2014 Workshop on Number Theory

July 20th - July 28th, 2014

Room 110Morningside Center of Mathematics

Organizers

·         Professor Ye Tian (Morningside Center of Mathematics)

·         Professor Yangbo Ye (The University of Iowa)

Morningside Center of Mathematics, Chinese Academy of Sciences

Speakers

·         Shuvra Gupta (The University of Iowa)

·         Joseph Hundley (Southern Illinois University)

·         Yeansu Kim (The University of Iowa)

·         Angel V. Kumchev (Towson University)

·         Yifeng Liu (Massachusetts Institute of Technology)

·         Guangshi Lyu (Shandong Universtiy)

·         Paul Savala (The Universtiy of Iowa)

·         Xinyi Yuan (University of California – Berkeley)

·         Deyu Zhang (Shandong Normal University)

·         Yitang Zhang (University of New Hampshire)

Topics

·         analytic number theory

·         Galois represetnations

·         automorphic L-functions and representations

·         Eisenstein series

·         arithmetic algebraic geometry

Schedule of Talks

 9:15-10:30 10:45-12:00 14:00-15:15 15:45-17:00 Jul. 20 Y. Zhang Kumchev Gupta Kim Jul. 21 Liu Kumchev Kim Gupta Jul. 22 Gupta Liu Kim Lyu Jul. 23 Liu Kim Free Jul. 24 Free Jul. 25 Hundley Liu Kumchev D. Zhang Jul. 26 Yuan Hundley Free Jul. 27 Kumchev Yuan Hundley Savala Jul. 28 Yuan Hundley Gupta Y. Zhang

Titles and Abstracts

·         Title: Galois Representations
Speaker: Shuvra Gupta
Abstract: We are going to talk about Galois Representations, some classical theorems including one which shows how these representations naturally arise from Hecke eigneforms. We are then going to see some applications of techniques from Galois Representations in answering special cases of the Inverse Galois Problem.

·         Title: Eisenstein series for GL(2) and GL(n)
Speaker: Joseph Hundley
Abstract: I thought that I would give some expository lectures in the same spirit as the textbook that I wrote with Goldfeld, covering some topics which were omitted from the book for reasons of space. I thought I would focus on Eisenstein series.

·         Title: Langlands Program: Eisensetein series and Langlands-Shahidi Method
Speaker: Yeansu Kim
Abstract: We will mainly study Langlands-Shahidi method and two main conjectures in Langlands program, i.e., local Langlands conjecture and Langlands functoriality conjecture. Briefly, the local Langlands conjecture, introduced by Langlands, asserts that there exists a `natural' bijection between two di erent sets of objects: Arithmetic (Galois or Weil-Deligne) side and analytic (representation theoretic) side. The Langlands-Shahidi method is to study L-functions that attached to generic representations that appear in the constant terms of Eisenstein series. Therefore, the Eisenstein series is also one of the main objects that we are going to study. We will also study Langlands functoriality conjecture which can be constructed in some cases using the Langlands-Shahidi method and the converse theorems. After I explain two main conjectures, we will explain recent results about local Langlands correspondence and the equality of L-functions through local Langlands correspondence in the generic case.
Outline:
1. Motivation and preliminaries (1.5 Lecture)
a. Local Langlands correspondence and Langlands functoriality conjecture
b. Basic theory of linear algebraic groups
2. Eisenstein series I (1 Lecture)
a. Basic theory of Eisenstein series
3. Eisenstein series II (0.5 Lecture)
a. Properties of L-functions from Langlands-Shahidi method
4. Applications (1 Lecture)
a. Converse theorem and functoriality
b. Recent results in the case of classical groups and GSpin groups

·         Title: Sieve methods and exponential sums: The interplay between combinatorics and harmonic analysis
Speaker: Angel V. Kumchev
Abstract: Exponential sums over primes have many applications in analytic number theory. The first estimates for such sums were obtained in the late 1930’s by I.M. Vinogradov, who used an elaborate combinatorial method to reduce the estimation of sums over primes to the estimation of “double sums.” By the early 1980’s, Vinogradov’s combinatorial method was supplanted in most applications by combinatorial identities discovered by R.C. Vaughan and D.R. Heath-Brown. Subsequent work has revealed that--when considered within the context of combinatorial sieve theory, those two approaches have much more in common than meets the eye. Moreover, the sieve viewpoint to such estimates leads to quantitative improvements in many applications, since it often allows for a more efficient deployment of the available harmonic analytic tools. In these lectures, I will explain the philosophy of the combinatorial sieve approach towards exponential sums over primes, including several applications.

·         Title: Periods and the Birch-Swinnerton-Dyer conjecture
Speaker: Yifeng Liu
Abstract: The Birch and Swinnerton-Dyer (BSD) conjecture is one of the most important conjectures in number theory. It predicts a very deep relation between the solution of an elliptic curve and its L-function. In this series of four lectures, we will prove a very special case about a rational elliptic curve E (satisfying certain assumptions). Namely, if the central L-value attached to E is non-vanishing, then the Mordell-Weil rank of E is zero and the p-part of its Tate-Shafarevich group is trivial for all but finitely many primes p. The proof we are going to present will be almost self-contained, and is not recorded in any literature as far as I know (of course, experts in this field would certainly know). It does not use Kolyvagin's Euler system, but a certain form of level raising instead. If time permits, we will discuss how to generalize the picture to motives of higher rank.

·         Title: Shifted convolution sum of cusp-forms with $\theta$-series
Speaker: Guangshi Lyu
Abstract: In recent joint work with Jie Wu and Wenguang Zhai, we introduce a simple approach to improve a recent result due to Luo, concerning a shifted convolution sum involving the Fourier coefficients of cusp forms with those of theta series.

·         Title: Computing Maass Forms Using Resonance
Speaker: Paul Savala
Abstract: In recent years there has been interest in computing the Fourier coecients of automorphic forms, especially for higher dimensions. Recent work in this area has focused on using a twisted functional equation to compute the rst few hundred Fourier coecients.In this talk I will discuss a di erent approach to computing these Fourier coecents. In particular we use the resonance properties of Maass forms to compute Fourier coecients of high index. Our work focuses on GL(2) forms as a stepping stone to generalizing to GL(3) and above.

·         Title: Introduction to the Tate-Shafarevich group
Speaker: Xinyi Yuan
Abstract: The goal of this lecture series is to introduce the Tate-Shafarevich group of elliptic curve over global fields, including the definition and basic properties, the finiteness conjecture and known cases, the role in the BSD conjecture.

·         Title: Zero density for automorphic L-functions
Speaker: Deyu Zhang
Abstract: In this talk, we introduce zero-density estimates for automorphic L-functions L(s,π) for GL_n, which is deduced from a bound for an integral power moment of L(s,π) on the critical line Re(s) = 1/2. In particular for the Riemann zeta function, classical zero density estimates are extended to short vertical strips. For g being a holomorphic or Maass eigenform for SL_2(Z), bounds for zero density for L(s,g) in short strips are proved, which extend Ivic's results on long strips. For a self-dual Hecke Maass eigenform f for SL_3(Z), estimates of zero density for L(s,f) in short and long strips are also proved.

·         Title: Recent progress on distribution of primes in arithmetic progressions
Speaker: Yitang Zhang
Abstract: It has been believed for long time that primes have uniform distribution in arithmetic progressions to moduli greater than the square root of the length of interval. Results of this type have important consequences, such as the bounded gaps between primes. In this talk we will describe the recent progress and some related problems.