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Drinfeld discriminant function and Fourier expansion of harmonic cochains

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Let \(F_{\infty }={{\mathbb {F}}_q}\left( \!\left( {1/T}\right) \!\right) \) be the completion of \({\mathbb {F}}_q(T)\) at 1/T. We develop a theory of Fourier expansions for harmonic cochains on the edges of the Bruhat–Tits building of \({{\,\textrm{PGL}\,}}_r(F_{\infty })\), \(r\ge 2\), generalizing an earlier construction of Gekeler for \(r=2\). We then apply this theory to study modular units on the Drinfeld symmetric space \(\Omega ^r\) over \(F_{\infty }\), and the cuspidal divisor groups of Satake compactifications of certain Drinfeld modular varieties. In particular, we obtain a higher dimensional analogue of a result of Ogg for classical modular curves \(X_0(p)\) of prime level.

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Acknowledgements

Part of this work was carried out while the first author was visiting the National Center for Theoretical Sciences in Hsinchu. He thanks the institute for its hospitality and excellent working conditions.

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Authors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Mihran Papikian

  2. Department of Mathematics, National Tsing Hua University, No. 101, Sec. 2, Kuang-Fu Rd., Hsinchu, 30013, Taiwan

    Fu-Tsun Wei

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  1. Mihran Papikian
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Correspondence to Fu-Tsun Wei.

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The first author was supported in part by a Collaboration Grant for Mathematicians from the Simons Foundation, Award No. 637364. The second author was supported by the National Science and Technology Council (Grant nos. 107-2628-M-007 -004- MY4 and 109-2115-M-007 -017 -MY5) and the National Center for Theoretical Sciences.

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Papikian, M., Wei, FT. Drinfeld discriminant function and Fourier expansion of harmonic cochains. Math. Ann. 388, 1379–1435 (2024). https://doi.org/10.1007/s00208-022-02549-8

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