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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 (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Forms, Modular, Eigenvectors | ||||
Official Date: | July 2012 | ||||
Dates: |
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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Publication Status: | Unpublished | ||||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 |
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