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Arithmetic aspects of the Burkhardt quartic threefold. (English) Zbl 1446.11111
Summary: We show that the Burkhardt quartic threefold is rational over any field of characteristic distinct from 3. We compute its zeta function over finite fields. We realize one of its moduli interpretations explicitly by determining a model for the universal genus 2 curve over it, as a double cover of the projective line. We show that the $$j$$-planes in the Burkhardt quartic mark the order 3 subgroups on the abelian varieties it parametrizes, and that the Hesse pencil on a $$j$$-plane gives rise to the universal curve as a discriminant of a cubic genus 1 cover.

##### MSC:
 11G10 Abelian varieties of dimension $$> 1$$ 11G18 Arithmetic aspects of modular and Shimura varieties 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14H10 Families, moduli of curves (algebraic) 14K10 Algebraic moduli of abelian varieties, classification
##### Keywords:
Burkhardt quartic threefold; Hesse pencil
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