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Algebraic Cycles, L-Values, and Euler Systems January 17, 2023 to May 26, 2023
Organizers Henri Darmon (McGill University), Ellen Eischen (University of Oregon), LEAD Benjamin Howard (Boston College), David Loeffler (UniDistance Suisse), Christopher Skinner (Princeton University), Sarah Zerbes (ETH Zürich), Wei Zhang (Massachusetts Institute of Technology)
Some Gaussian periods for the 255,255-th cyclotomic extension. Image credit: E. Eischen, based on earlier work by W. Duke, S. R. Garcia, T. Hyde, and R. Lutz
The fundamental conjecture of Birch and Swinnerton-Dyer relating the Mordell–Weil ranks of elliptic curves to their L-functions is one of the most important and motivating problems in number theory. It resides at the heart of a collection of important conjectures (due especially to Deligne, Beilinson, Bloch and Kato) that connect values of L-functions and their leading terms to cycles and Galois cohomology groups.  The study of special algebraic cycles on Shimura varieties has led to progress in our understanding of these conjectures. The arithmetic intersection numbers and the p-adic regulators of special cycles are directly related to the values and derivatives of L-functions, as shown in the pioneering theorem of Gross-Zagier and its p-adic avatars for Heegner points on modular curves. The cohomology classes of special cycles (and related constructions such as Eisenstein classes) form the foundation of the theory of Euler systems, providing one of the most powerful methods known to prove vanishing or finiteness results for Selmer groups of Galois representations.  The goal of this semester is to bring together researchers working on different aspects of this young but fast-developing subject, and to make progress on understanding the mysterious relations between L-functions, Euler systems, and algebraic cycles. For details on the image, see http://www.elleneischen.com/gaussianperiods.html. Bibliography
Keywords and Mathematics Subject Classification (MSC)
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Programmatic Workshops
January 19, 2023 - January 20, 2023 Connections Workshop: Algebraic Cycles, L-Values, and Euler Systems
January 23, 2023 - January 27, 2023 Introductory Workshop: Algebraic Cycles, L-Values, and Euler Systems
March 13, 2023 - March 17, 2023 Shimura Varieties and L-Functions