In this dissertation, we give a complete classification and list all identities of the form h = fg, where f , g and h are Hecke eigenforms of any weight with respect to Γ₁(N). This result extends the work of Ghate [Gha02] who considered this question for eigenforms with respect to Γ₁(N), with N square-free and f and g of weight 3 or greater. We remove all restrictions on the level N and the weights of f and g. For N = 1 there are only 16 eigenform identities, which are classically known. We first give a new proof of the level N = 1 case. We then give a proof which classifies all such eigenform identities for all levels N > 1. The identities fall into two categories. There are two infinite families of identities, given in Table 7.2. There are 209 other identities, listed (up to conjugacy) in Table 7.1. Thus any eigenform identity h = f g with respect to Γ₁(N) is either conjugate to an identity in Table 7.1 or takes the form of an identity described in Table 7.2.
| Identifer | oai:union.ndltd.org:arizona.edu/oai:arizona.openrepository.com:10150/228631 |
| Date | January 2012 |
| Creators | Johnson, Matthew Leander |
| Contributors | Joshi, Kirti, Joshi, Kirti, Cais, Bryden, Lux, Klaus, Thakur, Dinesh |
| 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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