中国科学院数学与系统科学研究院

数学研究所

中国科学院华罗庚数学重点实验室

华罗庚青年数学论坛

学术报告

报告人： 杨梓诠 博士（University of Wisconsin–Madison）
题  目：Arithmetic Deformation of Line Bundles
时  间：2023.09.21（星期四），09:00-10:00
地  点：腾讯会议：587-534-625  密码：0921
摘  要：In this talk, I address the following question: In an arithmetic family, can we obtain every line bundle in characteristic p by specializing those in characteristic zero? I will first explain that, when h^{2,0} = 1, being able to answer this questionhas important applications to the Tate/BSD conjecture, as a continuation of the previous talk. Then I will discuss a generalization when h^{2, 0} > 1. The main theorem is that, as long as the family has sufficiently big monodromy and Kodaira-Spencer map, thequestion generically has a positive answer. Examples include general hypersurface in P^3 or an elliptic surface over P^1. The method involves an application of theAx-Schanuel theorem in Hodge theory, recently established by Tsimerman and Baker. This generalization is based on a joint work with David Urbanik.
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报告人： 杨梓诠 博士（University of Wisconsin–Madison）
题  目：Twisted sheaves, Tate conjecture, and Good Reduction for K3's 
时  间：2023.09.28（星期四），09:00-10:00
地  点：腾讯会议：666-956-609   密码：0928
摘  要：I will discuss an isogeny theory for K3 surfaces which resembles that for abelian varieties. The theory is based on the notion of twisted sheaves and Fourier-Mukai equivalences. Then I will discuss several arithmetic applications of the theory, includinga Neron-Ogg-Shafarevich criterion for K3's over p-adic fields, and new proofs of the Tate conjecture in the ordinary case and the semisimplicity conjecture for K3's over finite fields. The point of the new proofs is that they use only the geometry of K3 surfaces,as opposed to resorting to abelian varieties in the end as in literature, and shed new light on how the Tate conjecture is related to certain finiteness statements in arithmetic geometry. This talk is in part based on a joint work with Daniel Bragg.