| 1. |
- Cooper, Simon, et al.
(författare)
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Hodge-Chern classes and strata-effectivity in tautological rings
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Given a connected, reductive group G over the finite field F of order p, a cocharacter μ of G over an algebraic closure of F and a stack X admitting a 'zip period morphism' to the stack of G-Zips of type μ, we study which classes in the Wedhorn-Ziegler tautological rings of X and its flag space Y are strata-effective, meaning that they are non-negative rational linear combinations of pullbacks of classes of zip (flag) strata closures. Two special cases are: (1) When X is the stack of G-Zips and the tautological rings coincide with the entire Chow rings (2) When X is the special fiber of an integral canonical model of a Hodge-type Shimura variety – in this case the strata are also known as Ekedahl-Oort strata. We focus on the strata-effectivity of three types of classes: (a) Effective tautological classes, (b) Chern classes of Griffiths-Hodge bundles and (c) Generically w-ordinary curves. We connect the question of strata-effectivity in (a) to the global section ‘Cone Conjecture’ of Goldring-Koskivirta. For every representation r of G, we conjecture that the Chern classes of the Griffiths-Hodge bundle associated to (G, μ, r) are all strata-effective. This provides a vast generalization of a result of Ekedahl-van der Geer that the Chern classes of the Hodge vector bundle on the moduli space of principally polarized abelian varieties in characteristic p are represented by the closures of p-rank strata. We prove several instances of our conjecture, including the case of Hilbert modular varieties, where the conjecture says that all monomials in the first Chern classes of the factors of the Hodge vector bundle are strata-effective. We prove results about each of (a), (b) and (c) which have applications to Shimura varieties and also in cases where no Shimura variety exists.
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| 2. |
- Cooper, Simon, 1997-
(författare)
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Intersection Theory on Zip Period Maps
- 2024
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of four papers, all motivated by questions about intersection theory on Shimura varieties in positive characteristic. The connection with intersection theory of flag varieties, made using the stack of G-Zips of type μ, is explored throughout. More generally, we work in the setting of intersection theory on spaces X admitting morphisms to the stack of G-Zips of type μ. These morphisms are termed 'zip period maps' in Paper III. The fundamental example of such an X is the special fibre of an integral canonical model of a Shimura variety of Hodge-type. Moreover, there is a notion of 'tautological ring' for any (smooth) zip period map which gives the usual tautological ring in the case of Shimura varieties.In Paper I the tautological ring of a Hilbert modular variety at an unramified prime is computed. The method generalises van der Geer's approach from the Siegel case and makes use of the properness of the non-maximal Ekedahl-Oort strata closures in this setting.The pushforward map in the Chow ring between Siegel flag varieties is computed in Paper II. Siegel flag varieties are projective varieties which are quotients of the symplectic group. They appear as the compact dual of the Siegel upper half plane. A conjecture exploring the connection between classes in Chow rings of flag varieties and classes in tautological rings of Shimura varieties is presented. The computation contained in this paper can be viewed as very basic evidence for this conjecture.In Paper III we develop various conjectures related to positivity in the tautological ring of a zip period map. The notion of strata-effective classes is introduced. Several conjectures are presented regarding classes which we expect to be strata-effective. These are proved in many cases, including for Hilbert modular varieties, which are more accessible for various group-theoretic reasons. A connection between strata-effectivity and the Cone Conjecture of Goldring-Koskivirta is developed and provides examples of tautological and effective classes which nevertheless fail to be strata-effective.In Paper IV we compute the Grothendieck group of the stack of G-Zips of type μ (as a ring) in the case where the derived group of G is simply connected.
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| 3. |
- Cooper, Simon, 1997-
(författare)
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Tautological rings of Shimura varieties
- 2022
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This licentiate thesis consists of two papers. In paper I the tautological ring of a Hilbert modular variety at an unramified prime is computed. The method of van der Geer in the case of A_{g} is extended to deal with the case of the Hilbert modular variety, which is more complicated. An example involving the unitary group is given which shows that this method cannot be used to compute the tautological rings of all Shimura varieties of Hodge type. In paper II we compute the pushforward map from a sub flag variety defined by a Levi subgroup to the Siegel flag variety. Specifically, this is the Levi factor of the parabolic associated with the maximal rational boundary component of the Siegel Shimura datum. The method involves an explicit understanding of the pullback map and an application of the self intersection formula.
