This is a weekly seminar focus on the research on number theory and arithmetic algebraic geometry. It usually takes place 10:30-11:30am each Tuesday or Wednesday at MCM 110.

If you have any questions, please contact with Prof. Yichao Tian (yichaot@math.ac.cn).

**Place: **MCM110

**Title: **Integral points on the Clebsch-Klein surfaces

**Abstract: **In this talk we present explicit bounds for the Weil height and the number of integral points on classical surfaces first studied by Clebsch (1871) and Klein (1873). Building on Hirzebruch's work in which he related these surfaces to a Hilbert modular surface, we deduced our bounds from a general result for integral points on coarse Hilbert moduli schemes. After explaining this deduction, we discuss the strategy of proof of the general result which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity, Masser-Wuestholz isogeny estimates, and results based on effective analytic estimates and/or Arakelov theory. Joint work with Arno Kret.

**Speaker:** Prof. Yongqi Liang (University of Science and Technology of China) **Time:** 10:30-11:30am, September 20, 2023 (Wednesday)**Place:** MCM110**Title:** 奇亏格超椭圆曲线的局部整体原则**Abstract:** Scharaschkin和Skorobogatov等人猜测Brauer-Manin障碍是定义在数域上的射影光滑曲线上的有理点不满足局部整体原则的唯一障碍。对于任意数域和任意给定奇自然数g，我们具体构造出无穷条亏格为g的由Brauer-Manin障碍解释的局部整体原则失效的曲线。我们的结论为上述猜想给出了具体的例证。这是我指导的博士生黄凯最近的工作。

**Speaker:** Prof. Meng Fai Lim (Central China Normal University)**Time:** 10:30-11:30am, September 26, 2023 (Tuesday)**Place:** MCM110**Title:** On fine Selmer groups**Abstract:** The fine Selmer group has been a much studied object in Iwasawa theory. One of its original roles was to provide a means of formulating (and proving) the main conjecture. On the other hand, the work of Coates and Sujatha initated a systematic study on this group on its own right. My talk will focus on this latter aspect. I shall begin with giving a brief survey on the work of Coates-Sujatha. After that, I will then report some of my work on the following two themes.

(1) A torsioness conjecture on the fine Selmer group over an arbitrary Zp-extension.

(2) A question of Greenberg concerning an explicit description of the structure of the fine Selmer group over the cyclotomic Zp-extension of Q and formulation of its variants.

**Speaker:** Prof. Bin Xu (Tsinghua University)**Time:** 10:30-11:30am, October 11, 2023 (Wednesday)**Place:** MCM110**Title:** Global A-packets for GSp(2n) and GSO(2n) in the nontempered case**Abstract**: Arthur (1989) conjectured that the discrete spectrum of automorphic representations of a connected reductive group over a number field can be decomposed into global A-packets, in terms of which he also conjectured a multiplicity formula. Arthur (2013) proved his conjecture for symplectic and orthogonal groups, in which case the global A-packets are parametrized by self-dual automorphic representations of general linear groups. In this talk, I will report some recent progress on this conjecture for general symplectic and general even orthogonal groups in the nontempered case.

**Speaker:** Prof. Thomas J. Haines (University of Maryland)**Time:** 10:30-11:30am, October 17, 2023 (Tuesday)**Place:** MCM110**Title:** Cellular pavings for convolution morphisms and applications**Abstract:** The recent work of Cass-van den Hove-Scholbach on the Geometric Satake Equivalence for Integral Motives required cellular pavings of fibers of convolution morphisms for affine Grassmannians over $\bbZ$. In this talk, I will explain that cellular pavings exist for all convolution morphisms attached to any partial affine flag varieties of Chevalley groups. I will explain the proof by a reduction to the full affine flag varieties, using some properties of "negative parahoric loop groups". Then I will mention some applications, such as a direct proof of Frobenius semisimplicity results for intersection cohomology groups of affine Schubert varieties over finite fields, and the "rationality over the base field" of the BBD decomposition theorem in related situations.

**Speaker:** Prof. Huayi Chen (Westlake University)**Time:** 10:30-11:30am, October 18, 2023 (Wednesday)**Place:** MCM110**Title:** Arakelov geometry over adelic curves**Abstract:** In a series of works in collaboration with Atsushi Moriwaki, we develop an Arakelov theory over any countable field equipped with an adelic structure. In this talk, I will explain examples and constructions of adelic structures over general countable fields, and also what we could obtain under such a framework of Arakelov geometry.

**Speaker:** Prof. Haining Wang (Fudan University)**Time:** 10:30-11:30am, October 25, 2023 (Wednesday)**Place:** MCM110**Title:** Flach system and ratio of integral periods**Abstract:** In this talk, I will report some integrality result on the ratio of the quaternionic distinguished period associated to a Hilbert modular form and the quaternionic Petersson norm associated to a modular form. These distinguished periods are closely related to the notion of distinguished representations that play a prominent role in the proof of the Tate conjecture for Hilbert modular surfaces by Langlands-Rapoport-Harder. Our proof is based on an Euler system argument initiated by Flach by producing elements in the motivic cohomologies of the quaternionic Hilbert-Blumenthal surfaces with control of their ramification behaviours. We show that these distinguished periods give natural bounds for certain subspaces of the Selmer groups of these quaternionic Hilbert–Blumenthal surfaces. The lengths of these subspaces can be determined by using the Taylor-Wiles method and can be related to the quaternionic Petersson norms of the modular forms.

