Average Heights of Abelian Varieties with Complex Multiplication

Explore the intricate relationship between periods and L-functions in the context of abelian varieties with complex multiplication. Delve into Pierre Colmez's groundbreaking conjecture from the 1990s, which proposes a precise connection between periods and the degree 1 coefficient of certain Artin L-functions' Taylor expansion at 0. Learn about Colmez's proof for the cyclotomic field case and discover how recent work by Tonghai Yang, Bruinier, Kudla, and Yang has led to proving a consequence of Colmez's conjecture for average periods of abelian varieties with complex multiplication by a fixed CM number field. Gain insights into this collaborative research effort with F. Andreatta, E. Goren, and B. Howard, presented as part of the International Congress of Basic Science 2024.