arXiv subjects

Siyan Daniel Li-Huerta

Publications and source records attributed to Siyan Daniel Li-Huerta.

8 recordsLinked to original sources

arXiv2026

Courbes et fibr\'es vectoriels en th\'eorie de Hodge $z$-adique globale

We study the global analogue of the Fargues-Fontaine curve over function fields $F$. We prove some foundational results about its moduli of $G$-bundles $\operatorname{Bun}_{G,F}$, which is a geometrization of the global Kottwitz set $B(F,G)$. For example, $\operatorname{Bun}_{G,F}$ plays the role of Igusa stacks over function fields. We use $\operatorname{Bun}_{G,F}$ to reformulate the global Langlands conjecture for $G$ over $F$ in terms of categorical local Langlands, refining conjectures of Arinkin-Gaitsgory-Kazhdan-Raskin-Rozenblyum-Varshavsky and Zhu. Finally, we verify this conjecture when $G$ is commutative. Along the way, we prove a GAGA theorem for smooth proper schemes over sousperfectoid spaces, which is of independent interest.

math.NT↗

arXiv2025

Global long root $A$-packets for $\mathsf{G}_2$: the dihedral case

Cuspidal automorphic representations $τ$ of $\mathrm{PGL}_2$ correspond to global long root $A$-parameters for $\mathsf{G}_2$. Using an exceptional theta lift between $\mathrm{PU}_3$ and $\mathsf{G}_2$, we construct the associated global $A$-packet and prove the Arthur multiplicity formula for these representations when $τ$ is dihedral and satisfies some technical hypotheses. We also prove that this subspace of the discrete automorphic spectrum forms a full near equivalence class. Our construction yields new examples of quaternionic modular forms on $\mathsf{G}_2$.

math.NT↗

arXiv2025

Gross's conjecture: the dihedral case

Quaternionic modular forms on $\mathsf{G}_2$ carry a surprisingly rich arithmetic structure. For example, they have a theory of Fourier expansions where the Fourier coefficients are indexed by totally real cubic rings. For quaternionic modular forms on $\mathsf{G}_2$ associated via functoriality with certain modular forms on $\mathrm{PGL}_2$, Gross conjectured in 2000 that their Fourier coefficients encode $L$-values of cubic twists of the modular form (echoing Waldspurger's work on Fourier coefficients of half-integral weight modular forms). We prove Gross's conjecture when the modular forms are dihedral, giving the first examples for which it is known.

math.NT↗

arXiv2025

The plectic conjecture over local fields

Using a mixed-characteristic incarnation of fusion, we prove an analog of Nekovář-Scholl's plectic conjecture for local Shimura varieties. We apply this to obtain results on the plectic conjecture for (global) Shimura varieties after restricting to a decomposition group. Along the way, we prove a $p$-adic uniformization theorem for the basic locus of abelian type Shimura varieties at hyperspecial level, which is of independent interest.

math.NT↗

arXiv2024

On close fields and the local Langlands correspondence

We prove that Fargues-Scholze's semisimplified local Langlands correspondence (for quasisplit groups) with $\overline{\mathbb{F}}_\ell$-coefficients is compatible with Deligne and Kazhdan's philosophy of close fields. From this, we deduce that the same holds with $\overline{\mathbb{Q}}_\ell$-coefficients after restricting to wild inertia, addressing questions of Gan-Harris-Sawin and Scholze. The proof involves constructing a moduli space of nonarchimedean local fields and then extending Fargues-Scholze's work to this context.

math.NT↗

arXiv2023

Local-global compatibility over function fields

We prove that V. Lafforgue's global Langlands correspondence is compatible with Fargues-Scholze's semisimplified local Langlands correspondence. As a consequence, we canonically lift Fargues-Scholze's construction to a non-semisimplified local Langlands correspondence for positive characteristic local fields. We also deduce that Fargues-Scholze's construction agrees with that of Genestier-Lafforgue, answering a question of Fargues-Scholze, Hansen, Harris, and Kaletha. The proof relies on a uniformization morphism for moduli spaces of shtukas.

math.NT↗

arXiv2021

The plectic conjecture over function fields

We prove the plectic conjecture of Nekovář-Scholl over global function fields $Q$. For example, when the cocharacter is defined over $Q$ and the structure group is a Weil restriction from a geometric degree $d$ separable extension $F/Q$, consider the complex computing $\ell$-adic intersection cohomology with compact support of the associated moduli space of shtukas over $Q_I$. We endow this with the structure of a complex of $\DeclareMathOperator{\Weil}{Weil}(\Weil(F)^d\rtimes\mathfrak{S}_d)^I$-modules, which extends its structure as a complex of $\Weil(Q)^I$-modules constructed by Arinkin-Gaitsgory-Kazhdan-Raskin-Rozenblyum-Varshavsky. We show that the action of $(\Weil(F)^d\rtimes\mathfrak{S}_d)^I$ commutes with the Hecke action, and we give a moduli-theoretic description of the action of Frobenius elements in $\Weil(F)^{d\times I}$.

math.NT↗

arXiv2021

The local Langlands correspondence for $\DeclareMathOperator{\GL}{GL}\GL_n$ over function fields

Let $F$ be a local field of characteristic $p>0$. By adapting methods of Scholze, we give a new proof of the local Langlands correspondence for $\GL_n$ over $F$. More specifically, we construct $\ell$-adic Galois representations associated with many discrete automorphic representations over global function fields, which we use to construct a map $\DeclareMathOperator{\rec}{rec}π\mapsto\rec(π)$ from isomorphism classes of irreducible smooth representations of $\GL_n(F)$ to isomorphism classes of $n$-dimensional semisimple continuous representations of $W_F$. Our map $\rec$ is characterized in terms of a local compatibility condition on traces of a certain test function $f_{τ,h}$, and we prove that $\rec$ equals the usual local Langlands correspondence (after forgetting the monodromy operator).

math.NT↗