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Anupam Saikia

Publications and source records attributed to Anupam Saikia.

17 recordsLinked to original sources
arXiv2025

Stability of 2-class groups in the $\mathbb{Z}_2$-extension of certain real biquadratic fields

Greenberg's conjecture on the stability of $\ell$-class groups in the cyclotomic $\mathbb{Z}_{\ell}$-extension of a real field has been proven for various infinite families of real quadratic fields for the prime $\ell=2$. In this work, we consider an infinite family of real biquadratic fields $K$. With some extensive use of elementary group theoretic and class field theoretic arguments, we investigate the $2$-class groups of the $n$-th layers $K_n$ of the cyclotomic $\mathbb{Z}_2$-extension of $K$ and verify Greenberg's conjecture. We also relate capitulation of ideal classes of certain sub-extensions of $K_n$ to the relative sizes of the $2$-class groups.

math.NT↗
arXiv2024

$\mathbb{Z}_2$-extension of real quadratic fields with $\mathbb{Z}/2\mathbb{Z}$ as $2$-class group at each layer

Let $K= \mathbb{Q}(\sqrt{d})$ be a real quadratic field with $d$ having three distinct prime factors. We show that the $2$-class group of each layer in the $\mathbb{Z}_2$-extension of $K$ is $\mathbb{Z}/2\mathbb{Z}$ under certain elementary assumptions on the prime factors of $d$. In particular, it validates Greenberg's conjecture on the vanishing of the Iwasawa $λ$-invariant for a new family of infinitely many real quadratic fields.

math.NT↗
arXiv2024

Unramified Iwasawa module of $\mathbb{Z}_2$-extension of certain quadratic fields with a bounded quotient

We consider an infinite family of real quadratic fields $k$ where the discriminant has three distinct odd prime factors, and the prime 2 splits. We show that the unramified Iwasawa module $X(k_{\infty})$ associated with the $\mathbb{Z}_2$-extension of $k$ has a bounded quotient. Thus, we also verify Greenberg's conjecture on the vanishing of Iwasawa invariants for such fields and obtain a finer structure for $X(k_{\infty})$.

math.NT↗
arXiv2023

A brief survey of recent results on Pólya groups

The Pólya group of an algebraic number field is a particular subgroup of the ideal class group. This article provides an overview of recent results on Pólya groups of number fields, their connection with the ring of integer-valued polynomials and touches upon some results on number fields having large Pólya groups. For the sake of completeness, we have included the proof of Zantema's theorem which laid the foundation to determine the Pólya groups of many finite Galois extensions over $\mathbb{Q}$. Towards the end of the article, we provide an elementary proof of a weaker version of a recent result of Cherubini et al.

math.NT↗
arXiv2023

On the structure and stability of ranks of $2$-class groups in cyclotomic $\mathbb{Z}_{2}$-extensions of certain real quadratic fields

For a real quadratic field $K= \mathbb{Q}(\sqrt{d})$ with discriminant $D_{K}$ having four distinct prime factors, we study the structure of the $2$-class group $A(K_{1})$ of the first layer $K_{1} = \mathbb{Q}(\sqrt{2},\sqrt{d})$ of the cyclotomic $\mathbb{Z}_{2}$-extension of $K$. With some suitably convenient assumptions on the rank and the order of $A(K_{1})$, we characterize $K$ for which the $2$-class group $A(K)$ is isomorphic to $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$. We infer that the $2$-ranks of the class groups in each layer stabilizes by virtue of a result of Fukuda. This also provides an alternate way to establish that the Iwasawa $μ$-invariant of $K$ vanishes. In some cases, we also provide sufficient conditions on the constituent prime factors of $D_{K}$ that imply $A(K) \simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$, $A(K_{1}) \simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/4\mathbb{Z}$ and $A(K^{\prime}) \simeq \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$, where $K^{\prime} = \mathbb{Q}(\sqrt{2d})$. This extends some results obtained by Mizusawa.

