数论讨论班

9月26日 (周五)

16:00–17:00

光华楼东主楼 1601

丁之元

(上海科技大学)

Title. Reducedness of twisted loop groups

Abstract. In this talk, we give an elementary proof of the reducedness of twisted loop groups. We use standard results about structures of algebraic groups and group schemes without referring to mixed characteristic.

10月17日 (周五)

16:00–17:00

SIMIS 1210

李时璋

(中科院晨兴数学中心)

Title. On the cohomology of rational \(p\)-adic local systems

Abstract. In \(p\)-adic geometry, there are more mysterious rational \(p\)-adic local systems than in algebraic/complex geometry, for example, the analytification of \(\mathbb{P}^1\) is no longer simply connected due to existence of a rank 2 rational \(p\)-adic local system. The cohomology of such local systems were studied previously by Kedlaya–Liu and more recently by Anschütz–Le Bras–Mann. In this talk, we shall see a (relatively) low-tech approach to reprove some of KL/ALBM results. This is a report of an ongoing work joint with Nizioł–Reinecke–Zavyalov.

复旦大学数学科学学院杰出学者讲坛

10月21日 (周二)

16:00–17:00

光华楼东主楼 2201

(Sorbonne Université)

Title. Infinitely Many Non-Hypergeometric Local Systems

Abstract. The Bombieri-Dwork conjecture predicts that an irreducible differential operator with a G-function solution comes from geometry, that is, encodes how periods vary in a pencil of algebraic varieties. This conjecture is completely open for operators of order at least 2. At the beginning of the 90s, Dwork proposed a strategy to establish the conjecture for G-operators of order 2, which would consist in proving that they are all pullbacks by a correspondence of some Gauss's hypergeometric differential operator. Sporadic counterexamples to this expectation were found by Krammer (1996) and Bouw-Möller (2010). I will present a joint work with Josh Lam and Yichen Qin where we prove that most G-operators of order 2 coming from geometry are not pullbacks of hypergeometric differential operators. A key ingredient to construct infinitely many counterexamples will be the André-Pink-Zannier conjecture for Shimura varieties, in the cases recently established by Richard and Yafaev.

10月28日 (周二)

16:00–17:00

SIMIS 1110

Title. Automorphic Green Currents and Applications

Abstract. Green currents gives the arithmetic intersection theory at the archimedean places. In this talk, we consider Green currents for Shimura varieties and its applications to special value formulae. This is based on a joint work with Ye Tian and Hang Yin.

11月4日 (周二)

16:00–17:00

SIMIS 1410

Title. On Swan classes for constructible étale sheaves

Abstract. For constructible étale sheaves on varieties over fields of positive characteristic, the Swan class extends the Swan conductor to a 0-cycle, serving as an invariant that measures ramification. Using different approaches, Kato-Saito and Saito have constructed two versions of the Swan class. In this talk, we introduce a new cohomological Swan class and investigate Saito's conjecture on its uniqueness. This is a joint work with Enlin Yang.

11月14日 (周五)

16:00–17:00

12月16日 (周二)

16:00–17:00