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Title: On certain families of special cycles on Shimura varieties
Authors: Jin, Zhaorong
Advisors: Skinner, Christopher
Contributors: Mathematics Department
Subjects: Mathematics
Issue Date: 2020
Publisher: Princeton, NJ : Princeton University
Abstract: In this two-part thesis, we study certain families of special cycles on Shimura varieties, which have interesting arithmetic applications. In the first part, extending the ideas of Darmon and Rotger, we construct a p-adic family of Hirzebruch-Zagier cycles on Shimura threefolds obtained from products of modular curves and Hilbert modular surfaces. This family gives rise to a big cohomology class residing in the Galois cohomology of a certain Lambda-adic Galois representation. We then establish a regulator formula for the cycles, which allows us to relate the big cohomology class to a twisted triple product p-adic L-function. As an application, we establish new instances of the equivariant BSD-conjecture in rank 0 and study the arithmetic of rational elliptic curves over quintic fields. In the second part, we follow the ideas of Loefller, Skinner and Zerbes to set up and conduct local automorphic computations on the algebraic group GSp4 x GL2, establishing technical results that should lead to the norm relations of an Euler system for GSp4 x GL2.
Alternate format: The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog:
Type of Material: Academic dissertations (Ph.D.)
Language: en
Appears in Collections:Mathematics

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