SIYAN DANIEL LI-HUERTA（李思严）

Quaternionic modular forms on G₂ 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 G₂ associated via functoriality with certain modular forms on PGL₂, 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.

We prove that Fargues–Scholze's semisimplified local Langlands correspondence (for quasisplit groups) with F_ℓ-coefficients is compatible with Deligne and Kazhdan's philosophy of close fields. From this, we deduce that the same holds with Q_ℓ-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.

Cuspidal automorphic representations τ of PGL₂ correspond to global long root A-parameters for G₂. Using an exceptional theta lift between PU₃ and G₂, 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 G₂.

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.

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.

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 ℓ-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 (Weil(F)^d 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 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×I.

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 ℓ-adic Galois representations associated with many discrete automorphic representations over global function fields, which we use to construct a map π→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).