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Bounds for Hecke Eigenforms and Their Allied L-functions
Zhang, Qing

2015, Doctor of Philosophy, Ohio State University, Mathematics.
We consider Hecke eigenforms and their allied L-functions from three aspects in this thesis.

First we generalize the Iwaniec's spectral large sieve estimates of Maass cusp form to the local version for all congruence groups of level q. Our approach is based on an inequality for a general bilinear form involving Kloosterman sums and Bessel functions. The exceptional eigenvalues emerge in the course of the proof.

In the second part, we extend Luo's result to prove a general optimal bound for L^4-norms of the dihedral Maass forms associated to Hecke's grossencharacters of a fixed real quadratic field. Given a fixed quadratic field with discriminant D, we remove the condition that the narrow class number of K is 1. The key ingredients are Watson and Ichino's formula and the local spectral large sieve inequality established in the first part.

Finally we obtain a long equation intended to establish an upper bound for the second moment of symmetric square L-functions. Petersson trace formula plays an important role and we study thoroughly an analogue of Estermann series using Hurwitz zeta function and establish its meromorphic extension and functional equation. This work provides a useful approach to the further study for the central value of the symmetric square L-functions.
Wenzhi Luo (Advisor)
Jim Cogdell (Committee Member)
Roman Holowinsky (Committee Member)
75 p.

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Zhang, Q. (2015). Bounds for Hecke Eigenforms and Their Allied L-functions. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Zhang, Qing. "Bounds for Hecke Eigenforms and Their Allied L-functions." Electronic Thesis or Dissertation. Ohio State University, 2015. OhioLINK Electronic Theses and Dissertations Center. 23 Nov 2017.

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Zhang, Qing "Bounds for Hecke Eigenforms and Their Allied L-functions." Electronic Thesis or Dissertation. Ohio State University, 2015. https://etd.ohiolink.edu/

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