山大数论讨论班

星期三，15:00-16:00，2021春季

中心校区明德楼C702，济南

组织者：刘建亚 和 黄炳荣

讨论班着重于数论中的新进展。报告人为校内外学者和学生。欢迎推荐报告。

报告：2021年春季学期  （星期三，15:00-16:00）

• 2021年03月02日，Jacques Benatar (Tel Aviv University) （注意报告时间的不同）

  题目：Moments of random trigonometric polynomials with multiplicative coefficients

  摘要：In this talk I will discuss some recent work, joint with Alon Nishry and Brad Rodgers, concerning the distribution of trigonometric polynomials

  Q_f,N(θ) = ∑_n≤N f(n) e(nθ).

  We will first discuss some results and conjectures for deterministic sequences f(n) and then move on to random coefficients f(n) = X(n) which are multiplicatively generated by a sequence of Rademacher or  teinhaus random variables. I will give a Salem-Zygmund type result for the distribution of Q_X,N and explain how it relates to an arithmetic point counting problem, via a moment method. Obtaining almost sure estimates for ||Q_X,N||_∞ presents additional challenges.

  腾讯会议，会议 ID：995 338 269

• 2021年03月10日，管彬 （山东大学）

  题目：Relative trace formula and averages of central values of triple product L-functions

  摘要：Feigon and Whitehouse studied central values of triple product L-functions averaged over newforms of weight 2 and prime level. They proved some exact formulas applying the results of Gross and Kudla which link central values of triple L-functions to classical "periods". In this talk, I will generalize their results for more cases using Jacquet's relative trace formula and Ichino's period formula, and apply these average formulas to the nonvanishing problem. 

  地点：明德楼C702

• 2021年03月17日，翟帅 （University of Cambridge）

  题目：Classical Diophantine problems and the Birch and Swinnerton-Dyer conjecture

  摘要：The Birch and Swinnerton-Dyer conjecture is one of the principal open problems of number theory today, which asserts that the size of the group of rational points on an elliptic curve is related to the behavior of the central value of the associated L-function. In this lecture, I will present a few classical Diophantine problems, and their connection to the conjecture of Birch and Swinnerton-Dyer.

• 2021年03月24日，魏菲 （清华大学）

  题目：Disjointness of Moebius from rigid dynamical systems

  摘要：In this talk, I will introduce some results about Sarnak’s Moebius Disjointness Conjecture for rigid dynamical systems, and the estimate involved in the average value of the Moebius function in short arithmetic progressions.

  腾讯会议，会议 ID：995 338 269

• 2021年03月31日，于树澄 （Uppsala University）

  题目：A quantitative Khintchine-Groshev Theorem with congruence conditions

  摘要：Given a non-increasing approximating function $\psi$, the classical Khintchine-Groshev Theorem gives a zero-one law on the Lebesgue measure of the set of $\psi$-approximable matrices. In this talk we will discuss a quantitative Khintchine-Groshev Theorem with an extra congruence condition. Our method replies on a classical lattice point counting result of Schmidt. This is joint work with Mahbub Alam and Anish Ghosh.

  腾讯会议，会议 ID：995 338 269

• 2021年04月07日，高鹏 （北京航空航天大学）

  题目：Bounds for moments of cubic and quartic Dirichlet L-functions

  摘要：Moments of L-functions at the central point over a family of characters of a fixed order have been extensively studied in the literature. Although many results are available on moments of quadratic Dirichlet L-functions, less is known for the moments of higher order Dirichlet L-functions. In this talk, I will explain how to apply the recent lower bounds principal of Heap and Soundararajan and an earlier method of Soundararajan with a refinement by Harper to obtain sharp bounds for moments of central values of cubic and quartic Dirichlet L-functions.  This is a joint work with L. Zhao.

  腾讯会议，会议 ID：995 338 269

• 2021年04月14、21、28日，无报告。（线下有数论方向短期课程。）

• 2021年05月05日，无报告。五一假期。

• 2021年05月12日，吴峙佑 （Max Planck Institute for Mathematics） 15:00 - 17:00

  题目：P-adic Hodge, prisms and Galois representations

  摘要：P-adic Hodge theory is an analogue of classical Hodge theory in complex geometry, it plays an important role in the study of Galois representations, which is fundamental in algebraic number theory. Recently, Peter Scholze and Bhargav Bhatt found a new perspective on p-adic Hodge theory, namely the theory of prisms. In a recent preprint, we prove that Galois representations and (φ,Γ)-modules can both be incorporated in this new framework, and we can recover the classical result that these two objects are equivalent. In this talk, I will explain this result together with the general background.

• 2021年05月19日，李俊贤 （Max Planck Institute for Mathematics）

  题目：Uniform Titchmarsh divisor problems

  摘要：The Titchmarsh divisor problem asks for an asymptotic evaluation of the average of the divisor function evaluated at shifted primes. We will discuss how strong error terms that are uniform in the shift parameters could be obtained using spectral theory of automorphic forms. We will also discuss the automorphic analogue of the Titchmarsh divisor problem. This is a joint work with E. Assing and V. Blomer.

  腾讯会议，会议 ID：995 338 269

• 2021年05月26日，无报告。硕士博士毕业论文答辩。

• 2021年06月02日，胡广伟 （山东大学）

  题目：The circle method and shifted convolution sums

  摘要：In this talk，we will consider some shifted convolution sums involving the Fourier coefficients of theta series. Here some approaches are devoted to generalize and improve previous results, and our results do not depend on the Ramanujan conjecture. This is a joint work with Guangshi Lü.

• 2021年06月09日，尹洪波 （山东大学）

  题目：Borcherds Product and CM Value of Lambda function

  摘要：In this talk, we will talk about Gross-Zagier's type factorization formulas for the CM value of special modular functions using the Borcherds product and Bruinier-Kudla-Yang's big CM value formula.  We will illustrate the method using the Lambda modular function which is a joint work with Tonghai Yang and Peng Yu. 

• 2021年06月16日和17号，齐治 （浙江大学）  时间：16日 16:00-17:00,  17日 10:00-11:00

  题目：Some results on the central L-values for GL(2) and GL(3)

  摘要：In these talks, I will talk about the moments of central L-values for GL(2) Maass forms and the subconex problem for self-dual GL(3) Maass forms over imaginary quadratic fields. Our main tools are the Kuznetsov trace formula for GL(2) and Voronoi summation formulae for GL(2) and GL(3) over number fields. My focus will be on the analysis on the complex numbers. These are joint works with Sheng-chi Liu.