[Paper Review] The moduli space of (1,11)-polarized abelian surfaces is unirational

Mark Gross, Sorin Popescu|ArXiv.org|Feb 2, 1999

Algebraic Geometry and Number Theory20 references4 citations

TL;DR

This paper proves that the moduli space of (1,11)-polarized abelian surfaces with level structure of canonical type is birational to Klein's cubic hypersurface in P⁴, establishing its unirationality. The result implies the non-existence of Γ₁₁-cusp forms of weight 3 and provides a new proof of the rationality of the moduli space A₉^lev.

ABSTRACT

We prove that the moduli space A_{11}^{lev} of (1,11) polarized abelian surfaces with level structure of canonical type is birational to Klein's cubic hypersurface: a^2b+b^2c+c^2d+d^2e+e^2a=0 in P^4. Therefore, A_{11}^{lev} is unirational but not rational, and there are no Gamma_{11}-cusp forms of weight 3. The same methods also provide an easy proof of the rationality of A_{9}^{lev}.

Motivation & Objective

• To determine the birational geometry of the moduli space A₁₁^lev of (1,11)-polarized abelian surfaces with level structure.
• To investigate whether this moduli space is rational or unirational, and to resolve the existence of cusp forms of weight 3 for the group Γ₁₁.
• To extend methods used in the rationality proof of A₉^lev to the more complex case of A₁₁^lev.
• To establish a geometric link between the moduli space and a well-known algebraic variety—Klein's cubic hypersurface.

Proposed method

• Construct a birational map between the moduli space A₁₁^lev and Klein's cubic hypersurface defined by a²b + b²c + c²d + d²e + e²a = 0 in P⁴.
• Utilize the theory of level structures and canonical polarizations on abelian surfaces to parametrize the moduli space.
• Apply techniques from algebraic geometry, including the study of theta characteristics and line bundles of type (1,11).
• Leverage the symmetry and singularities of Klein's cubic to analyze the rationality properties of the moduli space.
• Use the same methodological framework that previously proved A₉^lev rational to analyze A₁₁^lev.
• Employ computational algebraic tools (Macaulay2 code and PostScript files) to verify geometric constructions and symmetries.

Experimental results

Research questions

• RQ1 Is the moduli space A₁₁^lev of (1,11)-polarized abelian surfaces unirational?
• RQ2 What is the birational class of the moduli space A₁₁^lev, and how does it relate to known algebraic varieties?
• RQ3 Do there exist Γ₁₁-cusp forms of weight 3, and what does this imply about the geometry of A₁₁^lev?
• RQ4 Can the methods used to prove rationality of A₉^lev be extended to A₁₁^lev?
• RQ5 What is the significance of Klein's cubic hypersurface in the context of moduli spaces of abelian surfaces?

Key findings

• The moduli space A₁₁^lev is birational to Klein's cubic hypersurface in P⁴, defined by the equation a²b + b²c + c²d + d²e + e²a = 0.
• As a consequence, A₁₁^lev is unirational, though not rational, due to the non-rationality of Klein's cubic.
• There are no Γ₁₁-cusp forms of weight 3, as implied by the unirationality and the geometry of the moduli space.
• The same techniques used in this proof yield a new, simplified proof of the rationality of A₉^lev.
• The birational equivalence to a hypersurface with high symmetry provides a geometric explanation for the unirationality of A₁₁^lev.
• The result demonstrates a deep connection between moduli spaces of abelian surfaces and classical algebraic varieties like Klein's cubic.