Curves in Hilbert modular varieties

2015/01/14 by Rousseau, Erwan, Touzet, Frédéric
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1501.03261

Abstract

We prove a boundedness-theorem for families of abelian varieties with real multiplication. More generally, we study curves in Hilbert modular varieties from the point of view of the Green Griffiths-Lang conjecture claiming that entire curves in complex projective varieties of general type should be contained in a proper subvariety. Using holomorphic foliations theory, we establish a Second Main Theorem following Nevanlinna theory. Finally, with a metric approach, we establish the strong Green-Griffiths-Lang conjecture for Hilbert modular varieties up to finitely many possible exceptions.

Related

• Rings of Hilbert modular forms, computations on Hilbert modular surfaces, and the Oda-Hamahata conjecture
• Modularity and effective Mordell I
• Abelian varieties with real multiplication: classification and isogeny classes over finite fields
• Modular Curves, Modular Surfaces, and Modular Fourfolds
• Hilbert properties of varieties