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Eigenforms of half-integral weight
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Purkait, S. (Soma) (2012) Eigenforms of half-integral weight. PhD thesis, University of Warwick.
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WRAP_THESIS_Purkait_2012.pdf - Submitted Version Download (915Kb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b2583302~S1
Abstract
Let k be an odd integer and N a positive integer such that 4 | N. Let X be a Dirichlet character modulo N. Shimura decomposes the space of half-integral weight forms Sk/2(N,X) as Sk/2(N,X) = S0(N,X)oOΦSk/2(N,X,Φ) where Φ runs through the newforms of weight k-1 and level dividing N/2 and character X2; Sk/2(N,X,Φ) is the subspace of forms that are Shimura-equivalent to Φ; and S0(N,X) is the subspace generated by single-variable theta-series. We give an explicit algorithm for computing this decomposition. Once we have the decomposition, we can exploreWaldspurger's theorem expressing the critical values of the L-functions of twists of an elliptic curve in terms of the coefficients of modular forms of half-integral weight. Following Tunnell, this often allows us to give a criterion for the n-th twist of an elliptic curve to have positive rank in terms of the number of representations of certain integers by certain ternary quadratic forms.
| Item Type: | Thesis or Dissertation (PhD) |
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| Subjects: | Q Science > QA Mathematics |
| Library of Congress Subject Headings (LCSH): | Forms, Modular, Eigenvectors |
| Date: | July 2012 |
| Institution: | University of Warwick |
| Theses Department: | Mathematics Institute |
| Thesis Type: | PhD |
| Publication Status: | Unpublished |
| Supervisor(s)/Advisor: | Siksek, Samir |
| Sponsors: | University of Warwick |
| Extent: | v, 144 leaves |
| Language: | eng |
| URI: | http://wrap.warwick.ac.uk/id/eprint/50236 |
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