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.