Abstract
The vertical Sato-Tate conjectures gives expected trace distributions for for families of curves. We develop exact expression for the distribution associated to degree-4 representations of USp(4), SU(2) × SU(2) and SU(2) in the neighborhood of the extremities of the Weil bound. As a consequence we derive qualitative distinctions between the extremal traces arising from generic genus-2 curves and genus-2 curves with real or quaternionic multiplication. In particular we show, in a specific sense, to what extent curves with real multiplication dominate the contribution to extremal traces.
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