Langlands program and Ramanujan Conjecture: A survey

We present topics in the Langlands Program to graduate students and a wider mathematically mature audience. We study both global and local aspects in characteristic zero as well as characteristic p p . We look at modern approaches to the generalized Ramanujan conjecture, which is an open problem over number fields, and present known cases over function fields. In particular, we study automorphic L L -functions via the Langlands-Shahidi method. Hopefully, our approach to the Langlands Program can help guide the interested reader into an exciting field of mathematical research.

Average Bound Toward the Generalized Ramanujan Conjecture and Its Applications on Sato–Tate Laws for GL(n)

We give the 1st non-trivial estimate for the number of $GL(n)$ ($n\ge 3$) Hecke–Maass forms whose Satake parameters at any given prime $p$ fail the Generalized Ramanujan Conjecture and study some applications on the (vertical) Sato–Tate laws.

Optimal sup norm bounds for newforms on GL2 with maximally ramified central character

Recently, the problem of bounding the sup norms of {L^{2}}-normalized cuspidal automorphic newforms ϕ on {\mathrm{GL}_{2}} in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character χ of ϕ is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general χ. If the level N is a square, our result reduces to\|\phi\|_{\infty}\ll N^{\frac{1}{4}+\epsilon},at least under the Ramanujan Conjecture. In particular, when χ has conductor N, this improves upon the previous best known bound {\|\phi\|_{\infty}\ll N^{\frac{1}{2}+\epsilon}} in this setup (due to [A. Saha, Hybrid sup-norm bounds for Maass newforms of powerful level, Algebra Number Theory 11 2017, 1009–1045]) and matches a lower bound due to [N. Templier, Large values of modular forms, Camb. J. Math. 2 2014, 1, 91–116], thus our result is essentially optimal in this case.

Bounds for twisted symmetric square L-functions via half-integral weight periods

We establish the first moment bound $$\begin{align*}\sum_{\varphi} L(\varphi \otimes \varphi \otimes \Psi, \tfrac{1}{2}) \ll_\varepsilon p^{5/4+\varepsilon} \end{align*}$$ for triple product L-functions, where $\Psi $ is a fixed Hecke–Maass form on $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ and $\varphi $ runs over the Hecke–Maass newforms on $\Gamma _0(p)$ of bounded eigenvalue. The proof is via the theta correspondence and analysis of periods of half-integral weight modular forms. This estimate is not expected to be optimal, but the exponent $5/4$ is the strongest obtained to date for a moment problem of this shape. We show that the expected upper bound follows if one assumes the Ramanujan conjecture in both the integral and half-integral weight cases. Under the triple product formula, our result may be understood as a strong level aspect form of quantum ergodicity: for a large prime p, all but very few Hecke–Maass newforms on $\Gamma _0(p) \backslash \mathbb {H}$ of bounded eigenvalue have very uniformly distributed mass after pushforward to $\operatorname {\mathrm {SL}}_2(\mathbb {Z}) \backslash \mathbb {H}$ . Our main result turns out to be closely related to estimates such as $$\begin{align*}\sum_{|n| < p} L(\Psi \otimes \chi_{n p},\tfrac{1}{2}) \ll p, \end{align*}$$ where the sum is over those n for which $n p$ is a fundamental discriminant and $\chi _{n p}$ denotes the corresponding quadratic character. Such estimates improve upon bounds of Duke–Iwaniec.

The Ramanujan conjecture and its applications

In this paper, we review the Ramanujan conjecture in classical and modern settings and explain its various applications in computer science, including the explicit constructions of the spectrally extremal combinatorial objects, called Ramanujan graphs and Ramanujan complexes, points uniformly distributed on spheres, and Golden-Gate Sets in quantum computing. The connection between Ramanujan graphs/complexes and their zeta functions satisfying the Riemann hypothesis is also discussed. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.

From Ramanujan graphs to Ramanujan complexes

Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science. In recent years, a high-dimensional theory has emerged. In this paper, these developments are surveyed. After explaining their connection to the Ramanujan conjecture, we will present some old and new results with an emphasis on random walks on these discrete objects and on the Euclidean spheres. The latter lead to ‘golden gates’ which are of importance in quantum computation. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.