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| 4. |
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| 5. |
- Goldring, Wushi, et al.
(författare)
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Automorphic vector bundles with global sections on G-Zip((Z)over-bar)-schemes
- 2018
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Ingår i: Compositio Mathematica. - 0010-437X .- 1570-5846. ; 154:12
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Tidskriftsartikel (refereegranskat)abstract
- A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of G-zips of connected Hodge type; such schemes should include all Hodge-type Shimura varieties with hyperspecial level. We prove our conjecture for groups of type An 1, C 2, and Fp-split groups of type A 2 (this includes all Hilbert{Blumenthal varieties and should also apply to Siegel modular 3-folds and Picard modular surfaces). An example is given to show that our conjecture can fail for zip data not of connected Hodge type.
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| 6. |
- Goldring, Wushi, et al.
(författare)
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Quasi-constant characters : Motivation, classification and applications
- 2018
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Ingår i: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 339, s. 336-366
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Tidskriftsartikel (refereegranskat)abstract
- In [131, initially motivated by questions about the Hodge line bundle of a Hodge-type Shimura variety, we singled out a generalization of the notion of minuscule character which we termed quasi-constant. Here we prove that the character of the Hodge line bundle is always quasi-constant. Furthermore, we classify the quasi-constant characters of an arbitrary connected, reductive group over an arbitrary field. As an application, we observe that, if mu is a quasi-constant cocharacter of an F-p-group G, then our construction of group-theoretical Hasse invariants in loc. cit. applies to the stack G-Zip(mu), without any restrictions on p, even if the pair (G, mu) is not of Hodge type and even if mu is not minuscule. We conclude with a more speculative discussion of some further motivation for considering quasi-constant cocharacters in the setting of our program outlined in loc. cit.
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| 7. |
- Goldring, Wushi
(författare)
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Quasi-constant fundamental weights in terms of Levi Weyl groups
- 2020
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Ingår i: Journal of Algebra. - : Elsevier BV. - 0021-8693 .- 1090-266X. ; 559, s. 87-94
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Tidskriftsartikel (refereegranskat)abstract
- In previous joint work with J.-S. Koskivirta, we introduced the notion of quasi-constant character (of a maximal torus of a connected reductive group over a field); we showed that over an algebraically closed field it naturally unifies the notions minuscule and co-minuscule. In this note we characterize quasi-constant fundamental weights in terms of the Weyl group of the corresponding maximal Levi subgroup. Equivalently, purely in the language of root systems, the result characterizes special and co-special vertices of Dynkin diagrams in terms of the Weyl group of the corresponding maximal sub-root system.
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| 8. |
- Goldring, Wushi, et al.
(författare)
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Strata Hasse invariants, Hecke algebras and Galois representations
- 2019
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Ingår i: Inventiones Mathematicae. - : Springer Science and Business Media LLC. - 0020-9910 .- 1432-1297. ; 217:3, s. 887-984
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Tidskriftsartikel (refereegranskat)abstract
- We construct group-theoretical generalizations of the Hasse invariant on strata closures of the stacks G-Zip(mu). Restricting to zip data of Hodge type, we obtain a group-theoretical Hasse invariant on every Ekedahl-Oort stratum closure of a general Hodge-type Shimura variety. A key tool is the construction of a stack of zip flags G-ZipFlag(mu), fibered in flag varieties over G-Zip(mu). It provides a simultaneous generalization of the classical case homogeneous complex manifolds studied by Griffiths-Schmid and the flag space for Siegel varieties studied by Ekedahl-van der Geer. Four applications are obtained: (1) Pseudo-representations are attached to the coherent cohomology of Hodge-type Shimura varieties modulo a prime power. (2) Galois representations are associated to many automorphic representations with nondegenerate limit of discrete series Archimedean component. (3) It is shown that all Ekedahl-Oort strata in the minimal compactification of a Hodge-type Shimura variety are affine, thereby proving a conjecture of Oort. (4) Part of Serre's letter to Tate onmod p modular forms is generalized to general Hodgetype Shimura varieties.
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| 9. |
- Goldring, Wushi, et al.
