Geometric Langlands predicts an isomorphism between Whittaker coefficients of Eisenstein series and functions on the moduli space of
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Abstract -local systems. We prove this formula by interpreting Whittaker coefficients of Eisenstein series as factorization homology and then invoking Beilinson and Drinfeld’s formula for chiral homology of a chiral enveloping algebra.$\check {N}$ -
Abstract Kazhdan and Lusztig identified the affine Hecke algebra ℋ with an equivariant
-group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields$K$ with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from$F$ -theory to Hochschild homology and thereby identify ℋ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, the$K$ coherent Springer sheaf . As a result the derived category of ℋ-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu).In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of
into coherent sheaves on the stack of Langlands parameters.$\mathrm{GL}_{n}(F)$ -
Abstract In this paper, we introduce the notion of an Evertse–Ferretti Nevanlinna constant and compare it with the birational Nevanlinna constant introduced by the authors in a recent joint paper. We then use it to recover several previously known results. This includes a 1999 example of Faltings from his
Baker’s Garden article. We also extend the theory of these Nevanlinna constants to what we call “multidivisor Nevanlinna constants,” which allow the proximity function to involve the maximum of Weil functions for finitely many divisors. -
Abstract Let $S$ be a scheme and let $\pi : \mathcal{G} \to S$ be a ${\mathbb{G}}_{m,S}$-gerbe corresponding to a torsion class $[\mathcal{G}]$ in the cohomological Brauer group ${\operatorname{Br}}^{\prime}(S)$ of $S$. We show that the cohomological Brauer group ${\operatorname{Br}}^{\prime}(\mathcal{G})$ of $\mathcal{G}$ is isomorphic to the quotient of ${\operatorname{Br}}^{\prime}(S)$ by the subgroup generated by the class $[\mathcal{G}]$. This is analogous to a theorem proved by Gabber for Brauer–Severi schemes.
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We prove two variations of the classical gluing result of Beilinson–Bernstein–Deligne. We recast the problem of gluing in terms of filtered complexes in the total topos of a
-topos, in the sense of SGA 4, and prove our results using the filtered derived category.D Free, publicly-accessible full text available September 25, 2025 -
Let
be a smooth geometrically connected projective curve over the field of fractions of a discrete valuation ring$X$ , and$R$ a modulus on$\mathfrak {m}$ , given by a closed subscheme of$X$ which is geometrically reduced. The generalized Jacobian$X$ of$J_\mathfrak {m}$ with respect to$X$ is then an extension of the Jacobian of$\mathfrak {m}$ by a torus. We describe its Néron model, together with the character and component groups of the special fibre, in terms of a regular model of$X$ over$X$ . This generalizes Raynaud's well-known description for the usual Jacobian. We also give some computations for generalized Jacobians of modular curves$R$ with moduli supported on the cusps.$X_0(N)$ Free, publicly-accessible full text available May 1, 2025