NTS ABSTRACTFall2024

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Sept 5

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Sept 12

Non-reductive special cycles and arithmetic fundamental lemmas

We care about arithmetic invariants of polynomial equations e.g. L-functions, which (conjecturally) are often automorphic and related to special cycles on Shimura varieties (or Shimura sets) based on the relative Langlands program. Arithmetic fundamental lemmas reveal such relations in the p-adic local world. In this talk, I will study certain ``universal'’ non-reductive special cycles on local GL_n Shimura varieties, and give applications e.g. the proof of twisted arithmetic fundamental lemma for the tuple (U_n, GL_n, U_n). Time permitting, I will explain some global analogs where at least the (Betti) cohomology class of special cycles could be defined. It turns out that algebraic special cycles are often pullbacks of ``universal’’ non-algebraic cycles (e.g. from Kudla-Millson theory on non-Hermitian symmetric spaces).