Seminar

Special cycles on Shimura varieties

Prof. Dr. Viehmann, Prof. Dr. Liedtke, Dr. Ivanov

Zeit und Ort

Dienstags um 14 Uhr c.t. im Raum MI 02.06.020.

Inhalt

We want to study questions on the geometry of Shimura varieties arising in the context of Kudla's program of relating intersections of cycles on Shimura varieties to Eisenstein series. In the first part of the seminar we will study two (of the few exceptional) examples where explicit descriptions of the basic locus of Shimura varieties can be given. In the first case, where the local group is isomorphic to $GSp_4$, the irreducible components are just projective lines, and also their intersections can be described explicitly. The second example considers the case of Spin groups, of particular interest due to its applications to moduli of K3 surfaces. The description in these cases uses the recent definition of integral models for Shimura varieties of Hodge type by Kisin.

In the last part of the seminar we want to understand how this can be applied to compute intersections of special cycles on the corresponding Shimura varieties. For this we return to the first, much easier example. (In the second example such a theory is not yet available.)

Programm

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Literatur

U. Görtz, M. Rapoport (ed.): Argos-Seminar on intersection of modular correspondences, Asterisque 312 (2010).
Howard B., Pappas G.: Rapoport-Zink spaces for spinor groups, preprint, 2015, arxiv.org/abs/1509.03914.
Kudla S.: Central derivatives of Eisenstein series and height pairings, Ann. Math. 146 (1997), 545-646.
Kudla S., Rapoport, M.: Cycles on Siegel threefolds and derivatives of Eisenstein series, Ann. Sci. \'Ecole Norm. Sup. 33 (2000), 695-756.
Kisin M.: mod $p$ points on Shimura varieties of abelian type, preprint, 2013.
Kaiser C.: Ein getwistetes Fundamentales Lemma f\"ur die GSp_4, Dissertation, Bonn, 1997.
Wortmann D.: The µ-ordinary locus for Shimura varieties of Hodge type, preprint, 2013, arxiv.org/abs/1310.6444.