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Srilakshmi Krishnamoorthy

Publications and source records attributed to Srilakshmi Krishnamoorthy.

17 recordsLinked to original sources
arXiv2026

Consecutive pure fields of the form $\mathbb{Q}\left(\sqrt[l]{a}\right)$ with large class numbers

Let $l$ be a rational prime greater than or equal to $3$ and $k$ be a given positive integer. Under a conjecture due to Langlands and an assumption on upper bound for the regulator of fields of the form $\mathbb{Q}\left(\sqrt[l]a\right)$, we prove that there are atleast $x^{1/l-o(1)} $ integers $1\leq d\leq x$ such that the consecutive pure fields of the form $\mathbb{Q}\left(\sqrt[l]{d+1}\right), \dots ,\mathbb{Q}\left(\sqrt[l]{d+k}\right) $ have arbitrary large class numbers.

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arXiv2025

Modular degree and a conjecture of Watkins

Given an elliptic curve $E/\mathbb{Q}$ of conductor $N$, there exists a surjective morphism $ϕ_E: X_0(N) \to E$ defined over $\mathbb{Q}$. In this article, we discuss the growth of $\mathrm{deg}(ϕ_E)$ and shed some light on Watkins's conjecture, which predicts $2^{\mathrm{rank}(E(\mathbb{Q}))} \mid \mathrm{deg}(ϕ_E)$. Moreover, for any elliptic curve over $\mathbb{F}_q(T)$, we have an analogous modular parametrization relating to the Drinfeld modular curves. In this case, we also discuss growth and the divisibility properties.

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arXiv2025

On an indivisibility version of Iizuka's conjecture

Iizuka's conjecture predicts that, given $m \in \mathbb{N}$ and a prime $p$, there exists infinitely many integers $n$ such that the class numbers of \textit{all} of the following quadratic number fields, \[ \mathbb{Q}(\sqrt{n}),\ \mathbb{Q}(\sqrt{n+1}),\ \ldots,\ \mathbb{Q}(\sqrt{n+m}), \] are divisible by $p$. In this article, given $k$ and $m$, we study the proportion of $n$ such that the class numbers of \textit{none} of the successive fields \[ \mathbb{Q}(\sqrt{n}),\ \mathbb{Q}(\sqrt{n+1}),\ \ldots,\ \mathbb{Q}(\sqrt{n+m}), \] are divisible by \( 3^k \). Moreover, we study the proportion of imaginary biquadratic fields whose class numbers are not divisible by $3$.

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arXiv2025

Low degree extensions with Cyclic class group

Lenstra introduced the notion of the Euclidean ideal class, a generalization of the Euclidean domain that captures cyclic class groups. In this article, we establish the existence of Euclidean ideal classes in abelian quartic fields. As a corollary, we demonstrate that certain biquadratic fields with class number two possess a Euclidean ideal class. Additionally, we investigate the presence of Euclidean ideal classes in specific cubic and quadratic extensions.

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arXiv2024

Watkins's conjecture for elliptic curves with a rational torsion

Watkins's conjecture suggests that for an elliptic curve $E/\mathbb{Q}$, the rank of the group $E(\mathbb{Q})$ of rational points is bounded above by $ν_2 (m_E)$, where $m_E$ is the modular degree associated with $E$. It is known that Watkins's conjecture holds on average. This article investigates the conjecture over certain thin families of elliptic curves. For example, for prime $\ell$, we quantify the elliptic curves featuring a rational $\ell$-torsion that satisfies Watkins's conjecture. Additionally, the study extends to a broader context, investigating the inequality $\mathrm{rank}(E(\mathbb{Q}))+M\leq ν_2(m_E)$ for any positive integer $M$.

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arXiv2024

The divisibility of the class number of the imaginary quadratic fields $\mathbb{Q}(\sqrt{1-2m^k})$

Let $h_{(m,k)}$ be the class number of $\mathbb{Q}(\sqrt{1-2m^k}).$ We prove that for any odd natural number $k,$ there exists $m_0$ such that $k \mid h_{(m,k)}$ for all odd $m > m_0.$ We also prove that for any odd $m \geq 3,$ $k \mid h_{(m,k)}$ (when $k$ and $1-2m^k$ square-free numbers) and $p \mid h_{(m,p)}$ (except finitely many primes $p$). We deduce that for any pair of twin primes $p_1,p_2=p_1+2$, $p_1 \mid h_{(m,p_1)}$ or $p_2 \mid h_{(m,p_2)}.$ For any odd natural number $k$, we construct an infinite family of pairs of imaginary quadratic fields $\mathbb{Q}(\sqrt{d}), \mathbb{Q}(\sqrt{d+1})$ whose class numbers are divisible by $k$, which settles a generalized version of Iizuka's conjecture (cf : Conjecture 2.2) for the case $n=1$.

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arXiv2023

A collage of results on the divisibility and indivisibility of class numbers of quadratic fields

The investigation of the ideal class group $Cl_K$ of an algebraic number field $K$ is one of the key subjects of inquiry in algebraic number theory since it encodes a lot of arithmetic information about K. There is a considerable amount of research on many topics linked to quadratic field class groups notably intriguing aspect is the divisibility of the class numbers. This article discusses a few recent results on the divisibility of class numbers and the Izuka conjecture. We also discuss the quantitative aspect of the Izuka conjecture.

