arXiv subjects

Chandrakant Aribam

Publications and source records attributed to Chandrakant Aribam.

5 recordsLinked to original sources

arXiv2026

Variation of Iwasawa Invariants for Ordinary Representations

Let $K$ be a number field and $p$ be an odd prime. Greenberg introduced a natural topology on the space of all $\mathbb{Z}_p$-extensions of $K$ and established several boundedness results for the classical Iwasawa invariants. We extend this framework to compare the Iwasawa invariants of Selmer groups attached to an ordinary $p$-adic representation across $\mathbb{Z}_p$-extensions lying in a Greenberg neighbourhood in the sense of Greenberg. We also establish analogous results for the fine Selmer groups. Finally, in a neighbourhood of the cyclotomic $\mathbb{Z}_p$-extension, we provide evidence for the expected connection between the characteristic ideal of the Selmer group and the conjectural $p$-adic $L$-function introduced by Disegni.

math.NT↗

arXiv2023

Bipartite Euler Systems for certain Galois Representations

Let $E/\mathbb{Q}$ be an elliptic curve with ordinary reduction at a prime $p$, and let $K$ be an imaginary quadratic field. The anticyclotomic Iwasawa main conjecture, depending upon the sign of the functional equation of $L(E/K,s)$, predicts the behavior of Selmer group of $E/\mathbb{Q}$ along the anticyclotomic tower of $K$. Some of the crucial ideas of Bertolini and Darmon on this conjecture have been abstracted by Howard into an axiomatic set-up through a notion of Bipartite Euler systems, assuming that $E[p]$ is an irreducible representation of $G_{K}$. We generalize this work by assuming only $(E[p])^{G_K}=0$. We use the results of Howard, Nekovář and Castella \emph{et al}., along with those of Mazur and Rubin on Kolyvagin systems to show one divisibility of the anticyclotomic main conjecture, for both the signs. The other divisibility can be reduced to proving the nonvanishing of sufficiently many $p$-adic $L$-functions attached to a family of congruent modular forms.

math.NT↗

arXiv2022

Galois Cohomology for Lubin-Tate $(φ_q,Γ_{LT})$-modules over Coefficient rings

The classification of the local Galois representations using $(φ,Γ)$-modules by Fontaine has been generalized by Kisin and Ren over the Lubin-Tate extensions of local fields using the theory of $(φ_q,Γ_{LT})$-modules. In this paper, we extend the work of (Fontaine) Herr by introducing a complex which allows us to compute cohomology over the Lubin-Tate extensions and compare it with the Galois cohomology groups. We further extend that complex to include certain non-abelian extensions. We then deduce some relations of this cohomology with those arising from $(ψ_q,Γ_{LT})$-modules. We also compute the Iwasawa cohomology over the Lubin-Tate extensions in terms of $ψ_q$-operator acting on the étale $(φ_q,Γ_{LT})$-module attached to the local Galois representation. Moreover, we generalize the notion of $(φ_q,Γ_{LT})$-modules over the coefficient ring $R$ and show that the equivalence given by Kisin and Ren extends to the Galois representations over $R$. This equivalence allows us to generalize our results to the case of coefficient rings.

math.NT↗

arXiv2020

Noncommutative Iwasawa theory arising from Hecke algebras

Let $p$ be an odd prime and $f$ be a nearly ordinary Hilbert modular Hecke eigenform defined over a totally real field $F$. Let $\mathbb{I}$ be an irreducible component of the universal nearly ordinary or locally cyclotomic deformation of the representation of $\mathrm{Gal}_F$ that is associated to $f$. We study the deformation rings over a $p$-adic Lie extension $F_\infty$ that contains the cyclotomic $\mathbb{Z}_p$-extension of $F$. More precisely, we prove a control theorem about these rings. We introduce a category $\mathfrak{M}_{\mathcal H}^{\mathbb I}(\mathcal G)$, where $\mathcal G=\mathrm{Gal}(F_\infty/F)$ and $\mathcal H=\mathrm{Gal}(F_\infty/F_{cyc})$, which is the category of modules which are torsion with respect to a certain Ore set, which generalizes the Ore set introduced by Venjakob. For Selmer groups which are in this category, we formulate a Main conjecture in the spirit of Noncommutative Iwasawa theory. We then set up a strategy to prove the conjecture by generalizing work of Burns, Kato, Kakde, and Ritter and Weiss. This requires appropriate generalizations of results of Oliver and Taylor, and Oliver on Logarithms of certain $K$-groups, which we have presented here.

math.NT↗

arXiv2017

Root numbers and parity of local Iwasawa invariants

Given two elliptic curves $E_1$ and $E_2$ defined over the field of rational numbers, $\mathbb{Q}$, with good reduction at an odd prime $p$ and equivalent mod $p$ Galois representation, we compare the $p$-Selmer rank, global and local root numbers of $E_1$ and $E_2$ over number fields.

math.NT↗