Searcharxiv⌕ Search

arXiv subjects

Priyanka Majumder

Publications and source records attributed to Priyanka Majumder.

5 recordsLinked to original sources
arXiv2025

Bounds for canonical Green's functions at cusps

Let $Γ$ be a cofinite Fuchsian subgroup. The canonical Green's function associated with $Γ$ arises in Arakelov theory when establishing asymptotics for Arakelov invariants of the modular curve associated with some congruence subgroup of level $N$ with a positive integer $N$. More precisely, in the known cases, canonical Green's functions at certain cusps contribute to the analytic part of the asymptotics for the self-intersection of the relative dualizing sheaf. In this article, we prove canonical Green's function of a cofinite Fuchsian subgroup at cusps bounded by the scattering constants, the Kronecker limit functions, and the Selberg zeta function of the group $Γ$. Then as an application, we prove an asymptotic expression of the canonical Green's function associated with $Γ_0(N)$, for any positive integer $N$.

math.NT↗
arXiv2024

Modular forms with non-vanishing central values and linear independence of Fourier coefficients

In this article, we are interested in modular forms with non-vanishing central critical values and linear independence of Fourier coefficients of modular forms. The main ingredient is a generalization of a theorem due to VanderKam to modular symbols of higher weights. We prove that for sufficiently large primes $p$, Hecke operators $T_1, T_2, \ldots, T_D$ act linearly independently on the winding elements inside the space of weight $2k$ cuspidal modular symbol $\mathbb{S}_{2k}(Γ_0(p))$ with $k\geq 1$ for $D^2\ll p$. This gives a bound on the number of newforms with non-vanishing arithmetic $L$-functions at their central critical points and linear independence on the reductions of these modular forms for prime modulo $l\not=p$.

math.NT↗
arXiv2022

The intersection matrices of $X_0(p^r)$ and some applications

We compute intersection matrices for modular curves of the form $X_0(p^r)$ with $r \in \{3,4\}$ and as an application, we compute an asymptotic expression for the Arakelov self-intersection number of the relative dualizing sheaf of Edixhoven's minimal regular model for the modular curve $X_0(p^r)$ over $\qq$ with $r$ as above. This computation will be useful to understand an effective version of the Bogolomov conjecture for the stable models of modular curves $X_0(p^r)$ with $r \in \{3,4\}$ and obtain a bound on the stable Faltings height for those curves.

math.NT↗
arXiv2018

Estimates of the Bergman kernel on a hyperbolic Riemann surface of finite volume-II

In this article, we derive off-diagonal estimates of the Bergman kernel associated to the tensor-powers of the cotangent bundle defined on a hyperbolic Riemann surface of finite volume, when the distance between the points is less than injectivity radius. We then use these estimates to derive estimates of the Bergman kernel along the diagonal.

math.CV↗