We show that a conjectural extension of a fixed point formula in Arakelov geometry implies results about a tautological subring in the arithmetic Chow ring of bases of abelian schemes. Among the results are an Arakelov version of the Hirzebruch proportionality principle and a formula for a critical power of $\hat c_1$ of the Hodge bundle.
A fixed point formula of Lefschetz type in Arakelov geometry III: representations of Chevalley schemes and heights of flag varieties
Trace of abelian varieties over function fields and the geometric Bogomolov conjecture
Fixed point formula and loop group actions
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