On the monomials of Poincaré Series of negative index
Abstract
It is well known that an Eisenstein series Ek cannot be written as a product of two lower-weight Eisenstein series, except for E14=E4 E10= E42 E6= E6 E8. A similar result is also known for the Hecke eigenforms. It is also known that equalities among the monomials of the Eisenstein series can be reduced to the above-mentioned identities. Recently, the present author, along with E. Saha studied the possible monomial relations of the Poincaré cusp forms of index $1$. So far, the problem of studying the monomial relations has been restricted to the setup of holomorphic modular forms. For an even integer k≥ 4 and a negative integer m, the Poincaré series Gk(z,m) of weight k and index m is a weakly holomorphic modular form having a pole of order −m at ι∞. It is immediate that the Poincaré series Gk(z,m) for m<0 can not be written as a product of two lower-weight Poincaré series of the same index m. In this article, we investigate the possible equalities among the monomials of the Poincaré series Gk(z,m) for arbitrary m<0. In particular, we show that for any m<0 such that 0.06≤{−2mπ}≤0.99, two monomials composed of Gk(z,m) of the weights k≥ 50 are never equal, where {x} denotes the fractional part of a real number x. In view of Weyl's equidistribution criterion, at least 93% of m<0 satisfies the above inequality.
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BibTeX
@misc{kala2026monomials,
title = {{On the monomials of Poincaré Series of negative index}},
author = {Divyanshu Kala},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28350},
eprint = {2609.28350},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28350},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the monomials of Poincaré Series of negative index
AU - Divyanshu Kala
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28350
UR - https://arxiv.org/abs/2609.28350
AB - It is well known that an Eisenstein series E_k cannot be written as a product of two lower-weight Eisenstein series, except for E_14=E_4 E_10= E_4^2 E_6= E_6 E_8. A similar result is also known for the Hecke eigenforms. It is also known that equalities among the monomials of the Eisenstein series can be reduced to the above-mentioned identities. Recently, the present author, along with E. Saha studied the possible monomial relations of the Poincaré cusp forms of index $1$. So far, the problem of studying the monomial relations has been restricted to the setup of holomorphic modular forms. For an even integer k≥ 4 and a negative integer m, the Poincaré series G_k(z,m) of weight k and index m is a weakly holomorphic modular form having a pole of order −m at ι∞. It is immediate that the Poincaré series G_k(z,m) for m<0 can not be written as a product of two lower-weight Poincaré series of the same index m. In this article, we investigate the possible equalities among the monomials of the Poincaré series G_k(z,m) for arbitrary m<0. In particular, we show that for any m<0 such that 0.06≤{−2mπ}≤0.99, two monomials composed of G_k(z,m) of the weights k≥ 50 are never equal, where {x} denotes the fractional part of a real number x. In view of Weyl's equidistribution criterion, at least 93% of m<0 satisfies the above inequality.
ER -