On Mumford’s Construction of Degenerating Abelian Varieties

Moduli of Abelian Varieties

In this paper we generalize parts of Mumford’s theory of the equations defining abelian varieties. Using the concept of a strongly symmetric line bundle, which is weaker than Mumford’s concept of totally symmetric line bundle and is introduced here for the first time, we extend Mumford’s methods of obtaining equations to arbitrary levels and to ample strongly symmetric line bundles. The first t...

Non-archimedean canonical measures on abelian varieties

For a closed d-dimensional subvariety X of an abelian variety A and a canonically metrized line bundle L on A, Chambert-Loir has introduced measures c1(L|X) ∧d on the Berkovich analytic space associated to A with respect to the discrete valuation of the ground field. In this paper, we give an explicit description of these canonical measures in terms of convex geometry. We use a generalization o...

The Spectral Construction for a (1,8)-Polarized Fam- ily of Abelian Varieties

We extend the Spectral Construction, a technique used with great success to study and construct vector bundles on elliptically fibered varieties, to a special family of abelian surface fibered varieties. The results are motivated by requirements from Heterotic String Phenomenology where vector bundles with specified chern class are required to produce a realistic particle spectrum. Although onl...

Rigid local systems, Hilbert modular forms, and Fermat’s last theorem

1. Frey representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 1.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 1.2. Classification: The rigidity method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 1.3. Construction: Hyper...

On the varieties defined by Riemann-Mumford’s relations