several properties of the Selmer group and the $p$-adic $L$-function solving some of the open questions raised in the author's previous work. As applications, we have an infinite family of elliptic cuspforms where the cyclotomic Iwasawa main conjecture holds for every cuspform in the family.

This paper concerns the Galois theoretic behavior of the p-primary subgroup SelA(F)p of the Selmer group for an Abelian variety A defined over a number field F in an extension K/F such that the Galois group G(K/F) is a p-adic Lie group. Here p is any prime such that A has potentially good, ordinary reduction at all primes of F lying above p. The principal results concern the kernel and the cokernel of the natural map sK/F′ SelA(F′)p →  SelA(K)pG(K/F′) where F′ is any finite extension of F contained in K. Under various hypotheses on the extension K/F, it is proved that the kernel and cokernel are finite. More precise results about their structure are also obtained. The results are generalizations of theorems of B. Mazurand M. Harris.