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1. Special cycles, modularity and arithmetic intersection


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      Problem 1.1.

      [B. Howard] Is it possible to define a generating series of special cycles over Shimura varieties related to automorphic representations of higher weight? Do we have similar geometric/arithmetic modularity and Siegel-Weil formulas as in the Kudla program?


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          Problem 1.2.

          [Y. Luo, M. Mkrtchyan] Derived special cycles have been defined on integral models of certain Shimura varieties (resp. moduli of Shtukas) in recent works of Madapusi (resp. Feng-Yun-Zhang) in the good reduction case. Is it possible to extend their method and define derived special cycles in the bad reduction cases e.g. for some specific parahoric levels?


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              Problem 1.3.

              [Q. He, Z. Zhang] In recent proof of Kudla-Rapoport conjectures by Li–Zhang, numerical local modularity results for local Kudla–Rapoport divisors have been discovered and proved. Is it possible to formulate local modularity geometrically e.g. using moduli of p-adic Shtukas? Could the method of Feng-Yun-Zhang for global modularity be adapted to prove geometric local modularity?


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                  Problem 1.4.

                  [S. Leslie, Z. Yun] In the formulation of the arithmetic fundamental lemma, arithmetic intersection numbers are only formulated for regular semi-simple orbits because of proper intersections of cycles. Is it possible to define arithmetic intersection numbers in case of improper intersections of cycles (and formulate a similar analytic side)?


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                      Problem 1.5.

                      [M. Rapoport, Q. Li] In any arithmetic transfer conjectures, could we find perfect test transfer functions for (derived) orbital integrals to match arithmetic intersection numbers with no correction factors?


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                          Problem 1.6.

                          [M. Rapoport] In the recent work of Li–Rapoport–Zhang on arithmetic fundamental lemma conjecture for spherical Hecke algebras, there is a conjecture on commutativity of certain Hecke operators on Grothendieck groups of coherent sheaves on unitary Rapoport–Zink spaces. How to prove this conjecture?

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