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Explore advanced research in algebraic number theory through this mathematical lecture examining the Bloch–Kato Conjecture for four-dimensional symplectic Galois representations. Delve into the conjecture's prediction of relationships between Selmer ranks and orders of vanishing of L-functions for Galois representations arising from etale cohomology of algebraic varieties. Learn about significant results toward proving 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), which contribute to cohomology of classical Siegel threefolds. Discover the construction of auxiliary ramified Galois cohomology classes as the key proof technique, providing bounds on Selmer groups through level-raising congruences and the geometry of special cycles on Shimura varieties. Gain insights into cutting-edge research connecting algebraic geometry, number theory, and representation theory from Princeton University researcher Naomi Sweeting in this Institut des Hautes Etudes Scientifiques presentation.
