Shurui Liu

Publications

Pour les projets liés à l'informatique et à l'IA, voir Projets.

Mes intérêts de recherche principaux sont la théorie géométrique des représentations et le programme de Langlands catégorique.

Prépublications

Listées en ordre chronologique inverse.

  1. Modular Representation Theory via Crystalline D-modules. En cours, brouillon disponible sur demande, 2026.
    Résumé

    We develop a six-functor formalism of crystalline $\mathscr{D}$-modules (in the sense of Bezrukavnikov–Mirković–Rumynin) on smooth schemes and smooth Artin stacks in characteristic $p>0$ and then construct a crystalline Deligne–Lusztig induction functor from the finite Hecke category $\mathscr{D}^{\mathrm{ren}}(B\backslash G/B)$ to modular representations of a finite group of Lie type in defining characteristic. We show that this functor factors through a categorical trace via formalism in Zhu (2025). As a main result, we prove it is fully faithful and its monodromic enhancement (passing through $\mathscr{D}^{\mathrm{mon}}(U\backslash G/U)$) yields an equivalence of categories, and hence provides a framework to understand modular representations via the trace category of the finite (monodromic) Hecke category.

  2. Higher Period Integrals and Derivatives of L-functions (travail conjoint avec Zeyu Wang). Soumis, arXiv:2504.00275, 2025.
    Résumé

    We propose a geometric framework to produce a formula relating higher period integrals to higher central derivatives of $L$-functions over function fields, extending the framework of Ben-Zvi–Sakellaridis–Venkatesh to higher derivatives. For a strongly tempered affine smooth $G$-variety $X$, we give a geometric construction of the action of $L$-observables on the geometric period integral $\int_{X}\mathbb{L}_\sigma$ of a Hecke eigensheaf $\mathbb{L}_\sigma$ on $\mathrm{Bun}_G$. By taking a suitable version of Frobenius trace of this action, we recover higher central derivatives of the $L$-function attached to the dual symplectic representation. As an application, in the Rankin–Selberg case $(\mathrm{GL}_n\times\mathrm{GL}_{n-1},\mathrm{GL}_{n-1})$, we obtain a formula for higher derivatives of the Rankin–Selberg $L$-function. This provides a conceptual generalization of the higher Gross–Zagier formula of Yun–Zhang to higher-dimensional spherical varieties.

  3. Künneth Formula for Étale Fundamental Groups in Characteristic 0. Soumis, pdf, 2025.
    Résumé

    In this article, we provide a purely algebraic proof that the étale fundamental group is invariant under base change of algebraically closed fields of characteristic 0. As a corollary, we obtain the Künneth formula for étale fundamental groups in characteristic 0. We also explain how to adapt the proof (using alterations) to apply to prime-to-$p$ fundamental groups in characteristic $p>0$. These results are known to experts, but proofs do not seem to be documented in the literature, so for convenient citation by others we are providing them in a written form here.

IA pour les mathématiques

  1. Automated Conjecture Resolution with Formal Verification. Haocheng Ju*, Guoxiong Gao*, Jiedong Jiang*, Bin Wu*, Zeming Sun*, Shurui Liu*, Leheng Chen, Yutong Wang, Yuefeng Wang, Zichen Wang, Wanyi He, Peihao Wu, Liang Xiao, Ruochuan Liu, Bryan Dai, Bin Dong. [Arxiv] [Rethlas] [Archon].

* Contribution égale.