L-functions, including the Riemann zeta function, are one of the main research objects in analytic number theory. The estimation of moments of L-functions is a central problem in number theory and has important applications in automorphic forms, quantum chaos, etc.

The Rankin-Selberg problem is about improving the error term of the second moment of the Fourier coefficients of a GL(2) automorphic form proved by Rankin and Selberg in 1939/1940. This awarded work breaks this long-standing barrier for the first time in 2021 and proves a subconvexity exponent. The core of the proof is to transform the problem into moments of L-functions and connect it to the subconvexity problem of degree 3 L-functions, so as to solve the problem using the delta method.

Arithmetic quantum chaos studies chaotic systems with arithmetic structure, and arithmetic hyperbolic surfaces are one of the main models. The value distribution of eigenfunctions of its Laplace operator, namely Maass forms, in the semiclassical limit is one of the main research problems, including the random wave conjecture and the quantum fluctuation conjecture. This awarded work uses the estimation of moments of L-functions to solve the cubic moment problem of the Hecke-Maass forms and the quantum variance problem of the Eisenstein series. Compared with the quantum unique ergodicity (i.e., the second moment), this result obtains quantitative upper bounds in the cubic moments case.

Affine Deligne-Lusztig varieties are group-theoretic models for reductions of Shimura varieties, and play an important role in arithmetic geometry and Langlands program. The classification of their irreducible components is a fundamental open problem, which has essential applications in various important topics such as the Tate conjecture of Shimura varieties. To solve it, Miaofen Chen and Xinwen Zhu proposed a remarkable conjecture: the orbits of irreducible components are in canonical one-to-one correspondence with certain crystal basis of the Weyl module. By constructing a crystal structure on the irreducible components, this awarded work completely solved the conjecture of Chen and Zhu. Moreover, this awarded work obtained an algorithm to compute the stabilizer of each irreducible component. This in principle solves the classification problem of irreducible components of affine Deligne-Lusztig varieties.