Asian Journal of Mathematics

Volume 22 (2018)

Number 2

Special issue in honor of Ngaiming Mok (1 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

Explicit formulas for infinitely many Shimura curves in genus $4$

Pages: 381 – 390



Samuel Grushevsky (Department of Mathematics, Stony Brook University, Stony Brook, New York, U.S.A.)

Martin Möller (Institut für Mathematik, Goethe-Universität Frankfurt, Germany)


In this paper we construct infinitely many Shimura curves contained in the locus of Jacobians of genus four curves. All Jacobians in these families are $\mathbb{Z}/3$ covers of varying elliptic curves that appear in a geometric construction of Pirola, and include an example of a Shimura–Teichmüller curve that parameterizes Jacobians that are suitable $\mathbb{Z}/6$ covers of $\mathbb{P}^1$. We compute explicitly the period matrices of the Shimura curves we construct using the original construction of Shimura for moduli spaces of abelian varieties with automorphisms.


abelian variety, Jacobian, Shimura variety

2010 Mathematics Subject Classification

14G35, 14H40

Research of the first author is supported in part by National Science Foundation under the grants DMS-1201369 and DMS-1501265.

Research of the second author is supported in part by ERC-StG-257137.

Received 21 May 2016

Accepted 14 June 2017

Published 15 June 2018