Related papers: On the structure of signed Selmer groups

For an elliptic curve over the rational number field and a prime number $p$, we study the structure of the classical Selmer group of $p$-power torsion points. In our previous paper \cite{Ku6}, assuming the main conjecture and the…

Consider an abelian variety $A$ defined over a global field $K$ and let $L/K$ be a $\Z_p^d$-extension, unramified outside a finite set of places of $K$, with $\Gal(L/K)=\Gamma$. Let $\Lambda(\Gamma):=\Z_p[[\Gamma]]$ denote the Iwasawa…

This paper aims at studying the Iwasawa $\lambda$-invariant of the $p$-primary Selmer group. We study the growth behaviour of $p$-primary Selmer groups in $p$-power degree extensions over non-cyclotomic $\mathbb{Z}_p$-extensions of a number…

Let $p$ be a prime number, and let $K$ be a $p$-adic local field. We study a class of semistable $p$-adic Galois representations of $K$, which we call {\it triangulordinary} because it includes the ordinary ones yet allows non-\'etale…

Let p be an odd prime number, E an elliptic curve over a number field k, and F/k a Galois extension of degree twice a power of p. We study the Z_p-corank rk_p(E/F) of the p-power Selmer group of E over F. We obtain lower bounds for…

Let $E$ be an elliptic curve defined over a number field with good reduction at all primes above a fixed odd prime $p$, where at least one of which is a supersingular prime of $E$. In this paper, we will establish the algebraic functional…

Greenberg examined the local behavior of Iwasawa invariants as functions on the the set of all $\mathbb{Z}_p$-extensions of a number field $F$. Kleine later extended these ideas to explore the variation of Iwasawa invariants in the context…

Let $A$ be an abelian variety defined over a global field $F$ of positive characteristic $p$ and let $\calf/F$ be a $\Z_p^{\N}$-extension, unramified outside a finite set of places of $F$. Assuming that all ramified places are totally…

We extend Kobayashi's formulation of Iwasawa theory for elliptic curves at supersingular primes to include the case $a_p \neq 0$, where $a_p$ is the trace of Frobenius. To do this, we algebraically construct $p$-adic $L$-functions…

We study the behavior of Selmer groups of an elliptic curve $E/\mathbb{Q}$ in finite Galois extensions with prescribed Galois group. Fix a prime $\ell \geq 5$, a finite group $G$ with $\#G = \ell^n$, and an elliptic curve $E/\mathbb{Q}$…

Let $K$ be an imaginary quadratic field where $p$ is inert. Let $E$ be an elliptic curve defined over $K$ and suppose that $E$ has good supersingular reduction at $p$. In this paper, we prove that the plus/minus Selmer group of $E$ over the…

For an elliptic curve, we study how many Selmer groups are cotorsion over the anti-cyclotomic $\mathbb{Z}_p$-extension as one varies the prime $p$ or the quadratic imaginary field in question.

We reveal a new and refined application of (a weaker statement than) the Iwasawa main conjecture for elliptic curves to the structure of Selmer groups of elliptic curves of arbitrary rank. For a large class of elliptic curves, we obtain the…

Let $p$ be a prime number. We investigate a refined version of the Iwasawa main conjectures for rational elliptic curves (and more general Galois representations) over anticyclotomic $\mathbb Z_p$-extensions of imaginary quadratic fields,…

For a number field $F$ and an odd prime number $p,$ let $\tilde{F}$ be the compositum of all $\mathbb{Z}_p$-extensions of $F$ and $\tilde{\Lambda}$ the associated Iwasawa algebra. Let $G_{S}(\tilde{F})$ be the Galois group over $\tilde{F}$…

The Iwasawa $\mu$-invariant of the Selmer group of a residually reducible Galois representation arising from a Hecke eigencuspform is studied. Furthermore, certain Iwasawa-invariants refining the $\mu$-invariant are defined and analyzed. As…

Let p be a prime number and M a quadratic number field, M not equal to Q(\sqrt{p}) if p is congruent to 1 modulo 4. We will prove that for any positive integer d there exists a Galois extension F/Q with Galois group D_{2p} and an elliptic…

Fix two distinct odd primes $p$ and $q$. We study "$p\ne q$" Iwasawa theory in two different settings. Let $K$ be an imaginary quadratic field of class number 1 such that both $p$ and $q$ split in $K$. We show that under appropriate…

We study the rank of the $p$-Selmer group $Sel_p(A/k)$ of an abelian variety $A/k$, where $k$ is a function field. If $K/k$ is a quadratic extension and $F/k$ is a dihedral extension and the $\mathbb{Z}_p$-corank of $Sel_p (A/K)$ is odd, we…

Suppose $F$ is a finite unramified extension of $\mathbb{Q}_p$, and $G$ is the group of $F$-points of a split, connected, reductive group over $F$. Under a natural restriction on $p$, we determine the structure of the graded mod $p$ Iwasawa…