In this dissertation, we introduce the notion of Drinfeld modular forms with A-expansions, where instead of the usual Fourier expansion in tⁿ (t being the uniformizer at infinity), parametrized by n ∈ N, we look at expansions in tₐ, parametrized by a ∈ A = F(q)[T]. We construct an infinite family of such eigenforms. Drinfeld modular forms with A-expansions have many desirable properties that allow us to explicitly compute the Hecke action. The applications of our results include: (i) various congruences between Drinfeld eigenforms; (ii) interesting relations between the usual Fourier expansions and A-expansions, and resulting recursive relations for special families of forms with A-expansions; (iii) the computation of the eigensystems of Drinfeld modular forms with A-expansions; (iv) many examples of failure of multiplicity one result, as well as a restrictive multiplicity one result for Drinfeld modular forms with A-expansions; (v) the proof of diagonalizability of the Hecke action in 'non-trivial' cases; (vi) examples of eigenforms that can be represented as non-trivial' products of eigenforms; (vii) an extension of a result of Böckle and Pink concerning the Hecke properties of the space of cuspidal modulo double cuspidal forms for Γ₁(T) to the groups GL₂(F(q)[T]) and Γ₀(T).
| Identifer | oai:union.ndltd.org:arizona.edu/oai:arizona.openrepository.com:10150/222872 |
| Date | January 2012 |
| Creators | Petrov, Aleksandar Velizarov |
| Contributors | Thakur, Dinesh, Thakur, Dinesh, Cais, Bryden, Joshi, Kirti, Sharifi, Romyar |
| Publisher | The University of Arizona. |
| Source Sets | University of Arizona |
| Language | English |
| Detected Language | English |
| Type | text, Electronic Dissertation |
| Rights | Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. |
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