Year: 2026
Type: preprint
Abstract: 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 $\Slt=\Sold \oplus \Snew$ 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.
Source: arXiv (Cornell University)
Author: Dalal, Tarun
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213285126