Number Theory

An explicit understanding of the category of all (smooth, complex) representations of p-adic groups provides an important tool in the construction of an explicit and a categorical local Langlands correspondence and also has applications to the study of automorphic forms. The category of…

Abstract: The k-th power Weyl sum is S_k(N,a)=\sum_{n\le N} \exp(2\pi i an^k), where a is a real parameter. The classical bound takes the form O_{k,a,c}(N^c), for any c > 1-2^{1-k}, whenever a is well-approximable by rationals. This is best possible for k=2, and has not been improved for 100…

Abstract:  For many Diophantine equations or systems, the number of solutions within a box of side length N can grow like a power of N. Obtaining a nontrivial upper bound for the exponent is crucial for various problems. Recently, an analytic method called ``decoupling'' has been successful…

We will discuss joint work with Si-Ying Lee on generalizing the torsion-vanishing results of Caraiani-Scholze and Koshikawa for the cohomology of Shimura varieties. This is accomplished by combining a variety of geometric methods based on the Fargues-Fontaine curve. In the process, we also…

Abstract: The delta symbol developed by Duke-Friedlander-Iwaniec and Heath-Brown has played an important role in studying rational points on hypersurfaces of low degrees. We present a two dimensional delta symbol and apply it to establish a quantitative Hasse principle for a…

The successive minima of an order in a degree n number field are n real numbers encoding information about the Euclidean structure of the order. How many orders in degree n number fields are there with almost prescribed successive minima, fixed Galois group, and bounded discriminant? 

Abstract:  In this talk we are going to explore “real” zeros of holomorphic Hecke cusp of large weight on the modular surface. Ghosh and Sarnak established that the number of real zeros tends to infinity as the weight $k$ goes to infinity. To do so, they studied the behavior of holomorphic…

Abstract: The Serre weight conjectures for the modular curve imply an equivalence between mod p modular forms of weight 1 and 2-dimensional odd irreducible mod p representations which are unramified at p. This has a natural extension to quaternionic Shimura varieties. I will talk about work in…

We prove the infinitude of shifted primes p-1 without prime factors above p^{0.2844}. This refines p^{0.2961} from Baker and Harman in 1998. Consequently, we obtain an improved lower bound on the distribution of Carmichael numbers. Our main technical result is a new mean value theorem for primes…

The Breuil-Mezard Conjecture predicts the existence of hypothetical "Breuil-Mezard cycles" in the moduli space of mod p Galois representations of Gal_{Q_p} that should govern congruences between mod p automorphic forms. I will talk about joint work with Bao Le Hung on a new approach to the…