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Subconvexity Bounds and Simplified Delta Methods

Aggarwal, Keshav
http://rave.ohiolink.edu/etdc/view?acc_num=osu1555064743753817

Abstract Details

2019, Doctor of Philosophy, Ohio State University, Mathematics.
This doctoral thesis is drawn from the author's work as a graduate student investigating various recent methods to prove subconvex bound results for GL(2) and GL(3) L-functions, and their Dirichlet character twists. The main topic is the use of variants of the circle method to prove t-aspect bounds for GL(2) and GL(3) L-functions. Let f be a Hecke-Maass cusp form for $\Gamma_0(M)$ with nebentypus. We prove $L(1/2+it, f)\ll_{f,\epsilon} (1+|t|)^{1/3+\epsilon}$. Before the author's work, a t-aspect Weyl bound was known only for square-free M. We were able to use the uniform partition of circle as a delta method that allowed us to extend the result to any level M and any nebetypus. This is the first main result and is discussed in Chapter 4. The second main result of this thesis is a new t-aspect bound for SL(3,Z) Hecke-Maass cusp forms. Let $\pi$ be a Hecke-Maass cusp form for SL(3,Z). We revisit Munshi's proof of the t-aspect subconvex bound and are able to remove the `conductor lowering' trick from his proof. This simplification along with a better stationary phase analysis allows us to get a new subconvex bound, $L(1/2+it, \pi) \ll_{\pi, \epsilon} (1+|t|)^{3/4-3/40+\epsilon}$. This is discussed in Chapter 5. An important tool used in the above two proofs is a Voronoi summation formula. In the case f is a holomorphic cusp form of weight k>1 that has CM by an imaginary quadratic field K, we indicate how one can use a two-dimensional Poisson summation formula to prove a Voronoi summation formula for f. Calculations for certain special cases are presented in Chapter 3.
Roman Holowinsky (Advisor)
114 p.
Mathematics
Circle method; Subconvex bound problem; Voronoi summation formula

Recommended Citations

Citations

  • Aggarwal, K. (2019). Subconvexity Bounds and Simplified Delta Methods [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1555064743753817

    APA Style (7th edition)

  • Aggarwal, Keshav. Subconvexity Bounds and Simplified Delta Methods. 2019. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1555064743753817.

    MLA Style (8th edition)

  • Aggarwal, Keshav. "Subconvexity Bounds and Simplified Delta Methods." Doctoral dissertation, Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1555064743753817

    Chicago Manual of Style (17th edition)

osu1555064743753817
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© 2019, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.