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The first part of this talk examines the isomorphism classes M_{\O_K}(\Phi) of simple
principally polarized abelian varieties of dimensions g = 2, 3 over number fields,
with complex multiplication (CM) of type (K, \Phi). A critical aspect of understanding
the structure of \M_{\O_K}(\Phi) lies in analyzing the Shimura class group CK of K and
the reflex-type-norm map N_{\Phi^r} within C_K. According to Shimura's Main Theorem of
CM, the orbits of M_{\O_K}(\Phi) under the action of the Galois group \Gal(\bar Q |
K^r) correspond to the elements of the quotient C_K/N_{\Phi^r}. In this part of the
talk, we will define all the relevant structures involved in the characteristic zero
case, including the Shimura class group, the reflex-type-norm maps, and their
interactions with CM abelian varieties.
The second part of this talk, a joint project with P. Kutas (ELTE), G. Lorenzon, and W. Castryck (KU Leuven), focuses on oriented principally polarized superspecial abelian surfaces in positive characteristic $p$. This section explores how insights from M_{\O_K}(\Phi) in characteristic zero can inform our understanding of these structures in characteristic p. We will also introduce some challenges of descending the Shimura class group action from characteristic zero to positive characteristic. |