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.

It is well known that every finite subgroup of GLd(Q[ell ]) is conjugate to a subgroup of GLd(Z[ell ]). However, this does not remain true if we replace general linear groups by symplectic groups. We say that G is a group of inertia type of G is a finite group which has a normal Sylow-p-subgroup with cyclic quotient. We show that if [ell ]>d+1, and G is a subgroup of Sp2d(Q[ell ]) of inertia type, then G is conjugate in GL2d(Q[ell ]) to a subgroup of Sp2d(Z[ell ]). We give examples which show that the bound is sharp. We apply these results to construct, for every odd prime [ell ], isogeny classes of Abelian varieties all of whose polarizations have degree divisible by [ell ]2. We prove similar results for Euler characteristic of invertible sheaves on Abelian varieties over fields of positive characteristic.

We establish one direction of a conjecture by Lakshmibai and Sandhya which describes combinatorially the singular locus of a Schubert variety. We prove that the conjectured singular locus is contained in the singular locus.

Let Γ be one of the N2-dimensional bicovariant first-order differential calculi on the quantum groups Oq(N) or Spq(N), where q is not a root of unity. We show that the second antisymmetrizer exterior algebra sΓ∧ is the quotient of the universal exterior algebra uΓ∧ by the principal ideal generated by θ∧θ. Here ∧ denotes the unique up to scalars bi-invariant 1-form. Moreover, θ∧θ is central in uΓ∧ and uΓ∧ is an inner differential calculus.

We formulate a generalization of Givental–Kim's quantum hyperplane principle. This is applied to compute the quantum cohomology of a Calabi–Yau 3-fold defined as the rank 4 locus of a general skew-symmetric 7×7 matrix with coefficients in P6. The computation verifies the mirror symmetry predictions of Rødland [25].

We prove that the multiplicity of an arbitrary dominant weight for an irreducible highest weight representation of the affine Kac–Moody algebra A(1)r is a polynomial in the rank r. In the process we show that the degree of this polynomial is less than or equal to the depth of the weight with respect to the highest weight. These results allow weight multiplicity information for small ranks to be transferred to arbitrary ranks.

Let S be a scheme, and let G be a finite, flat, commutative group scheme over S. In this paper we show that (subject to a mild technical assumption) every primitive class in Pic(G) is realisable. This gives an affirmative answer to a question of Waterhouse. We also discuss applications to locally free classgroups and to Selmer groups of Abelian varieties.