Strong Ramanujan bound for traces of Hecke operators on Drinfeld cusp forms

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Let A=Fq[T]A=\mathbb{F}_q[T], let p⊴A\mathfrak p\trianglelefteq A be a maximal ideal of degree dd, and let n≥1n\geq 1. For integers k,lk,l, write Tp\mathbf{T}_{\mathfrak p} for the corresponding Hecke operator on the Drinfeld cusp-form space Sk,l\mathrm{S}_{k,l}. Strong Ramanujan bound. One has

deg⁡Tr⁡(Tpn∣S⁡k,l)≤nd(k−(q+1))2.\deg \operatorname{Tr}\left(\mathbf{T}_{\mathfrak p}^n\mid\operatorname{S}_{k,l}\right)\leq \frac{nd\left(k-(q+1)\right)}{2}.

This is presented as a strong form of the Ramanujan bound for Drinfeld modular forms. The source gives no resolution status, so the assertion remains open in this database.

References

Primary source

Sjoerd de Vries, “Traces of Hecke operators on Drinfeld modular forms for GL_2(F_q[T])”, arXiv:2407.04555 (2026).

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