Bruhat Tits buildings, harmonic cocycles and an Eichler-Shimura map in the function field setting
Let K be a global function field, ∞ a fixed place of K, K_∞ the corresponding completion and Γ ⊂ GL_r(K) be a principal congruence subgroup. In this thesis we establish an equivariant homotopy equivalence between subregions of the Bruhat-Tits building for GL_r(K_∞), which are unstable with respect t...
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| Formato: | Book/Monograph Tesis |
| Lenguaje: | inglés |
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Heidelberg
28 Jul. 2026
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| DOI: | 10.11588/heidok.00039110 |
| Materias: | |
| Acceso en línea: | Resolving-System, kostenfrei: https://nbn-resolving.org/urn:nbn:de:bsz:16-heidok-391108 Resolving-System, kostenfrei: https://doi.org/10.11588/heidok.00039110 Verlag, kostenfrei, Volltext: http://www.ub.uni-heidelberg.de/archiv/39110 Langzeitarchivierung Nationalbibliothek, kostenfrei: https://d-nb.info/1414756720/34 
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| Notas de Autor: | put forward by Sriram Chinthalagiri Venkata, M.Sc. ; supervisor: Prof. Dr. Gebhard Böckle |
| Sumario: | Let K be a global function field, ∞ a fixed place of K, K_∞ the corresponding completion and Γ ⊂ GL_r(K) be a principal congruence subgroup. In this thesis we establish an equivariant homotopy equivalence between subregions of the Bruhat-Tits building for GL_r(K_∞), which are unstable with respect to Γ, and the Tits buildings for GL_r(K), for r ≥ 2 an integer. In rank 2 this is due to Serre and in higher ranks we adapt the proof strategy of a similar result of Grayson-Quillen in [Gra82]. The role of simplices admitting non-trivial Harder-Narasimhan filtrations resp. semistable simplices in [Gra82] is played by Γ-unstable resp. Γ-stable simplices in our case. Using this we generalize a result of Teitelbaum in [Tei91] for r = 2, about reconstruction of Γ-invariant harmonic cocycles from their values on the Γ-stable edges, to higher ranks by combining a novel technique of constructing harmonic cocycles on Bruhat-Tits buildings from the Steinberg module using modular symbols à la Ash-Rudolph [AR79]. All this is covered in Part I, at the end of which we consider the simplifications occurring in some of the results in the case K = F_q(t) and ∞ being the usual place at infinity, by generalizing to higher ranks Serre’s ideas in [Ser80] regarding the action of GL2(F_q[t]) on the Bruhat-Tits tree. In Part II we consider a geometric application of the results in Part I namely the construction of prototype of an Eichler-Shimura map in higher rank motivated from the work of Böckle in [Boe02]. |
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| Descripción Física: | Online Resource |
| DOI: | 10.11588/heidok.00039110 |