A Conjectural Reciprocity Law for Realizations of Motives

Explore a deep dive into advanced algebraic geometry and number theory in this 1 hour 28 minute lecture by Dustin Clausen at the Institut des Hautes Etudes Scientifiques (IHES). Delve into the concept of motives over a scheme S, understanding how they capture the cohomology of smooth proper S-schemes through linear algebra. Examine various "realizations" of motives and their relationships across different schemes. Discover how these realizations connect through comparison isomorphisms in certain cases, similar to Hodge theory. Investigate a proposed subtle relationship between realizations for general schemes, drawing parallels to classical reciprocity laws in number theory. Gain insights into cutting-edge research at the intersection of algebraic geometry, motives, and number theory.