Publications
18 results found
Ben-Zvi D, Chen H, Helm D, et al., 2024, Coherent Springer theory and the categorical Deligne-Langlands correspondence, Inventiones Mathematicae, Vol: 235, Pages: 255-344, ISSN: 0020-9910
Kazhdan and Lusztig identified the affine Hecke algebra H with an equivariant K-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 F with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from K-theory to Hochschild homology and thereby identify Hwith the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, the 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 GLnpF q into coherent sheaves on the stack of Langlands parameters.
Dat J-F, Helm D, Kurinczuk R, et al., 2023, FINITENESS FOR HECKE ALGEBRAS OF p-ADIC GROUPS, JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, ISSN: 0894-0347
Allen PB, Calegari F, Caraiani A, et al., 2023, Potential automorphy over CM fields, Annals of Mathematics, Vol: 197, Pages: 897-1113, ISSN: 0003-486X
Let F be a CM number field. We prove modularity lifting theoremsfor regular n-dimensional Galois representations over F without any selfduality condition. We deduce that all elliptic curves E over F are potentially modular, and furthermore satisfy the Sato–Tate conjecture. As anapplication of a different sort, we also prove the Ramanujan Conjecture forweight zero cuspidal automorphic representations for GL2(AF ).
Helm D, 2022, An ordinary abelian variety with an etale self-isogeny of p-power degree and no isotrivial factors, Mathematics Research Letters, Vol: 29, Pages: 445-454, ISSN: 1073-2780
We construct, for every prime p, a function field K of characteristicp and an ordinary abelian variety A over K, with no isotrivial factors, thatadmits an ´etale self-isogeny φ : A → A of p-power degree. As a consequence, we deduce that there exist ordinary abelian varieties over function fields whose groups of points over the maximal purely inseparable extension is not finitely generated, answering in the negative a question of Thomas Scanlon.
Allen PB, Calegari F, Caraiani A, et al., 2022, Potential automorphy over CM fields, Publisher: arXiv
Let $F$ be a CM number field. We prove modularity lifting theorems forregular $n$-dimensional Galois representations over $F$ without anyself-duality condition. We deduce that all elliptic curves $E$ over $F$ arepotentially modular, and furthermore satisfy the Sato--Tate conjecture. As anapplication of a different sort, we also prove the Ramanujan Conjecture forweight zero cuspidal automorphic representations for$\mathrm{GL}_2(\mathbf{A}_F)$.
Helm D, 2020, Curtis Homomorphisms and the integral Bernstein center for GLn, Algebra and Number Theory, ISSN: 1937-0652
We describe two conjectures, one strictly stronger than the other,that give descriptions of the integral Bernstein center for GLn(F) (that is, the center of the category of smooth W(k)[GLn(F)]-modules, forFap-adic field and k an algebraically closed field of characteristic`different from p) in terms of Galois theory. Moreover, we show that the weak version of the conjecture(form≤n), together with the strong version of the conjecture form < n,implies the strong conjecture for GLn. In a companion paper [HM] we show that the strong conjecture forn−1 implies the weak conjecture forn; thus the two papers together give an inductive proof of both conjectures. The upshot is a description of the Bernstein center in purely Galois theoretic terms; previous work of the author shows that this description implies the conjectural “local Langlands correspondence in families” of [EH].
Helm DF, Moss G, 2018, Converse theorems and the local Langlands correspondence in families, Inventiones Mathematicae, Vol: 214, Pages: 999-1022, ISSN: 0020-9910
We prove a descent criterion for certain families of smooth representations of GLn(F) (F a p-adic field) in terms of the γ-factors of pairs constructed in Moss (Int Math Res Not 2016(16):4903–4936, 2016). We then use this descent criterion, together with a theory of γ-factors for families of representations of the Weil group WF (Helm and Moss in Deligne–Langlands gamma factors in families, arXiv:1510.08743v3, 2015), to prove a series of conjectures, due to the first author, that give a complete description of the center of the category of smooth W(k)[GLn(F)]-modules (the so-called “integral Bernstein center”) in terms of Galois theory and the local Langlands correspondence. An immediate consequence is the conjectural “local Langlands correspondence in families” of Emerton and Helm (Ann Sci Éc Norm Supér (4) 47(4):655–722, 2014).
