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2. Arithmetic intersection theory beyond classical unitary Shimura varieties


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      Problem 2.1.

      [A. Mihatsch] Similar to Kudla program, we could also define a generating series of (derived) complex multiplication cycles on Shimura varieties and conjecture its modularity. Is it possible to compute arithmetic intersection of the generating series with special divisors / Hodge bundle and prove its numerical modularity when the Shimura variety has low dimension e.g. on Siegel threefold or Hilbert modular surface?


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          Problem 2.2.

          [K. Madapusi Pera] Could we formulate and prove arithmetic Siegel–Weil formulas of special cycles on unitary Shimura varieties for U(a,b) (a, b>1) where there are no special divisors?


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              Problem 2.3.

              [S. Leslie] Could we formulate and prove arithmetic (resp. higher) Siegel–Weil formulas on Shimura varieties (resp. moduli of Shtukas) for exceptional groups?


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                  Problem 2.4.

                  [A. Mihatsch] Could we describe the reduced locus of Rapoport–Zink spaces (integral local Shimura varieties) for inner forms of general linear groups? It will be useful for studying arithmetic transfer conjectures in the context of linear arithmetic fundamental lemmas.


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                      Problem 2.5.

                      [S. Kudla] Study non-algebraic locally symmetric spaces for $O(4,4)$ and natural special cycles on them. What kind of structures do we expect?

                          Cite this as: AimPL: Arithmetic intersection theory on Shimura varieties, available at http://aimpl.org/intersectshimura.