On the diagonalizability of the Atkin U-operator for Drinfeld cusp forms

Andrea Bandini; Maria Valentino

On the diagonalizability of the Atkin U-operator for Drinfeld cusp forms

Number Theory 2017-10-05 v2

Authors: Andrea Bandini , Maria Valentino

Abstract

We study the diagonalizability of the Atkin $U$ -operator acting on Drinfeld cusp forms for $\Gamma_1(t)$ and $\Gamma(t)$ using Teitelbaum's interpretation as harmonic cocycles. We prove $U$ is diagonalizable for small weights and explicitly compute the eigenvalues. We also formulate a conjecture, supported by numerical search and proofs in some special cases, about non diagonalizability of $U$ in even characteristic.

Keywords

Atkin U-operator diagonalizability Drinfeld cusp forms harmonic cocycles eigenvalues even characteristic

Cite

@article{arxiv.1702.08801,
  title  = {On the diagonalizability of the Atkin U-operator for Drinfeld cusp forms},
  author = {Andrea Bandini and Maria Valentino},
  journal= {arXiv preprint arXiv:1702.08801},
  year   = {2017}
}

Comments

This paper has been withdrawn and substituted with the two new ones arXiv:1710.01036 and arXiv:1710.01038 containing new developments

On the diagonalizability of the Atkin U-operator for Drinfeld cusp forms

Abstract

Keywords

Cite

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