Helm D, Tian Y, Xiao L, 2017, Tate cycles on some unitary Shimura varieties mod, Algebra and Number Theory, Vol: 11, Pages: 2213-2288, ISSN: 1937-0652
Let F be a real quadratic field in which a fixed prime p is inert, and E0 be an imaginary quadratic field in which p splits; put E=E0F. Let X be the fiber over Fp2 of the Shimura variety for G(U(1,n−1)×U(n−1,1)) with hyperspecial level structure at p for some integer n≥2. We show that under some genericity conditions the middle-dimensional Tate classes of X are generated by the irreducible components of its supersingular locus. We also discuss a general conjecture regarding special cycles on the special fibers of unitary Shimura varieties, and on their relation to Newton stratification.
Helm DF, 2016, Whittaker models and the integral Bernstein center for GL_n, Duke Mathematical Journal, Vol: 165, Pages: 1597-1628, ISSN: 1547-7398
We establish integral analogues of results of Bushnell and Henniart for spaces of Whittaker functions arising from the groups GLn(F)GLn(F) for FF a pp-adic field. We apply the resulting theory to the existence of representations arising from the conjectural “local Langlands correspondence in families” and reduce the question of the existence of such representations to a natural conjecture relating the integral Bernstein center of GLn(F)GLn(F) to the deformation theory of Galois representations.
Helm D, 2016, The Bernstein Center of the category of smooth W(k)[GLn(F)]-modules, Forum of Mathematics, Sigma, Vol: 4, ISSN: 2050-5094
We consider the category of smooth -modules, where is a -adic field and is an algebraically closed field of characteristic different from . We describe a factorization of this category into blocks, and show that the center of each such block is a reduced, -torsion free, finite type -algebra. Moreover, the -points of the center of a such a block are in bijection with the possible ‘supercuspidal supports’ of the smooth -modules that lie in the block. Finally, we describe a large explicit subalgebra of the center of each block and give a description of the action of this algebra on the simple objects of the block, in terms of the description of the classical ‘characteristic zero’ Bernstein center of Bernstein and Deligne [Le ‘centre’ de Bernstein, in Representations des groups redutifs sur un corps local, Traveaux en cours (ed. P. Deligne) (Hermann, Paris), 1–32].
Emerton M, Helm D, 2014, THE LOCAL LANGLANDS CORRESPONDENCE FOR GL<i><sub>n</sub></i> IN FAMILIES, ANNALES SCIENTIFIQUES DE L ECOLE NORMALE SUPERIEURE, Vol: 47, Pages: 655-722, ISSN: 0012-9593
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- Citations: 37
Helm D, 2013, ON THE MODIFIED MOD <i>p</i> LOCAL LANGLANDS CORRESPONDENCE FOR GL<sub>2</sub>(Q<sub>l</sub>), MATHEMATICAL RESEARCH LETTERS, Vol: 20, Pages: 489-500, ISSN: 1073-2780
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- Citations: 3
Helm D, Katz E, 2012, Monodromy Filtrations and the Topology of Tropical Varieties, CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, Vol: 64, Pages: 845-868, ISSN: 0008-414X
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- Citations: 18
Helm D, 2012, A geometric Jacquet-Langlands correspondence for U(2) Shimura varieties, ISRAEL JOURNAL OF MATHEMATICS, Vol: 187, Pages: 37-80, ISSN: 0021-2172
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- Citations: 11
Helm D, Voloch JF, 2011, Finite descent obstruction on curves and modularity, BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, Vol: 43, Pages: 805-810, ISSN: 0024-6093
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- Citations: 3
Helm D, 2010, TOWARDS A GEOMETRIC JACQUET-LANGLANDS CORRESPONDENCE FOR UNITARY SHIMURA VARIETIES, DUKE MATHEMATICAL JOURNAL, Vol: 155, Pages: 483-518, ISSN: 0012-7094
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- Citations: 8
Helm D, 2010, ON l-ADIC FAMILIES OF CUSPIDAL REPRESENTATIONS OF GL(2)(Q(p)), MATHEMATICAL RESEARCH LETTERS, Vol: 17, Pages: 805-822, ISSN: 1073-2780
Helm D, Osserman B, 2008, Flatness of the linked Grassmannian, PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 136, Pages: 3383-3390, ISSN: 0002-9939
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- Citations: 6
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