Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.00720

Multiplicative Hecke operators and their application II

Chang Heon Kim·Gyucheol Shin
View original source ↗Read source document ↗Export metadata ↓

Abstract

Inspired by Borcherds' questions, Guerzhoy constructed a new type of Hecke operators $\mathcal{T}(p)$, called the multiplicative Hecke operators, which acts on the space of meromorphic modular forms on the full modular group ${\rm SL}(\Z)$. By Kim and Shin, this result was extended in two directions: to higher levels and to $\mathcal{T}(n)$ with a positive integer $n$. In this paper, building on the results by Kim and Shin, we further generalize the result in another direction by considering alternative infinite product expansions of meromorphic modular forms. As an application, we demonstrate how multiplicative Hecke operators relate both the divisor of modular forms and traces of singular moduli. Additionally, we prove the existence of a modular form with nonintegral coefficients whose poles or zeros are only supported at the cusps and which is not a multiplicative Hecke eigenform.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Chang Heon Kim, Gyucheol Shin. 2025-08-31. Multiplicative Hecke operators and their application II. https://arxiv.org/abs/2509.00720

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case

Given a cusp form $f$ which is supersingular at a fixed prime $p$ away from the level, and a Coleman family $F$ through one of its $p$-stabilisations, we construct a $2$-variable meromorphic $p$-adic $L$-function for the symmetric square of $F$. We prove that this new $p$-adic $L$-function interpolates values of complex imprimitive symmetric square $L$-functions, for the various specialisations of the family $F$. We use this $p$-adic $L$-function to prove a $p$-adic factorisation formula, expressing the geometric $p$-adic $L$-function attached to the Rankin--Selberg convolution of $f$ with itself as a the product of the $p$-adic symmetric square $L$-function of $f$ and a Kubota-Leopoldt $L$-function. This extends a result of Dasgupta in the ordinary case.

math.NT↗
arXiv2026

Exceptional poles of archimedean Rankin-Selberg L-functions for irreducible generic representations of GL(n,R)

For irreducible generic representations $π_1$ and $π_2$ of $\operatorname{GL}_n(\mathbb R)$, we prove that the notions of exceptional pole of type $1$ and type $2$ coincide at every level. When both representations are in general position, we use this identification to express the Rankin--Selberg $L$-function $L(s,π_1\timesπ_2)$ in terms of the exceptional $L$-factors attached to the irreducible constituents of their derivatives.

math.NT↗