Subconvexity, Orbit Method, and Microlocal Analysis

Organizers: Aditya Ghosh, Dorian Goldfeld, Austin Lei, Alan Zhao

The first result in subconvexity was proved in the 1920s by Hardy-Littlewood-Weyl for the Riemann Zeta function: $$|\zeta(1/2 + it) | \ll (1 + |t|)^{1/6}$$

Since then, the question has been generalised to L-functions of modular forms and Maass forms of higher rank.

For a Maass form or an automorphic form $f$, the general subconvexity problem looks like $$L(f, s)\ll C(f,s)^{1/4 - \delta} $$ where $C(f,s)$ is the conductor of $f$, which measures the complexity of $f$. The case of $\delta=0 $ is the convexity result, which follows from the Phragmen-Lindelof principle. Any result for $\delta$ is consequently termed as a subconvex bound.

Over time, obtaining subconvex bounds has become a testing ground for results in Analytic Number theory and the theory of integral representation. Subconvex bounds have been used to prove equidistribution of lattice points and Quantum Unique Ergodicity. In the course of the seminar we will investigate these various approaches, especially the recent results by Michel-Venkatesh and Paul Nelson which incorporate microlocal analysis and the orbit method to prove subconvex bounds for high rank.

Here is a tentative, nonexhaustive list of topics we plan to discuss at some point during the seminar:

• Kuznetsov Trace Formula and Voronoi Summation
• Delta Method
• Amplification Method
• Relative Trace Formula
• Gindikin-Karpelevich Method
• The Nelson-Venkatesh Orbit Method and Microlocal Analysis
• The $\text{GL}(2)$ Supnorm Problem
• Subconvexity for the unitary group
• Triple Product L-function Subconvexity
• $\text{GL}(n)$ Subconvexity