**Speaker:** Prof. Chia-Fu Yu (Institute of Mathematics, Academia Sinica of Taiwan)**Time:** 10:30-11:30am, November 1, 2023 (Wednesday)**Place:** MCM110**Title:** On the supersingular locus of Shimura varieties for quaternionic unitary groups**Abstract:** We study a Shimura variety attached to a unitary similitude group of a skew-Hermitian form over a totally indefinite quaternion algebra over a totally real number field. We give a necessary and sufficient condition for the existence of skew-Hermitian self-dual lattices, and show that the superspecial locus in the fiber at p of the associated Shimura variety is non-empty. We also give an explicit formula for the number of irreducible components of the supersingular locus when p is odd and unramified in the quaternion algebra. This talk is the joint work with Yasuhiro Terakado and Jiangwei Xue.

**Speaker:** Prof. Mao Sheng (University of Science and Technology of China)**Time:** 10:30-11:30am, November 8, 2023 (Wednesday)**Place:** MCM110**Title:** P-adic Simpson correspondence via exponential twisting**Abstract:** In this talk, I shall explain the work of G. Faltings on a p-adic Simpson correspondence, using exponential twisting. This reinterpretation of the correspondence makes clearer analogy between the p-adic Simpson correspondence of Faltings and the char p Simpson correspondence due to Ogus-Vologodsky. This is a joint work with Zhaofeng Yu.

**Speaker:** Prof. Enlin Yang (Peking University)**Time:** 10:30-11:30am, November 15, 2023 (Wednesday)**Place:** MCM110**Title:** Cohomological Milnor formula for constructible etale sheaves**Abstract:** In this talk, we will sketch the construction of non-acyclicity classes for constructible etale sheaves on (not necessarily smooth) varieties, which is defined in a recent joint work with Yigeng Zhao. This cohomological class is supported on the non-locally acyclicity locus. As applications, we show that the Milnor formula and Bloch's conductor formula can be reformulated in terms of the functorial properties of non-acyclicity classes. Based on this formalism, we propose a Milnor type formula for non-isolated singularities.

**Speaker:** Prof. Haoyu Hu (Nanjing University)**Time:** 10:30-11:30am, November 22, 2023 (Wednesday)**Place:** MCM110**Title:** Betti numbers of étale sheaves**Abstract:** The calculation of Betti numbers of étale sheaves has many applications in number theory. For example, the study of exponential sums requires Betti numbers of Artin-Schreier type sheaves on affine spaces. In this talk, we discuss a boundedness result for Betti numbers of étale sheaves with wild ramifications on affine and smooth schemes. This is a joint work with Jean-Baptiste Teyssier.

**Speaker:** Prof. Jinbo Ren (Xiamen University）**Time:** 10:30-11:30am, December 6, 2023 (Wednesday)**Place:** MCM110**Title:** Bounded Generation: a diophantine approximation approach**Abstract:** An abstract group is said to have the {\it bounded generation} property (BG) if it can be written as a product of finitely many cyclic subgroups. Being a purely combinatorial notion, bounded generation has close relation with many group theoretical problems including semisimple rigidity, Kazhdan's property (T) and Serre's congruence subgroup problem.

This talk is devoted to explain how to use the Laurent's theorem in Diophantine approximation to prove that an infinite $S$-arithmetic subgroup of an anisotropic linear algebraic group $G$ over a number field $K$ {\it never} has (BG). If time allows, I will also introduce our newly obtained asymptotic formula for counting the elements of a (BG) set inside $GL_n(K)$ ($K$ is a number field) when ordered by heights, together with some applications of this formula.

The novelty of this project relies on the deep subspace theorem by Schlickewei-Schmidt as well as the theory of generic elements by Prasad-Rapinchuk. This is joint work with Corvaja, Demeio, Rapinchuk and Zannier.

**Speaker:** Prof. Lei Fu (Tsinghua University）**Time:** 10:30-11:30am, December 13, 2023 (Wednesday)**Place:** MCM110**Title:** Exponential sums on reductive groups**Abstract:** For a family of irreducible representations of a reductive group, we can define a family of exponential sums. We introduce the l-adic hypergeometric sheaf to study these exponential sums. Under the homogeneity condition, we calculate the rank and smooth locus of this sheaf. We apply these result to the estimation of exponential sums.

**Speaker:** Prof. Dingxin Zhang (Tsinghua University)**Time:** 10:30-11:30am, December 20, 2023 (Wednesday)**Place:** MCM110**Title:** Dwork operator and applications to rigid cohomology**Abstract:** Over a finite field, Dwork offered a link between the zeros and poles of a variety's zeta function and the spectra of a completely continuous operator. I will explain the utility of this operator in extracting arithmetic properties of the Frobenius eigenvalues in rigid cohomology. This includes identifying Frobenius eigenvalues invisible in the zeta function, and pinpointing the maximum radius of the domain where the zeta function is free from poles and zeros. (Joint work with Daqing Wan.)

**Speaker:** Prof. Yang Cao (Shandong University) **Time:** 10:30-11:30am, December 26, 2023 (Tuesday)**Place:** MCM110**Title:** Unramified cohomology and descent methods to weak approximation**Abstract:** On one hand, the unramified cohomology is stably birational invariant and provides a criterion for rationality; on the other hand, weak approximation of algebraic varieties (the density of rational points in local points) is also stably birational invariant, and one of its basic tools is Colliot-Thélène and Sansuc's descent methods. In this talk, I will try to use unramified cohomology to construct new descent methods.

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Phone: 86-10-82541165 Email: mcmadmin@math.ac.cn