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Subconvexity Problems using the delta method

Mejia Cordero, Julian Alonso
http://orcid.org/0000-0001-7134-3957
http://rave.ohiolink.edu/etdc/view?acc_num=osu1658449732266246

Abstract Details

2022, Doctor of Philosophy, Ohio State University, Mathematics.
By looking into the direct applications of the delta method developed by Duke, Friedlander and Iwaniec, we explore new methods to attain subconvexity bounds for $L$-functions. We focus mainly on the $GL(2)$ $L$-functions and their twists in the level aspect. Motivated by the challenge of R. Munshi, to establish $GL(2)$ level aspect subconvexity without appealing to moments, we establish the following initial result. Let $f$ be a holomorphic Hecke newform of level $P_1$ and $\chi$ a character modulo $P_2$. Assuming $P_1\sim P_2^\eta$, we obtain the subconvexity bound $$L(f\otimes \chi,1/2)\ll \Q^{\epsilon+1/4-h(\eta)}, \hbox{ for }\frac{1}{2}<\eta<1 $$ with $$h(\eta)=\left\{\begin{array}{cc} \frac{1-\eta}{40\eta+20} & \hbox{ if }\,\frac{11}{16}\leq \eta<1\\ \frac{2\eta-1}{48\eta+24}& \hbox{ if }\,\frac{1}{2}<\eta\leq \frac{11}{16}. \end{array}\right.$$ This result is not as strong as recent results given by Rizwanur Khan by using moment methods, but the emphasis of this work is to avoid moment methods and try to obtain a subconvexity bound applying the delta method directly in the same fashion Munshi did to obtain subconvexity results in different aspects.
Roman Holowinsky (Advisor)
Ghaith Hiary (Committee Member)
James Cogdell (Committee Member)
115 p.
Mathematics
Number Theory; L-functions; Subconvexity bounds

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Citations

  • Mejia Cordero, J. A. (2022). Subconvexity Problems using the delta method [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1658449732266246

    APA Style (7th edition)

  • Mejia Cordero, Julian. Subconvexity Problems using the delta method. 2022. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1658449732266246.

    MLA Style (8th edition)

  • Mejia Cordero, Julian. "Subconvexity Problems using the delta method." Doctoral dissertation, Ohio State University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=osu1658449732266246

    Chicago Manual of Style (17th edition)

osu1658449732266246
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