arXiv subjects

Koustav Mondal

Publications and source records attributed to Koustav Mondal.

2 recordsLinked to original sources

arXiv2026

Special $L$-values of certain CM weight three Hecke eigenforms

Ramanujan's theory of elliptic functions to alternative bases connects modular forms with hypergeometric series and has led to applications such as the modularity of certain hypergeometric Galois representations. In this paper, we relate special values of $L$-functions of certain CM Hecke eigenforms to Ramanujan's alternative bases via the modularity of hypergeometric Galois representations associated with hypergeometric series ${}_{3}F_{2}\!\left[ \genfrac{}{}{0pt}{}{\frac{1}{2} \ \frac{1}{d} \ \frac{d-1}{d}}{\ 1 \ \ \ \ 1} ;\ t \right]$, $d=2$, $3$, $4$, and $6$, arising from tensor products of CM elliptic curves over real quadratic fields. We also give a complete classification of these type of hypergeometric Galois representations.

math.NT↗

arXiv2025

Relating elliptic curve point-counting and solutions of quadratic forms with congruence conditions

In this paper, we analyze the theta series associated to the quadratic form $Q(\mathbf{x}) := x_1^2 + x_2^2 + x_3^2 + x_4^2$ with congruence conditions on $x_i$ modulo $2, 3, 4$, and $6$. By employing special operators on modular, non-holomorphic Eisenstein series of weight $2$, we construct a basis for the Eisenstein space for levels $2^k$ (with $k \le 7$), $3^{\ell}$ (with $\ell \le 3$), and $p$, where $p>3$ is an odd prime. Using the relation between the trace of Frobenius on an elliptic curve and the Fourier coefficients of the cusp-form part of the theta series corresponding to $Q$, we establish a relation between the number of integer solutions to the equation $Q(\mathbf{x}) = p$ and the number of $\mathbb{F}_p$-rational points on the associated elliptic curve under certain congruence conditions on $p$.

math.NT↗