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Bounds for Hecke Eigenforms and Their Allied L-functions

Zhang, Qing
http://rave.ohiolink.edu/etdc/view?acc_num=osu1429704681

Abstract Details

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.
Mathematics
large sieve inequality; dihedral Maass form; symmetric square L-function

Recommended Citations

Citations

  • Zhang, Q. (2015). Bounds for Hecke Eigenforms and Their Allied L-functions [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1429704681

    APA Style (7th edition)

  • Zhang, Qing. Bounds for Hecke Eigenforms and Their Allied L-functions. 2015. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1429704681.

    MLA Style (8th edition)

  • Zhang, Qing. "Bounds for Hecke Eigenforms and Their Allied L-functions." Doctoral dissertation, Ohio State University, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=osu1429704681

    Chicago Manual of Style (17th edition)

osu1429704681
879
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This open access ETD is published by The Ohio State University and OhioLINK.