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Eigenforms of halfintegral weight
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Purkait, S. (Soma) (2012) Eigenforms of halfintegral weight. PhD thesis, University of Warwick.

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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
halfintegral 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 k1 and level dividing N/2 and
character X2; Sk/2(N,X,Φ) is the subspace of forms that are Shimuraequivalent
to Φ; and S0(N,X) is the subspace generated by singlevariable thetaseries.
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 Lfunctions of twists of an elliptic curve
in terms of the coefficients of modular forms of halfintegral weight. Following
Tunnell, this often allows us to give a criterion for the nth 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) 

Subjects:  Q Science > QA Mathematics 
Library of Congress Subject Headings (LCSH):  Forms, Modular, Eigenvectors 
Official 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Publication Status:  Unpublished 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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