Bounding Self-Dual L-functions: The Conrey-Iwaniec Method Revisited by Yongxiao Lin

DISCUSSION MEETING
L-FUNCTIONS, CIRCLE-METHOD AND APPLICATIONS (HYBRID)

ORGANIZERS: Soumya Das (IISc, Bengaluru, India), Ritabrata Munshi (ISI-Kolkata, India) and Saurabh Kumar Singh (IIT - Kanpur, India)
DATE: 27 June 2022 to 01 July 2022
VENUE: Ramanujan Lecture Hall and Online

The circle method originated in a paper of S. Ramanujan and G. H. Hardy on the partition function. This method has evolved with time and has seen many interesting applications. The classical applications of the circle method are to the Waring’s problem, to the ternary Gold-bach problem, and to count rational points on varieties. The modern applications of this method are to the subconvexity problem on various L-functions and to the shifted convolution problem. Also, the circle method is a powerful analytical tool to study correlations between two arithmetical functions and it is very flexible to use. The analytic study of L-functions is a central theme in analytic number theory, and it has many arithmetical consequences. The growth of L-functions (few classes of L-functions) can be understood by studying a correlation problem using the circle method. We hope that this method will continue to have many more interesting applications. The aim of this programe is to explore this method and look into its future.

0:00:00 Bounding Self-Dual L-functions: The Conrey-Iwaniec Method Revisited
0:01:12 Contents
0:01:44 Maass forms
0:02:49 L-functions
0:03:58 Analytic Conductor
0:05:04 The subconvexity problem
0:05:46 Subconvexity on GL1
0:07:20 The GL2 case: twist aspect
0:08:54 A fundamental work of Conrey-Iwaniec
0:11:36 Remarks on the Weyl bound
0:12:33 A Motohashi-type formula
0:15:09 Remarks on the Weyl bound: II
0:16:27 GL3: Xiaoqing Li's work (spectral aspect)
0:18:40 Xiaoqing Li's work
0:20:42 Main result
0:22:43 Other results: without self-duality assumption
0:25:39 Remarks: Motohashi-type formula
0:28:49 Some tools
0:31:21 Some tools (cont.)
0:32:28 Proof sketch
0:35:28 Proof sketch (cont.)
0:39:16 Remarks on subconvexity exponent
0:41:24 GL3: q-aspect case
0:43:39 Comparison: Bloomer vs Conrey-Iwaniec
0:44:42 Comparison: Bloomer vs Conrey-Iwaniec Conrey-Iwaniec (F = 1 0 1 0 1)
0:45:06 Remarks on Bloomer
0:46:05 GL3 revisited: q composite
0:47:28 Sketch for L(. . . , I/2) q5/4-o, self-dual
0:50:09 Thank you
0:50:17 Q&A