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On Decomposition of Drinfeld cusp forms of level t
Abstract
In this article, we first prove that the Hecke operator Tt has no eigenform in Sla with eigenvalue −tk/2, when the characteristic of the base field is odd. Furthermore, if dim Slt is even, we show that Tt has no eigenform in Sla with eigenvalue tk/2. As a consequence, we prove that the direct sum decomposition Slt=Sold ⊕ Snew holds when dim Slt is even. This proves the conjecture \cite[Conjecture 1.1(3)]{BV19a} of Bandini and Valentino for an infinite family of cusp forms. In particular, for any weight k, there exists at least one type m (there are only two possible non-trivial values of m) for which the conjecture \cite[Conjecture 1.1(3)]{BV19a} is true.
Related
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- Drinfeld modular forms of higher rank from a lattice-oriented point of view
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BibTeX
@misc{dalal2026decomposition,
title = {{On Decomposition of Drinfeld cusp forms of level $t$}},
author = {Tarun Dalal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14167},
eprint = {2609.14167},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14167},
}
RIS
TY - UNPB
TI - On Decomposition of Drinfeld cusp forms of level t
AU - Tarun Dalal
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.14167
UR - https://arxiv.org/abs/2609.14167
AB - In this article, we first prove that the Hecke operator T_t has no eigenform in Sla with eigenvalue −t^k/2, when the characteristic of the base field is odd. Furthermore, if dim Slt is even, we show that T_t has no eigenform in Sla with eigenvalue t^k/2. As a consequence, we prove that the direct sum decomposition Slt=Sold ⊕ Snew holds when dim Slt is even. This proves the conjecture \cite[Conjecture 1.1(3)]{BV19a} of Bandini and Valentino for an infinite family of cusp forms. In particular, for any weight k, there exists at least one type m (there are only two possible non-trivial values of m) for which the conjecture \cite[Conjecture 1.1(3)]{BV19a} is true.
ER -