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NOTES ON ATKIN–LEHNER THEORY FOR DRINFELD MODULAR FORMS

Part of: Arithmetic algebraic geometry Discontinuous groups and automorphic forms Topological fields

Published online by Cambridge University Press:  15 November 2022

TARUN DALAL Open the ORCID record for TARUN DALAL [Opens in a new window]  and
NARASIMHA KUMAR Open the ORCID record for NARASIMHA KUMAR [Opens in a new window]
TARUN DALAL
Affiliation:
Department of Mathematics, Indian Institute of Technology Hyderabad, Kandi, Sangareddy 502285, India e-mail: ma17resch11005@iith.ac.in
NARASIMHA KUMAR*
Affiliation:
Department of Mathematics, Indian Institute of Technology Hyderabad, Kandi, Sangareddy 502285, India
*
e-mail: narasimha@math.iith.ac.in
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Abstract

We settle a part of the conjecture by Bandini and Valentino [‘On the structure and slopes of Drinfeld cusp forms’, Exp. Math. 31(2) (2022), 637–651] for $S_{k,l}(\Gamma _0(T))$ when $\mathrm {dim}\ S_{k,l}(\mathrm {GL}_2(A))\leq 2$. We frame and check the conjecture for primes $\mathfrak {p}$ and higher levels $\mathfrak {p}\mathfrak {m}$, and show that a part of the conjecture for level $\mathfrak {p} \mathfrak {m}$ does not hold if $\mathfrak {m}\ne A$ and $(k,l)=(2,1)$.

Keywords

Drinfeld cusp formsoldformsnewformsAtkin–Lehner operatorsHecke operatorscommutativitydiagonalisability

MSC classification

Primary: 11F52: Modular forms associated to Drinfel'd modules
Secondary: 11F25: Hecke-Petersson operators, differential operators (one variable) 11G09: Drinfel'd modules; higher-dimensional motives, etc. 12J25: Non-Archimedean valued fields
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 108 , Issue 1 , August 2023 , pp. 50 - 68
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first author thanks University Grants Commission, INDIA for the financial support provided in the form of a Research Fellowship to carry out this research work at IIT Hyderabad. The second author’s research was supported by the SERB grant MTR/2018/000137.

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