2023年科大合工大数论会议
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2023 USTC-HFUT Number Theory Conference
会议信息
| 时间： | 2023年7月2日至2023年7月5日 |
| 地点： | 中国科学技术大学第二教学楼2503 |
| 主办单位： | 中国科学技术大学数学科学学院 |
| 合肥工业大学数学学院 | |
| 会务联系： | 张神星 zhangshenxing@hfut.edu.cn |
| 杨金榜 yjb@ustc.edu.cn |
报告安排
| 7月3日 09:00-09:50 许跃 (晨兴数学中心) Joint distribution of 2∞-class groups of quadratic fields In 2017, A. Smith made a breakthrough in the study of the distributions of 2∞-class groups and 2∞-Selmer groups, as predicted by Cohen-Lenstra-Gerth and Delaunay, respectively. Recently, Koymans and Pagano have extended Smith's method to prove Stevenhagen's conjecture for the negative Pell equation. Building upon their work, we investigate the joint distribution of 2∞-class groups in quadratic fields. This talk is based on joint work with Xiuwu Zhu. |
| 7月3日 10:00-10:50 蔡立 (首都师范大学) On the archimedean arithmetic smooth matching We will firstly talk about the relative trace formula approach to the Gross-Zagier formula, especially the archimedean arithmetic smooth matching. Then we discuss a general problem reducing the semi-global comparison to the local one. The talk is based on two joint works: one with Ye Tian, Xinyi Yuan and Wei Zhang, and the another one with Ye Tian. |
| 7月3日 11:00-11:50 许宾 (四川大学) Concrete constructions of automorphic representations and central values of 𝐿-functions The central values of 𝐿-functions play an important role in number theory and automorphic representation theory. In this talk, we will recall some relations among global packets, automorphic periods, and automorphic 𝐿-functions. Then, in the case of PGL(2), we introduce a new approach to show the existence of different quadratic twists with non-zero central 𝐿-values. The new approach is based on concrete constructions of automorphic representations. This talk is based on a joint work with Baiying Liu. |
| 7月3日 14:00-14:50 丁一文 (北京大学) Change of weights for locally analytic representations of GL2 (ℚ𝑝) Let 𝐷1⊂𝐷2 be (φ,Γ)-modules of rank 2 over the Robba ring. Let π(𝐷1),π(𝐷2) be the associated locally analytuc representations of GL2(ℚ𝑝) via 𝑝-adic local Langlands correspondence. We describe the relation of π(𝐷1) and π(𝐷2). |
| 7月3日 15:10-16:00 王善文 (中国人民大学) Monodromies Monodromies appear naturally in theory of log p-divisible group, Grothendieck’s panachee extensions and the theory of semi-stable representations. We establish the relation among these three monodromies. This is a joint work in progress with A. Bertapelle and H. Zhao. |
| 7月3日 16:30-17:20 范洋宇 (首都师范大学) Canonical local periods in families Period integrals are basic objects in automorphic representation theory. According to the conjecture of Sakellaridis-Venkatesh, period integrals can be decomposed into the product of 𝐿-value and the so-called canonical local periods. To this regard, canonical local periods are generalizations of local zeta integrals. In this talk, motivated by the construction of 𝑝-adic 𝐿-functions, we will study behaviors of these canonical local periods in families of representations. This is based on a joint work with Li Cai. |
| 7月4日 09:30-10:20 王浩然 (清华大学) On mod 𝑝 cohomology of Shimura curves The mod 𝑝 local Langlands correspondence is well-understood for GL(2,ℚ𝑝) by Breuil, Colmez, Emerton, Paskunas, et. al.. In order to understand the mod 𝑝 Langlands program for higher rank groups, the study of the mod 𝑝 cohomology of Shimura varieties seems to be an important approach. We will report some recent joint works with Yongquan Hu on the study of cohomology of Shimura curves. |
| 7月4日 10:50-11:40 Ashay Burungale (UT Austin) Zeta elements over imaginary quadratic fields Let 𝐸 be an elliptic curve over ℚ of conductor 𝑁. For a prime 𝑝∤2𝑁 split in an imaginary quadratic field 𝐾, the talk will outline the existence of 𝑝-adic zeta element for 𝐸 over 𝐾, and its applications to the arithmetic of 𝐸 over ℚ. (Joint with C. Skinner, Y. Tian and X. Wan.) |
| 7月4日 14:00-14:50 尹洪波 (山东大学) The cube sum problem One old question in number theory is to determine whether an integer can be written as the sum of two nonzero rational cubes. The Sylvester conjecture predicts that for every prime 𝑝≡4,7,8 mod 9, the answer is positive. This conjecture is quite open and only has some partial results. In this talk, I will introduce the background of cube sum problem and some recent progress. |
| 7月4日 15:10-16:00 舒杰 (同济大学) On certain families of Fermat Jacobians of positive ranks We present certain families of Fermat Jacobians with positive ranks. |
| 7月4日 16:30-17:20 谢建峰 (北京大学) The growth of Tate-Shafarevich groups in cyclic extensions Let 𝐾 be a global field and 𝑝 be a prime number with 𝑝≠char 𝐾. A classical theorem in algebraic number theory asserts that when 𝐿 varies in ℤ/𝑝ℤ-extensions of 𝐾, the 𝑝-rank of the class group of 𝐿 is unbounded. It is expected that similar unboundenss result also holds for other arithmetic objects. Based on Cassels-Poitou-Tate sequence, K. Česnavičius proved that for fixed abelian variety 𝐴 over 𝐾, the 𝑝-Selmer group of 𝐴 over 𝐿 is unbounded when 𝐿 varies in ℤ/𝑝ℤ-extensions of 𝐾. And he raised a further problem: in the same setting, does the Tate-Shafarevich group of 𝐴 also grow unboundedly? Using a machinery developed by B. Mazur and K. Rubin, we give a positive answer to this problem. This is a joint work with Yi Ouyang. |