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arXiv2023

Some New Congruences for $\ell$-Regular Multipartitions

For a positive integer $n$, let $B_{\ell_1,\dots,\ell_r}(n)$ denote the number of $(\ell_1,\ell_2,\cdots,\ell_r)$-regular multipartitions of $n$. If $\ell_1=\ell_2=\cdots=\ell_r=\ell$, then we denote $B_{\ell_1,\dots,\ell_r}(n)$ as $B_\ell^{(r)}(n)$. In this paper, we prove several infinite families of congruences satisfied by $B_\ell^{(r)}(n)$ for different values of $\ell$ and $r$.

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arXiv2022

On the existence of a non-principal Euclidean ideal class in biquadratic fields with class number two

Lenstra introduced the notion of a Euclidean ideal class, which is a generalization of the Euclidean domain. Lenstra also proved that the Euclidean ideal in a number field $K$ implies that the class group of $K$ is cyclic. We construct a family of biquadratic fields with a Euclidean ideal whenever the class number is 2. This extends the families given by Graves, Hsu, Chattopadhyay, and Muthukrishnan.

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arXiv2022

The Eisenstein and winding elements of modular symbols for odd square-free level

We explicitly write down the Eisenstein elements inside the space of modular symbols for Eisenstein series with integer coefficients for the congruence subgroups $Γ_0(N)$ with $N$ odd square-free. We also compute the winding elements explicitly for these congruence subgroups. This gives an answer to a question of Merel in these cases. Our results are explicit versions of the Manin-Drinfeld Theorem [Thm. 6]. These results are the generalization of the paper [1] results to odd square-free level.

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arXiv2021

On a conjecture of Sun

A number of the form $x(x+1)/2$ where $x$ is an integer is called a triangular number. Suppose, $N(a_1,\cdots,a_k;n)$ and $T(a_1,\cdots,a_k;n)$ denote the number of ways $n$ can be expressed as $\sum_{i=1}^k a_ix_i^2$ and $\sum_{i=1}^k a_i\frac{x_i(x_i+1)}{2}$, respectively. Z.-H. Sun, in \cite{4}, conjectured some relations between $T(a,b,c;n)$ and $N(a,b,c;8n+a+b+c)$. In this paper, we prove these conjectures using theta function identities. Moreover, we add some new triplets $(a,b,c)$ satisfying these conjectures.

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arXiv2020

On zero-sum subsequences of length exp(G)

Let $G$ be a finite abelian group. Let $g(G)$ be the smallest positive integer $t$ such that every subset of cardinality $t$ of the group $G$ contains a subset of cardinality $\mathrm{exp}(G)$ whose sum is zero. In this paper, we show that if X is a subset of $\mathbb{Z}^2_{2n}$ with cardinality $4n+1$ and $2n$ or $2n-1$ elements of $X$ have the same first coordintes, then $X$ contains a zero sum subset. As an application of our results we prove that $g(\mathbb{Z}^2_6) = 13.$ This settles Gao-Thangaduri's conjecture for the case $n=6.$ We also prove some results towards the general even $n$ cases of the conjecture.

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arXiv2016

A note on the Fourier coefficients of a Cohen-Eisenstein series

We prove a formula for the coefficients of a weight $3/2$ Cohen-Eisenstein series of square-free level $N$. This formula generalizes a result of Gross and in particular, it proves a conjecture of Quattrini. Let $l$ be an odd prime number. For any elliptic curve $E$ defined over $\mathbb{Q}$ of rank zero and square-free conductor $N$, if $l \mid |E(\mathbb{Q})|$, under certain conditions on the Shafarevich-Tate group $III_D$, we show that $l$ divides $|III_D|$ if and only if $l$ divides the class number $h(-D)$ of $\mathbb{Q}(\sqrt{-D}).$

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arXiv2016

On sign changes for almost prime coefficients of half-integral weight modular forms

For a half-integral weight modular form $f = \sum_{n=1}^{\infty} a_f(n)n^{\frac{k-1}{2}} q^n$ of weight $k = l +\frac{1}{2}$ on $Γ_0(4)$ such that $a_f(n)$ ($n$ $\in$ $\mathbb{N}$) are real, we prove for a fixed suitable natural number $r$ that $a_f(n)$ changes sign infinitely often as $n$ varies over numbers having at most $r$ prime factors, assuming the analog of the Ramanujan conjecture for half-integral weight forms.

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arXiv2015

The Eisenstein elements of modular symbols for level product of two distinct odd primes

We explicitly write down the Eisenstein elements inside the space of modular symbols for Eisenstein series with integer coefficients for the congruence subgroups Γ_0 (pq) with p and q distinct odd primes, giving an answer to a question of Merel in these cases. We also compute the winding elements explicitly for these congruence subgroups. Our results are explicit versions of the Manin-Drinfeld Theorem.

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arXiv2015

Lifting Congruences to weight 3/2

Given a congruence of Hecke eigenvalues between newforms of weight $2$, we prove, under certain conditions, a congruence between corresponding weight-$3/2$ forms.

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