math.NT↗
arXiv2022

On the $p$-rationality of consecutive quadratic fields

In 2016, in the work related to Galois representations, Greenberg conjectured the existence of multi-quadratic $p$-rational number fields of degree $2^{t}$ for any odd prime number $p$ and any integer $t \geq 1$. Using the criteria provided by him to check $p$-rationality for abelian number fields, certain infinite families of quadratic, biquadratic and triquadratic $p$-rational fields have been shown to exist in recent years. In this article, for any integer $k \geq 1$, we build upon the existing work and prove the existence of infinitely many prime numbers $p$ for which the imaginary quadratic fields $\mathbb{Q}(\sqrt{-(p - 1)}),\ldots,\mathbb{Q}(\sqrt{-(p - k)})$ and $\mathbb{Q}(\sqrt{-p(p - 1)}),\ldots, \mathbb{Q}(\sqrt{-p(p - k)})$ are all $p$-rational. This can be construed as analogous results in the spirit of Iizuka's conjecture on the divisibility of class numbers of consecutive quadratic fields. We also address a similar question of $p$-rationality for two consecutive real quadratic fields by proving the existence of infinitely many $p$-rational fields of the form $\mathbb{Q}(\sqrt{p^{2} + 1})$ and $\mathbb{Q}(\sqrt{p^{2} + 2})$. The result for imaginary quadratic fields is accomplished by producing infinitely many primes for which the corresponding consecutive discriminants have large square divisors and the same for real quadratic fields is proven using a result of Heath-Brown on the density of square-free values of polynomials at prime arguments.

math.NT↗
arXiv2022

Construction of an infinite family of elliptic curves of 2-selmer rank 1 from heron triangles

Given any positive integer n, it is well known that there always exist triangles with rational sides a, b and c such that the area of the triangle is n. Assuming finiteness of the Shafarevich-Tate group, we first construct a family of infinitely many Heronian elliptic curves of rank exactly 1 from Heron triangles of a certain type. We also explicitly produce a separate family of infinitely many Heronian elliptic curves with 2-Selmer rank lying between 1 and 3.

math.NT↗
arXiv2021

On the $p$-ranks of the ideal class groups of imaginary quadratic fields

For a prime number $p \geq 5$, we explicitly construct a family of imaginary quadratic fields $K$ with ideal class groups $Cl_{K}$ having $p$-rank ${\rm{rk}_{p}(Cl_{K})}$ at least $2$. We also quantitatively prove, under the assumption of the $abc$-conjecture, that for sufficiently large positive real numbers $X$ and any real number $\varepsilon$ with $0 < \varepsilon < \frac{1}{p - 1}$, the number of imaginary quadratic fields $K$ with the absolute value of the discriminant $d_{K}$ $\leq X$ and ${\rm{rk}_{p}(Cl_{K})} \geq 2$ is $\gg X^{\frac{1}{p - 1} - \varepsilon}$. This improves the previously known lower bound of $X^{\frac{1}{p} - \varepsilon}$ due to Byeon and the recent bound $X^{\frac{1}{p}}/(\log X)^{2}$ due to Kulkarni and Levin.

math.NT↗
arXiv2021

Totally real bi-quadratic fields with large Pólya groups

For an algebraic number field $K$ with ring of integers $\mathcal{O}_{K}$, an important subgroup of the ideal class group $Cl_{K}$ is the {\it Pólya group}, denoted by $Po(K)$, which measures the failure of the $\mathcal{O}_{K}$-module $Int(\mathcal{O}_{K})$ of integer-valued polynomials on $\mathcal{O}_{K}$ from admitting a regular basis. In this paper, we prove that for any integer $n \geq 2$, there are infinitely many totally real bi-quadratic fields $K$ with $|Po(K)| = 2^{n}$. In fact, we explicitly construct such an infinite family of number fields. This extends an infinite family of bi-quadratic fields with Pólya group $\mathbb{Z}/2\mathbb{Z}$ given by the authors in \cite{self-ja}. This also provides an infinite family of bi-quadratic fields with class numbers divisible by $2^{n}$.

math.NT↗
arXiv2021

Non-Pólya bi-quadratic fields with an Euclidean ideal class

For an integral domain $R$, the {\it ring of integer-valued polynomials} over $R$ consists of all polynomials $f(X) \in R[X]$ such that $f(R) \subseteq R$. An interesting case to study is when $R$ is a Dedekind domain, in particular when $R$ is the ring of integers of an algebraic number field. An algebraic number field $K$ with ring of integers $\mathcal{O}_{K}$ is said to be a Pólya field if the $\mathcal{O}_{K}$-module of integer-valued polynomials on $K$ admits a regular basis. Associated to $K$ is a subgroup $Po(K)$ of the ideal class group $Cl_{K}$, known as the {\it Pólya group of $K$}, that measures the failure of $K$ from being a Pólya field. In this paper, we prove the existence of three pairwise distinct totally real bi-quadratic fields, each having Pólya group isomorphic to $\mathbb{Z}/2\mathbb{Z}$. This extends the previously known families of number fields considered by Heidaryan and Rajaei in \cite{rajaei-jnt} and \cite{rajaei}. Our results also establish that under mild assumptions, the possibly infinite families of bi-quadratic fields having a non-principal Euclidean ideal class, considered in \cite{self-jnt}, fail to be Pólya fields.

