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We define an enrichment of the logarithmic derivative of the zeta function of a variety over a finite field to a power series with coefficients in the Grothendieck–Witt group. We show that this enrichment is related to the topology of the real points of a lift. For cellular schemes over a field, we prove a rationality result for this enriched logarithmic derivative of the zeta function as an analogue of part of the Weil conjectures. We also compute several examples, including toric varieties, and show that the enrichment is a motivic measure.
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We prove two finiteness results for reductions of Hecke orbits of abelian varieties over local fields: one in the case of supersingular reduction and one in the case of reductive monodromy. As an application, we show that only finitely many abelian varieties on a fixed isogeny leaf admit CM lifts, which in particular implies that in each fixed dimensiongonly finitely many supersingular abelian varieties admit CM lifts. Combining this with the Kuga–Satake construction, we also show that only finitely many supersingular K3surfaces admit CM lifts. Our tools includep-adic Hodge theory and group-theoretic techniques.more » « less
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Abstract We prove an inequality between the conductor and the discriminant for all hyperelliptic curves defined over discretely valued fields $$K$$ with perfect residue field of characteristic not $$2$$. Specifically, if such a curve is given by $$y^{2} = f(x)$$ with $$f(x) \in \mathcal{O}_{K}[x]$$, and if $$\mathcal{X}$$ is its minimal regular model over $$\mathcal{O}_{K}$$, then the negative of the Artin conductor of $$\mathcal{X}$$ (and thus also the number of irreducible components of the special fiber of $$\mathcal{X}$$) is bounded above by the valuation of $$\operatorname{disc}(f)$$. There are no restrictions on genus of the curve or on the ramification of the splitting field of $$f$$. This generalizes earlier work of Ogg, Saito, Liu, and the second author.more » « less
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