arXiv subjects

Jishu Das

Publications and source records attributed to Jishu Das.

6 recordsLinked to original sources

arXiv2025

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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arXiv2024

Shortest nonzero lattice points in a totally real multi-quadratic number field and applications

Let $F$ be a multi-quadratic totally real number field. Let $\sigma_1,\dots, \sigma_r$ denote its distinct embeddings. Given $s \in F,$ we give an explicit formula for $\| \sigma(s)\|$ and $\sum_{i<j} \sigma_i(s)\sigma_j(s),$ where $\| \sigma(s)\|=\sqrt{\sum_{i=1}^r(\sigma_i(s))^2}.$ Let $\mathfrak{M}$ be a fractional ideal in $F$ and $\min\left( \mathfrak{M}\right):=\min\{\|\sigma(s)\| \, | \, s \in \mathfrak{M}, s\neq 0 \}.$ The set of shortest nonzero lattice points for $\mathfrak{M}$ is given by $\{s\in \mathfrak{M} : \| \sigma(s)\|=\min(\mathfrak{M}) \}.$ We provide shortest nonzero lattice points for $\mathfrak{M}$ in terms of rational solutions to a given Diophantine equation. As an application, we get a refined asymptotic for the Petersson trace formula for the space of Hilbert cusp forms. We then use the refined asymptotic to obtain a lower bound analogue to a theorem by Jung and Sardari.

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arXiv2024

A lower bound for classical Kloosterman sums and an application

We present a lower bound for the classical Kloosterman sum $S(a,b;c)$ where $(ab,c)=1$ and $c$ is an odd integer. We apply this lower bound for Kloosterman sums to derive an explicit lower bound in Petersson's trace formula, subject to a given condition. Consequently, we achieve a modified version of a theorem by Jung and Sardari, where weight $k$ and level $N$ are permitted to vary independently. Using this modified version, we get a lower bound for a weighted trace of the Hecke operator $T_n$ acting on the space $S_k(N)$, of cusp forms of weight $k$ and level $N$ with $(n,N)=1$.

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arXiv2023

A lower bound for the discrepancy in a Sato-Tate type measure

Let $S_k(N)$ denote the space of cusp forms of even integer weight $k$ and level $N$. We prove an asymptotic for the Petersson trace formula for $S_k(N)$ under an appropriate condition. Using the non-vanishing of a Kloosterman sum involved in the asymptotic, we give a lower bound for discrepancy in the Sato-Tate distribution for levels not divisible by $8$. This generalizes a result of Jung and Sardari for squarefree levels. An analogue of the Sato-Tate distribution was obtained by Omar and Mazhouda for the distribution of eigenvalues $\lambda_{p^2}(f)$ where $f$ is a Hecke eigenform and $p$ is a prime number. As an application of the above-mentioned asymptotic, we obtain a sequence of weights $k_n$ such that discrepancy in the analogue distribution obtained by Omar and Mazhouda has a lower bound.

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arXiv2023

A central limit theorem for Hilbert modular forms

For a prime ideal $\mathfrak{p}$ in a totally real number field $L$ with the adele ring $\mathbb{A}$, we study the distribution of angles $\theta_\pi(\mathfrak{p})$ coming from Satake parameters corresponding to unramified $\pi_\mathfrak{p}$ where $\pi_\mathfrak{p}$ comes from a global $\pi$ ranging over a certain finite set $\Pi_{\underline{k}}(\mathfrak{n})$ of cuspidal automorphic representations of GL$_2(\mathbb{A})$ with trivial central character. For such a representation $\pi$, it is known that the angles $\theta_\pi(\mathfrak{p})$ follow the Sato-Tate distribution. Fixing an interval $I\subseteq [0,\pi]$, we prove a central limit theorem for the number of angles $\theta_\pi(\mathfrak{p})$ that lie in $I$, as $\mathrm{N}(\mathfrak{p})\to\infty$. The result assumes $\mathfrak{n}$ to be a squarefree integral ideal, and that the components in the weight vector $\underline{k}$ grow suitably fast as a function of $x$.

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arXiv2023

A discrepancy result for Hilbert modular forms

Let $F$ be a totally real number field and $r=[F :\mathbb{Q}].$ Let $A_k(\mathfrak{N},\omega) $ be the space of holomorphic Hilbert cusp forms with respect to $K_1(\mathfrak{N})$, of weight $k=(k_1,\dots,k_r)$ such that $k_j>2$ for all $j$, and with central Hecke character $\omega$. For integral ideals $\mathfrak{N}$ and $\mathfrak{n}$ in $F$ such that $( \mathfrak{n}, \mathfrak{N}) = 1$, we study the Petersson trace formula for the Hecke operator $T_{\mathfrak{n}}$ acting on the space $A_k(\mathfrak{N},\omega)$. We present asymptotic estimates for the terms of the Petersson formula as $k_0\rightarrow\infty,$ where $k_0=\min(k_1,\dots,k_r)$. As an application, we obtain a weighted discrepancy bound for the distribution of the eigenvalues of the Hecke operator $T_{\mathfrak{p}}$ (for a fixed prime ideal $\mathfrak{p}$) acting on the space $A_k(\mathfrak{N},1),$ when $F$ has narrow class number $1$, and the ideal $\mathfrak{N}$ is generated by (rational) integers. This generalizes a discrepancy result previously obtained by Jung and Sardari in the context of classical cusp forms.

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