摘要： ?
We will discuss the Gross-Zagier theorem for modular curves and ?introduce its high dimensional generalizations, both over number ?fields and over function fields.
The main objects of the talks ?are special values of L-functions and their connection to arithmetic ?intersection numbers of modular cycles, i.e., special algebraic cycles ?on the moduli space of abelian varieties with certain additional ?structures.
In the end we discuss the arithmetic fundamental ?lemma conjecture which connects intersection numbers of modular cycles ?over p-adic fields to lattice-counting problems.
?报告人简介： ?
张伟，2004年毕业于北京大学数学科学学院，获学士学位。2009年获美国哥伦比亚大学博士学位。曾任美国哥伦比亚大学教授，现为美国麻省理工学院教授。
他与恽之玮因对Gan-Gross-Prasad猜想和函数域上L-函数的高阶导数公式做出的贡献同获2017科学突破奖之数学新视野奖。他在2018国际数学家大会上所作报告题目为“Periods, ?Cycles, and L -functions: A Relative Trace Formula ?Approach”。受Gross-Zagier和Waldspurger公式的启发，他回顾了关于自守周期积分的猜想和定理，志村簇上的特殊代数闭链，以及它们对于L-函数及其导数的中心值的联系。 ?着重于数域上的整体Gan-Gross-Prasad猜想和算术版本，讨论了相对迹公式和算术基本引理猜想等研究成果。
Wei ?Zhang joined the MIT Math Faculty as Professor in September, 2017. He ?received a BS from Peking University in 2004 and a PhD from Columbia ?University in 2009, both degrees in mathematics.
Zhang works in ?number theory, automorphic forms, and arithmetic geometry. His ?research program involves fundamental objects such as L-functions, ?which appear in the Riemann hypothesis and its generalizations, and ?are central to the Langlands program. He has been developing the ?theory of relative trace formula to connect L-functions with ?algebro-geometric objects, notably the discovery of "arithmetic ?fundamental lemmas". In the function field case, he and Zhiwei ?Yun have discovered geometric interpretations for the higher ?derivatives of L-functions and received the 2018 New Horizons ?Breakthrough Prize in Mathematics. In 2018, he gave an invited address ?with title “Periods, Cycles, and L -functions: A Relative Trace ?Formula Approach” at ICM 2018 in Rio de Janeiro. In another direction, ?Zhang and his collaborators have made significant advances on ?conjectures subsequent to the Gross--Zagier theorem on elliptic ?curves, including a proof of the Birch and Swinnerton-Dyer conjecture ?for approximately 66% of elliptic curves.