数论与算术几何高级研讨会日程
北京师范大学
2026年7月7-9日
7月7日（周二）：北京师范大学后主楼1124 |
时间 |
报告人 |
报告题目 |
09:10-10:10 |
扶磊 |
Arithmetic D-modules and exponential sums for reductive groups |
10:30-11:30 |
袁新意 |
On Vojta’s proof of the Mordell conjecture |
14:00-15:00 |
吴晨彦 |
Hecke algebras and Langlands parameters for Bernstein blocks of the metaplectic groups |
15:10-16:10 |
李时璋 |
u-power torsions of prismatic cohomology |
16:30-17:30 |
杨梓诠 |
Special loci on M_g in positive characteristic |
7月8日（周三）：北京师范大学后主楼1124 |
09:00-10:00 |
胡永泉 |
On mod p Jacquet-Langlands correspondence for GL(2,Q_p) |
10:20-11:20 |
郗平 |
Recent developments of Burgess’s method |
14:00-15:00 |
梁永祺 |
Inverse Brauer problem and the smallest Brauer subgroup obstructing the Hasse principle |
15:10-16:10 |
高鹏 |
Large values of quadratic Hecke L-functions of the Gaussian field |
16:30-17:30 |
曹阳 |
Kottwitz exact sequence over p-adic function fields |
7月9日（周四）：北京师范大学后主楼1124 |
09:00-10:00 |
于鹏 |
To be announced |
10:20-11:20 |
崔沛仪 |
l-Modular Blocks of SL_n(F) |
报告摘要
（按姓氏笔画排序）
• To be announced
于鹏（中国人民大学）
• Arithmetic D-modules and exponential sums for reductive groups
扶磊（清华大学）
For a family of representations of a reductive group, we define a Laurent polynomial on the group. The exponential sum associated to this Laurent polynomial is called the hypergeometric exponential sum. We introduce an arithmetic hypergeometric D-module to study the exponential sum. We prove it is overholonomic, determine the open set where it is an overconvergent F-isocrystal, and estimate its rank. We apply these results to the estimation of the exponential sum. This is a joint work with Xuanyou Li and Chenhan Liu.
• u-power torsions of prismatic cohomology
李时璋（中国科学院数学与系统科学研究院）
We shall report a previous work with Ofer Gabber concerning bounds on u-power torsions of Breuill--Kisin prismatic cohomology of Bhatt--Scholze.
• Special loci on M_g in positive characteristic
杨梓诠（香港中文大学）
We introduce new techniques to study special loci in moduli spaces and give several applications, focusing in particular on the moduli space M_g of curves. These applications include Zariski density results for split Jacobians, non-density results for supersingular loci, and lifting results for algebraic cycles, all under appropriate numerical conditions. Our methods generalize to more-or-less arbitrary moduli spaces mapping to Hodge-type Shimura varieties (and even some settings beyond the Shimura case), and relate to questions raised by Oort and de Jong. This is joint work with David Urbanik.
• Hecke algebras and Langlands parameters for Bernstein blocks of the metaplectic groups
吴晨彦（墨尔本大学）
We study the Bernstein blocks of representations of the metaplectic group (which covers the symplectic group) in the p-adic setting. We show that they are equivalent to the categories of right modules over certain extended Hecke algebras with parameters and that these parameters can be expressed in terms of the Langlands parameters of the supercuspidal support of the Bernstein block. Via the Hecke algebras, we find that these Bernstein blocks, when suitably combined, can be expressed in terms of unipotent Bernstein blocks of quasisplit classical groups, where the representations theory and the Langlands parameters are well known. This is a joint work with Volker Heiermann.
•On mod p Jacquet-Langlands correspondence for GL(2,Q_p)
胡永泉（中国科学院数学与系统科学研究院）
The classical Jacquet-Langlands (J-L) correspondence relates complex smooth representations of GL(n) and that of its inner forms. In this talk I will discuss an analogue of the J-L correspondence for representations with mod p coefficients, which was initiated by Scholze around 2015. I will review the mod p J-L correspondence in the case of GL(2,Q_p) and present a result concerning the full faithfulness of this correspondence. This is joint work in progress with Vytautas Paskunas.
• Recent developments of Burgess’s method
郗平（西安交通大学）
Starting from 1950’s, D. A. Burgess has developed an ingenious approach to estimate incomplete character sums, appealing to Weil’s proof on Riemann Hypothesis for curves over finite fields. This allows him to break the barrier of Pólya–Vinogradov and thus works nontrivially for suitably short character sums. Burgess’s method has been simplified and generalized in the subsequent decades thanks to Karatsuba, Friedlander–Iwaniec, Fouvry–Michel, et al, and it turns out to be very powerful in many applications of Fourier analysis to number theory. In this talk, I will give a brief introduction to the relevant history, and present our recent work on estimates for multilinear character sums and exponential sums, as well as applications to distributions of primes. This is based on several joint works with Etienne Fouvry and Igor E. Shparlinski, and with Junren Zheng.
• On Vojta’s proof of the Mordell conjecture
袁新意（北京大学）
The Mordell conjecture was originally proved by Faltings in 1983, and a second proof was given by Vojta in 1989. In this talk, I will describe Vojta’s proof with a simplification by using my arithmetic Siu inequality.
• Large values of quadratic Hecke L-functions of the Gaussian field
高鹏（北京航空航天大学）
In this talk, we apply the long resonator method to exhibit large values of the family of quadratic Hecke L-functions of the Gaussian field under the generalized Riemann hypothesis.
• Kottwitz exact sequence over p-adic function fields
曹阳（山东大学）
(joint work with Ting-Yu Lee) The classical Kottwitz exact sequence studies the local-global principle for Galois cohomology with coefficients in connected linear algebraic groups. In this talk, we consider this local-global problem over a p-adic function field, i.e., the function field of a smooth curve over a p-adic field. We will show that a new phenomenon occurs even for semisimple simply connected groups of type A, and this leads to a new cohomological obstruction theory over p-adic function fields.
• l-Modular Blocks of SL_n(F)
崔沛仪（清华大学）
It is well-known that l-modular blocks show great difference comparing to complex blocks of p-adic groups. In this talk, I will introduce some examples of depth zero blocks and all the l-modular blocks of SL_n(F) from the perspective of establishing nonsplit projective objects. We will explore the technical challenges in associating an l-modular block with a depth-zero block and consider a natural connection between them.
• Inverse Brauer problem and the smallest Brauer subgroup obstructing the Hasse principle
梁永祺（中国科学技术大学）
Brauer groups play an important role in the study of the Hasse principle for existence of rational points on algebraic varieties defined over number fields. We answer affirmatively to the "inverse Brauer problem": Any finite abelian group is the Brauer group (modulo the constant part) of a certain smooth projective variety. Furthermore, we also prove that any given non-zero finite abelian group can be the smallest Brauer subgroup obstructing the Hasse principle for a certain algebraic variety. This is a joint work with Yufan Liu.
（本研讨会由国家自然科学基金数学天元基金资助。联系人：陆晴）
