Arithmetic aspects of the Burkhardt quartic threefold

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 one cover.

[1]  M. Arslan.  Integral cohomology of the Siegel modular variety of degree two and level three , 2006 .

[2]  Gerard van der Geer.  Note on abelian schemes of level three , 1987 .

[3]  Limits of Pluri–Tangent Planes to Quartic Surfaces , 2013, 1304.7463.

[4]  Damiano Testa,et al.  Descent via (3,3)-isogeny on Jacobians of genus 2 curves , 2014, 1401.0580.

[5]  Jean-François Mestre,et al.  Construction de courbes de genre 2 à partir de leurs modules , 1991 .

[6]  G. M.,et al.  Quartic Surfaces with Singular Points , 1917, Nature.

[7]  Point Sets and Allied Cremona Groups. , 1915, Proceedings of the National Academy of Sciences of the United States of America.

[8]  S. Weintraub,et al.  The Siegel modular variety of degree two and level four , 1998 .

[9]  E. V. Flynn,et al.  Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2: Index rerum et personarum , 1996 .

[10]  David Eisenbud,et al.  Classical Algebraic Geometry , 2019, Oberwolfach Reports.

[11]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[12]  Arthur B. Coble,et al.  Associated sets of points , 1922 .

[13]  I. Dolgachev,et al.  On isogenous principally polarized abelian surfaces , 2007, 0710.1298.

[14]  H. Burkhardt.  Untersuchungen aus dem Gebiete der hyperelliptischen Modulfunctionen , 1890 .

[15]  A locus with 25920 linear self-transformations , 1946 .

[16]  Bruce Hunt,et al.  The Geometry of some special Arithmetic Quotients , 1996 .

[17]  A. J. Jong,et al.  On the Burkhardt quartic , 1990 .

[18]  S. Weintraub,et al.  Janus-like algebraic varieties , 1994 .

[19]  Qing Liu,et al.  Algebraic Geometry and Arithmetic Curves , 2002 .

[20]  Serge Lang,et al.  Abelian varieties , 1983 .

[21]  Noam D. Elkies.  The identification of three moduli spaces , 1999 .

[22]  H. Baker.  An Introduction to the Theory of Multiply Periodic Functions , 2007 .

[23]  G. B. Mathews,et al.  Kummer's Quartic Surface , 1990 .

[24]  Andrew Snowden,et al.  A description of the outer automorphism of S6, and the invariants of six points in projective space , 2007, J. Comb. Theory A.

[25]  J. Todd.  On a quartic primal with forty-five nodes, in space of four dimensions , 1936 .

[26]  Eberhard Freitag,et al.  The Burkhardt Group and Modular Forms , 2004 .

[27]  Bernd Sturmfels,et al.  Tropicalization of Classical Moduli Spaces , 2013, Math. Comput. Sci..

[28]  Projective models of Picard modular varieties , 1992 .