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Explore advanced topics in algebraic number theory through this comprehensive lecture series from the International Centre for Theoretical Sciences. Delve into the intricate connections between elliptic curves and special values of L-functions across multiple detailed presentations by leading mathematicians. Study non-vanishing properties of modular forms at CM points, examine the Brumer-Stark conjecture through extensive multi-part lectures, and investigate average ranks of elliptic curves with geometric and statistical approaches. Learn about motivic L-functions, recent developments in the Beilinson-Bloch-Kato conjecture, and Euler systems including Beilinson-Flach elements. Explore Iwasawa theory for elliptic curves at Eisenstein primes, parametrization of Selmer groups, and special values of zeta functions. Discover geometry-of-numbers techniques in arithmetic statistics, p-adic L-functions for various groups including GSp(4)×GL(2) and U(3)×U(2), and computational aspects of local volumes and mass formulas. Examine advanced topics including p-adic Artin formalism, modular forms of weight one, Serre duality on character varieties, multivariate (φ,Γ)-modules, and Iwasawa main conjectures for universal families. Investigate bounds for L-functions, structure of Selmer groups, non-vanishing theorems for elliptic curve families, locally potentially equivalent Galois representations, and ergodic approaches to equidistribution results, providing deep insights into modern arithmetic geometry and number theory.
