数论讨论班

3月13日 (周五)

16:00–17:00

光华楼东主楼 2001

吴志翔

(中国科学技术大学)

Title: Breuil-Strauch conjecture for the Lubin-Tate space of dimension $1$.

Abstract: In the $p$-adic local Langlands program, the Breuil-Strauch conjecture predicts that certain representations of $\mathrm{GL}_2(\mathbb{Q}_p)$, arising from equivariant vector bundles on the one-dimensional Drinfeld space, are closely related to two-dimensional $p$-adic de Rham Galois representations. This conjecture was proved by Dospinescu-Le Bras. In this talk, we present an analogous picture for the one-dimensional Lubin-Tate space. In this setting, the cohomology of equivariant vector bundles naturally produces representations of the unit group of the non-split quaternion algebra over $\mathbb{Q}_p$, which may be viewed as the Jacquet-Langlands counterpart of $\mathrm{GL}_2(\mathbb{Q}_p)$. Using global methods introduced by Lue Pan, we explain how these representations relate to $p$-adic Galois representations and clarify certain features of the $p$-adic Jacquet-Langlands correspondence. This is joint work with Benchao Su and Zhenghui Li.

3月20日 (周五)

16:00–17:00

光华楼东主楼 1801

Title. A survey on the Carlitz-Wan conjecture

Abstract. The Carlitz-Wan conjecture is one of the most famous problems in the history of permutation polynomials and exceptional polynomials. In 1963, Carlitz and his student Cavior posed a special case of the Carlitz-Wan conjecture as a question. In 1991, Wan（万大庆）posed and proved in some cases a generalization of this question, which became known as the Carlitz–Wan conjecture. In 1994, Lenstra first proved the Carlitz–Wan conjecture in full generality. In 1997, Guralnick and Muller generalized the Carlitz–Wan conjecture to exceptional covers of curves. Recently, we studied the exceptional extensions of local fields, which led to different proofs of the result by Guralnick and Muller. In this talk, I will survey the history and various proofs of the Carlitz-Wan conjecture.

4月10日 (周五)

13:30–14:30

SCMS 102）

Title. Coherent cohomological dimension of Siegel modular varieties and the modularity of formal Siegel modular forms

Abstract. In this talk, we provide an upper bound of the coherent cohomological dimension of the Siegel modular variety. As a corollary, we show that the boundary of the compactified Siegel modular variety satisfies the Grothendieck-Lefschetz condition. This implies, in particular, that every formal Siegel modular forms of genus 𝑔 ≥ 2 and cogenus 1 is classical.

4月10日 (周五)

14:50–15:50

SCMS 102

Title. Finitely presented log-regular rings over rank 1 valuation rings

Abstract. The theory of log-regular rings, introduced by Kazuya Kato, has become a cornerstone of logarithmic geometry and $p$-adic Hodge theory. By combining the property of fs-monoids and the commutative algebra of regular local rings, Kato's framework provides a notion of "smoothness" for schemes with singularities, such as semistable reduction models. In this talk, we present an extension of Kato's log-regularity to the setting of finitely presented algebras over a general rank 1 valuation ring $\mathcal{O}$. In addition, we establish the essential properties of this class of rings, specifically proving their normality, Hartogs' lemma, and the rigidity (uniqueness) of the log structure.

4月10日 (周五)

16:10–17:10

SCMS 102

Title. Monodromies associated to (log) $p$-divisible groups

Abstract. Let $R$ be a Henselian DVR with residue field $k$ of characteristic $p>0$ and fraction field $K$, and let $S$ be the standard log trait associated to $R$. In a joint work with Bertapelle and Wang, we established that log $p$-divisible groups over $S$ correspond to $p$-divisible groups with semi-stable reduction over $K$, and also to semi-stable Galois $\mathbb{Z}_p$-representations with Hodge-Tate weights in $\{0,1\}$ (assuming $k$ is perfect and $K$ is a finite extension of $W(k)[1/p]$). In this talk, we discuss mainly the compatibility of monodromies along the correspondences from the joint work.