Compositio Mathematica

The Moduli Space of (1,11)-Polarized Abelian Surfaces is Unirational

Mark Gross a1 and Sorin Popescu a2
a1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, U.K. E-mail:
a2 Department of Mathematics, Columbia University, New York, NY 10027, U.S.A. E-mail:

Article author query
gross m   [Google Scholar] 
popescu s   [Google Scholar] 


We prove that the moduli space $\cal A$11lev of <$>(1,11)-polarized Abelian surfaces with level structure of canonical type is birational to Klein's cubic hypersurface in P4. Therefore, $\cal A$11lev is unirational but not rational, and there are no Γ11-cusp forms of weight 3. The same methods also provide an easy proof of the rationality of $\cal A$9lev.

Key Words: Abelian surfaces; moduli; Kleine's cubic hypersurface.