arXiv subjects

Sudipa Mondal

Publications and source records attributed to Sudipa Mondal.

11 recordsLinked to original sources

arXiv2026

Towards a generalized Maeda conjecture for modular forms with quadratic nebentypus

We establish a lower bound for the number of non-CM Galois orbits of newforms in $S_k(N,Ψ)$ with non-trivial quadratic nebentypus $Ψ$ for sufficiently large weights. Extending the work of Dieulefait, Pacetti, and Tsaknias in the trivial nebentypus setting, we analyze the restrictions imposed by the quadratic character on local inertial types and determine the number of admissible Galois orbits of such types. We further prove that Atkin-Li pseudo-eigenvalues are Galois equivariant and hence, up to a natural equivalence relation, define a global Galois invariant. Together with existence results for newforms having prescribed local behavior, these invariants yield a lower bound for the number of non-CM Galois orbits by counting compatible pairs of local-global invariants. Finally, computations in small weights show that this lower bound is not always attained, indicating that certain local equivalences are not realized globally by Galois conjugation over the coefficient field of the modular form.

math.NT↗

arXiv2026

Monsky Matrix and 2-Selmer rank

In this article, we produce infinite families of non-congruent numbers in the residue class of $1,2,$ and $3$ modulo $8$ with arbitrarily many triples or quadruples prime factors. In short, we use Monsky matrix to show that the $2$-Selmer rank of the corresponding congruent number elliptic curve is zero. We also establish some quantitative results to conclude that each such family contains infinitely many non-congruent numbers.

math.NT↗

arXiv2026

$2$-Selmer groups, $2$-class groups, and congruent numbers

In this article, we study necessary conditions for certain square-free integers to be congruent numbers. Our method uses divisibility properties of class numbers of related imaginary quadratic fields. We first consider positive square-free integers of the form $n = p_1 p_2 \cdots p_t q,$ where each prime $p_i \equiv 5 \pmod{8}$ and $q \equiv 7 \pmod{8}$. We show that if such an integer $n$ is a congruent number, then the class number $h(-n)$ of the quadratic field $\mathbb{Q}(\sqrt{-n})$ satisfies a specific divisibility condition. Furthermore, we provide quantitative lower bounds on the number of non-congruent numbers of this form. Next, we study integers of the form $n = p_1 p_2 \cdots p_t q,$ with $p_i \equiv 5 \pmod{8}$ and $q \equiv 3 \pmod{8}$. Assuming that $n$ is a congruent number, we obtain a congruence modulo powers of $2$ between the class numbers of the fields $\mathbb{Q}(\sqrt{-n})$ and $\mathbb{Q}\!\left(\sqrt{-p_1 p_2 \cdots p_t}\right)$.

math.NT↗

arXiv2026

A necessary condition for a congruent number of the form $8k+3$

A positive square-free integer is called a \textit{congruent number} if it arises as the area of a right triangle with rational side lengths. Let $ n = p_1p_2 \cdots p_t q $ be a square-free integer, where each $ p_i \equiv 1 \pmod{8} $ and $ q \equiv 3 \pmod{8} $, with the $ p_i $ and $ q $ being distinct primes. In this article, we present a congruence relation modulo powers of 2 between the 2-part of the class numbers of $ \mathbb{Q}(\sqrt{-n}) $ and $ \mathbb{Q}(\sqrt{-p_1p_2 \cdots p_t}) $, under the assumption that $ n $ is a congruent number, using a modified Rédei matrix.

math.NT↗

arXiv2024

Relative $p$-class groups and $p$-Selmer groups

Let $E$ be an elliptic curve with $j$-invariant $0$ or $1728$ and let $\widetilde{E}$ be a $k^{th}$ twist of $E$. We show that for any prime $p$ of good reduction of $\widetilde{E}$, a degree $k$ relative $p$-class group and the root number of $\widetilde{E}$ determines the dimension of the $p$-Selmer group of $\widetilde{E}$. As a consequence, we construct families of large rank $p$-class group. We also relate congruent number and cube sum problem with relative $p$-class group.

math.NT↗

arXiv2024

On the change of epsilon factors for symmetric square transfers under twisting and applications

