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Explore a mathematical lecture presenting a recent proof demonstrating that for every quadratic extension of number fields K/F, there exists an abelian variety over F with positive rank that maintains the same rank over K. Learn how this result, achieved through collaboration with Alpoge, Bhargava, and Ho, connects to Hilbert's tenth problem by showing that such varieties can be constructed as Jacobians of curves of the form y^2 = x^p + n. Discover how combining this finding with Shlapentokh's previous work leads to a negative answer for Hilbert's tenth problem over the ring of integers of any number field, a result also recently established through different methods by Koymans-Pagano. Examine the broader implications of K-rank stability and what current research reveals about the proportion of elliptic curves over F that exhibit this property, gaining insight into cutting-edge developments in algebraic number theory and Diophantine equations.