math.NT↗
arXiv2020

Simultaneous indivisibility of class numbers of pairs of real quadratic fields

For a square-free integer $t$, Byeon \cite{byeon} proved the existence of infinitely many pairs of quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{tD})$ with $D > 0$ such that the class numbers of all of them are indivisible by $3$. In the same spirit, we prove that for a given integer $t \geq 1$ with $t \equiv 0 \pmod {4}$, a positive proportion of fundamental discriminants $D > 0$ exist for which the class numbers of both the real quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{D + t})$ are indivisible by $3$. This also addresses the complement of a weak form of a conjecture of Iizuka in \cite{iizuka}. As an application of our main result, we obtain that for any integer $t \geq 1$ with $t \equiv 0 \pmod{12}$, there are infinitely many pairs of real quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{D + t})$ such that the Iwasawa $λ$-invariants associated with the basic $\mathbb{Z}_{3}$-extensions of both $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{D + t})$ are $0$. For $p = 3$, this supports Greenberg's conjecture which asserts that $λ_{p}(K) = 0$ for any prime number $p$ and any totally real number field $K$.

math.NT↗
arXiv2019

On a conjecture Mordell

A conjecture of Mordell states that if $p$ is a prime and $p$ is congruent to $3$ mod $4$, then $p$ does not divide $y$ where $(x,y)$ is the fundamental solution to $x^{2}-py^{2}=1$. The conjecture has been verified for primes not exceeding $10^{7}$. In this article, we show that Mordell's conjecture holds for four conjecturally infinite families of primes.

math.NT↗
arXiv2017

Bounding Hilbert coefficients of parameter ideals

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ and depth R$\geq d-1$. Let $Q$ be a parameter ideal of $R$. In this paper, we derive uniform lower and upper bounds for the Hilbert coefficient $e_i(Q)$ under certain assumptions on the depth of associated graded ring $G(Q)$. For $2\leq i\leq d $, we show that (1) $e_i(Q)\leq 0$ provided depth $G(Q)\geq d-2$ and (2) $e_i(Q)\geq -λ_R(H_{\mathfrak{m}}^{d-1}(R))$ provided depth $G(Q)\geq d-1$. It is proved that $e_3(Q)\leq 0$. Further, we obtain a necessary condition for the vanishing of the last coefficient $e_d(Q)$. As a consequence, we characterize the vanishing of $e_2(Q)$. Our results generalize \cite[Theorem 3.2]{goto-ozeki} and \cite[Corollary 4.5]{Lori}.

math.AC↗
arXiv2010

Little Dragon Two: An efficient Multivariate Public Key Cryptosystem

In 1998 [8], Patarin proposed an efficient cryptosystem called Little Dragon which was a variant a variant of Matsumoto Imai cryptosystem C*. However Patarin latter found that Little Dragon cryptosystem is not secure [8], [3]. In this paper we propose a cryptosystem Little Dragon Two which is as efficient as Little Dragon cryptosystem but secure against all the known attacks. Like Little Dragon cryptosystem the public key of Little Dragon Two is mixed type that is quadratic in plaintext and cipher text variables. So the public key size of Little Dragon Two is equal to Little Dragon Cryptosystem. Our public key algorithm is bijective and can be used for both encryption and signatures.

cs.CR↗
arXiv2009

On units generated by Euler systems

In the context of cyclotomic fields, it is still unknown whether there exist Euler systems other than the ones derived from cyclotomic units. Nevertheless, we first give an exposition on how norm-compatible units are generated by any Euler system, following work of Coates. Then we prove that the units obtained from Euler systems and the cyclotomic units generate the same $\mathbb{Z}_{p}$-module for any odd prime $p$. The techniques adopted for the Iwasawa theoretic proof in latter part of this article originated in Rubin's work on main conjectures of Iwasawa theory.

math.NT↗