(författare)
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Stratifications of Flag Spaces and Functoriality
- 2019
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Ingår i: International mathematics research notices. - : Oxford University Press (OUP). - 1073-7928 .- 1687-0247. ; 2019:12, s. 3646-3682
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Tidskriftsartikel (refereegranskat)abstract
- We define stacks of zip flags, which form towers above the stack of G-zips of Moonen, Pink, Wedhorn and Ziegler in [14-16]. A stratification is defined on the stack of zip flags, and principal purity is established under a mild assumption on the underlying prime p. We generalize flag spaces of Ekedahl-Van der Geer [4] and relate them to stacks of zip flags. For large p, it is shown that strata are affine. We prove that morphisms with central kernel between stacks of G-zips have discrete fibers. This allows us to prove principal purity of the zip stratification for maximal zip data. The latter provides a new proof of the existence of Hasse invariants for Ekedahl-Oort strata of good reduction Shimura varieties of Hodge-type, first proved in [8].
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| 10. |
- Goldring, Wushi
(författare)
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The Griffiths bundle is generated by groups
- 2019
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Ingår i: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 375:3-4, s. 1283-1305
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Tidskriftsartikel (refereegranskat)abstract
- First the Griffiths line bundle of a Q-VHS V is generalized to a Griffiths character grif(G, mu, r) associated to any triple (G, mu, r), where G is a connected reductive group over an arbitrary field F, mu is an element of X-*(G) is a cocharacter (over (F) over bar) and r : G -> GL(V) is an F-representation; the classical bundle studied by Griffiths is recovered by taking F = Q, G the Mumford-Tate group of V, r : G -> GL(V) the tautological representation afforded by a very general fiber and pulling back along the period map the line bundle associated to grif(G, mu, r). The more general setting also gives rise to the Griffiths bundle in the analogous situation in characteristic p given by a scheme mapping to a stack of G-Zips. When G is F-simple, we show that, up to positive multiples, the Griffiths character grif(G, mu, r) (and thus also the Griffiths line bundle) is essentially independent of r with central kernel, and up to some identifications is given explicitly by -mu. As an application, we show that the Griffiths line bundle of a projective G-Zip(mu)-scheme is nef.
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| 11. |
- Reppen, Stefan, 1992-
(författare)
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Hasse invariants on Shimura varieties and moduli of G-bundles
- 2024
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of four papers, referred to in the text as Paper I, II, III and IV, respectively. In Paper I and II, we study the classical Hasse invariant on the geometric special fiber of several Hodge and abelian-type Shimura varieties. We explicitly compute the order of vanishing of the Hasse invariant and describe it in terms of the Bruhat and Ekedahl-Oort stratifications. We compute the conjugate line position (referred to as the a-number by van der Geer and Katsura, respectively Oort) and thus show that it agrees with the order of the Hasse invariant. This equality was obtained by Ogus for certain families of Calabi-Yau varieties mod p, and we thus dub it as Ogus' principle. We give a group-theoretical generalization of this principle via the theory of G-zips, and argue that this theory is a suitable framework within which to understand the principle.Paper III and IV concern (moduli of) G-bundles over smooth projective curves over an algebraically closed field, for G a reductive group. In Paper III we introduce the notion of essentially finite (EF) G-bundles, which generalizes the notion of EF vector bundles, studied initially by Weil and more generally by Nori. We state some elementary characterizations of such bundles, and we prove that they are semistable of torsion degree. Let MGef denote the subset of EF G-bundles in the moduli space of degree 0 semistable G-bundles. We show that in characteristic 0, MGef is not dense if G has semisimple rank 1 and the curve has genus g ≥ 2. This is in contrast to the case of line bundles in arbitrary characteristic, and vector bundles of arbitrary rank in positive characteristic. In Paper IV we first study the moduli stack BunG of G-bundles for nonconnected reductive group schemes G over the curves. We prove that the semistable locus of BunG admits a projective good moduli space. To this end we prove a ``decomposition theorem'' stating that BunG is a finite disjoint union of substacks Xi, each of which admits a finite torsor BunGi à Xi for some connected reductive group schemes Gi. We expand the nondensity result in Paper III and show that if the base field has characteristic 0 and the curve has genus g ≥ 2, then dim(MGef) ≤ g.rk(G). In particular, MGef is not dense unless G is a torus. To this end we give a generalization of Jordan's classical result on finite subgroups of GL(n).
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