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Explore advanced research in algebraic number theory through this mathematical lecture examining mod p local-global compatibility for partial weight one Hilbert modular forms. Delve into the study of mod p Hilbert modular forms for totally real fields F where p is unramified, focusing on forms of level prime to p and weight (k, l) with integer tuples k and l. Learn about the Diamond-Sasaki association of two-dimensional mod p Galois representations to mod p Hilbert modular Hecke eigenforms, and understand the local-global compatibility conjecture that predicts crystalline lifts with explicitly determined Hodge-Tate weights at places above p. Discover a proof demonstrating that LGC for regular p-bounded weights (entries of k between 2 and p+1) implies LGC in the partial weight one p-bounded case (entries of k between 1 and p+1). Examine the innovative approach combining scheme-theoretic intersection computations on the Emerton-Gee stack with weight-changing arguments on quaternionic Shimura varieties, utilizing restriction to Goren-Oort strata. This presentation represents joint work in progress with Brandon Levin and David Savitt, offering insights into cutting-edge developments in the intersection of algebraic geometry, number theory, and representation theory.
