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Automorphic L-Functions and Their Derivatives

Liu, Shenhui
http://orcid.org/0000-0002-8560-4772
http://rave.ohiolink.edu/etdc/view?acc_num=osu1499965951825371

Abstract Details

2017, Doctor of Philosophy, Ohio State University, Mathematics.
In this dissertation we investigate automorphic L-functions and their derivatives at the central point of the critical strip, by the method of moments and/or the mollification method à la Selberg. In Chapter 1, we introduce basic concepts and facts of the families of automorphic forms considered in this work and state the main results. In Chapter 2, we study the average and nonvanishing of the central L-derivative values of L(s,f) and L(s,f_{K_D}) for f in an orthogonal Hecke eigenbasis H_{2k} of weight 2k cusp forms for SL(2,Z) for large odd k. Here f_{K_D} is the base change of f to an imaginary quadratic field K_D = Q(vD) with fundamental discriminant D. We prove asymptotic formulas for the first and second moments of L(1/2,f), as well as the first moment of L(1/2,f_{K_D}), over H_{2k} as odd k→∞. Further, we employ mollifiers to establish that for sufficiently large k there are positive proportion of Hecke eigenforms f in H_{2k} with nonzero L(1/2,f). We also give applications of our results to Heegner cycles of high weights of the modular curve X_0(1). In Chapter 3, we establish an asymptotic formula with arbitrary power saving for the first moment of L(1/2,sym²f) for f∈H_{2k} as k→∞, where L(s,sym²f) denotes the symmetric square L-function of f. We extract two secondary main terms from the best known error term O(k^{-1/2}) in the asymptotic formula for the first moment of L(1/2,sym²f). Specifically, the secondary main terms involve central values of Dirichlet L-functions of characters χ_{-4} and χ_{-3} and depend on the values of k (mod 2) and k (mod 3), respectively. In Chapter 4 we study the central L-values of Maass forms of weight 0 for SL(2,Z) and establish a positive-proportional nonvanishing result of such values in the aspect of large spectral parameter in short intervals, which is qualitatively optimal in view of Weyl’s law. As an application of this result and a formula of Katok–Sarnak, we give a nonvanishing result on the first Fourier coefficients of Maass forms of weight 1/2 for Γ_0(4) in the Kohnen plus space.
Wenzhi Luo (Advisor)
James Cogdell (Committee Member)
Roman Holowinsky (Committee Member)
129 p.
Mathematics
automorphic L-functions, central value, derivative, moments, nonvanishing, mollifiers

Recommended Citations

Citations

  • Liu, S. (2017). Automorphic L-Functions and Their Derivatives [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1499965951825371

    APA Style (7th edition)

  • Liu, Shenhui. Automorphic L-Functions and Their Derivatives. 2017. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1499965951825371.

    MLA Style (8th edition)

  • Liu, Shenhui. "Automorphic L-Functions and Their Derivatives." Doctoral dissertation, Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu1499965951825371

    Chicago Manual of Style (17th edition)

osu1499965951825371
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This open access ETD is published by The Ohio State University and OhioLINK.