Let us consider the symmetric square transfer of the automorphic representation $π$ associated to a modular form $f \in S_k(N,ε)$. In this article, we study the variation of the epsilon factor of ${\mathrm{sym}}^2(π)$ under twisting in terms of the local Weil-Deligne representation at each prime $p$. As an application, we detect the possible types of the symmetric square transfer of the local representation at $p$. Furthermore, as the conductor of ${\mathrm{sym}}^2(π)$ is involved in the variation number, we compute it in terms of $N$.

math.NT↗

arXiv2023

Contribution of symmetric power transfers to the cuspidal cohomology of ${\rm GL_n}$

Let $π$ be a cuspidal automorphic representation of ${\mathrm {GL}}_2(\mathbb{A}_\mathbb{Q})$. Newton and Thorne have proved that for every $n\geq 1$, the symmetric power lifting ${\mathrm {sym}^n(π)}$ is automorphic if $π$ is attached to a non-CM Hecke eigenform. In this article, we establish an asymptotic estimate of the number of cuspidal automorphic representations of ${\mathrm {GL}}_{n+1}(\mathbb{A}_\mathbb{Q})$ which contribute to the cuspidal cohomology of ${\mathrm {GL}}_{n+1}$ and are obtained by symmetric $n$th transfer of cuspidal representations of ${\mathrm {GL}}_2(\mathbb{A}_\mathbb{Q})$. Here we fix the weight and vary the level. This generalises the previous works done for ${\mathrm {GL}}_3$ and ${\mathrm {GL}}_4$.

math.NT↗

arXiv2023

Two properties of symmetric cube transfers of modular forms

In this article, we study two important properties of ${\rm{sym}}^3$ transfers of the automorphic representation $π$ associated to a modular form. First we compute the conductor of ${\rm{sym}}^3(π)$. Then we detect the types of local automorphic representations at bad primes by the variation of the epsilon factors of symmetric cube transfer of the representation $π$ attached to a cusp form $f$. Here we twist the modular forms by a specific quadratic character. From this variation number, for each prime $p$, we classify all possible types of symmetric cube transfers of the local representations $π_p$. For ${\rm{sym}}^3$ transfer, the most difficult prime is $p=3$.

math.NT↗

arXiv2021

On the growth of cuspidal cohomology of ${\rm GL}_4$

In this article, we establish an asymptotic estimate on the number of cuspidal automorphic representations of ${\rm GL}_4(\mathbb A_{\mathbb Q})$ which contribute to the cuspidal cohomology of ${\rm GL}_4$ and are obtained from symmetric cube transfer of automorphic representations of ${\rm GL}_2(\mathbb A_{\mathbb Q})$ of a given weight and with varying level structure. This generalises the recent work of C. Ambi [2020] about the similar problem for ${\rm GL}_3$.

math.NT↗

arXiv2021

Automorphic tensor products and cuspidal cohomology of the ${\rm GL}_4$

In this article, we establish an asymptotic lower bound estimate on the contribution of cuspidal automorphic representations of ${\rm GL}_4(\mathbb A_{\mathbb Q})$ to cuspidal cohomology of the ${\rm GL}_4$ which are obtained from automorphic tensor product of two automorphic representations of ${\rm GL}_2(\mathbb A_{\mathbb Q})$ of given weights and with varying level structure. In the end, we also prove that the symmetric cube of a representation of ${\rm GL}_2$ and the automorphic tensor product of two representations of ${\rm GL}_2$ can not be equal (up to a twist by a character of ${\rm GL}_1$) to each other, under the suitable assumptions on the representations being cuspidal and cohomological.

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arXiv2020

Powers in the wreath product of $G$ with $S_n$

In this paper we compute powers in the wreath product $G\wr S_n$, for any finite group $G$. For $r\geq 2$, a prime, consider $ω_r: G\wr S_n\to G\wr S_n$ defined by $g \mapsto g^r$. Let $P_{r}(G\wr S_n)=\frac{|ω_r(G\wr S_n)|}{|G|^n n!}$, be the probability that a randomly chosen element in $G\wr S_n$ is a $r^{th}$ power. We prove, $P_r(G\wr S_{n+1})=P_r(G\wr S_n)$ for all $n\not \equiv -1(\text{mod } r)$ if, order of $G$ is coprime to $r$. We also give a formula for the number of conjugacy classes that are $r^{th}$ powers in $G\wr S_n$.

math.GR↗