Multiplicative zeta function and logarithmic point counting over finite fields

2017/05/02 by Braunling, Oliver
#11M38 #14C15 #Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.1705.01192

Abstract

The zeta function of a motive over a finite field is multiplicative with respect to the direct sum of motives. It has beautiful analytic properties, as were predicted by the Weil conjectures. There is also a multiplicative zeta function, which instead respects the tensor product of motives. There is no analogue of the Weil conjectures, and we give a sufficient criterion for an analytic continuation to exist. This happens, for example, for cellular varieties, abelian varieties, or genus g > 1 curves with a supersingular Jacobian.

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