Asian Journal of Mathematics

Volume 22 (2018)

Number 4

Special issue in honor of Ngaiming Mok (3 of 3)

Guest Editors: Jun-Muk Hwang, Korea Institute for Advanced Study; Yum-Tong Siu, Harvard University; Wing-Keung To, National University of Singapore; Stephen S.-T. Yau, Tsinghua University; Sai-Kee Yeung, Purdue University

Curves in Hilbert modular varieties

Pages: 673 – 690



Erwan Rousseau (Institut Universitaire de France & Aix Marseille Université, Marseille, France)

Frédéric Touzet (IRMAR, Université de Rennes, France)


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.


Hilbert-modular varieties, Green–Griffiths–Lang conjecture, entire curves, foliations, Second Main Theorem

2010 Mathematics Subject Classification

Primary 32Q45, 37F75. Secondary 11F41.

Received 3 May 2016

Accepted 1 June 2017

Published 20 September 2018