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This 50-minute conference talk explores the Bloch–Kato Conjecture for four-dimensional symplectic Galois representations. Discover how Naomi Sweeting presents new results toward this conjecture in ranks 0 and 1 for self-dual Galois representations derived from Siegel modular forms on GSp(4) with parallel weight (3,3). Learn about the crucial construction of auxiliary ramified Galois cohomology classes that provide bounds on Selmer groups, with these ramified classes originating from level-raising congruences and the geometry of special cycles on Siegel threefolds. Understand the essential role of torsion-vanishing results for cohomology of Shimura varieties, attributed to Caraiani–Scholze, Koshikawa, and Lee–Hamann. This presentation is part of the 2025 Simons Collaboration on Perfection in Algebra, Geometry and Topology Annual Meeting.
