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”Shimura curves”
http://www.math.columbia.edu/~chaoli/
Chao Li's homepage
http://www.math.columbia.edu/~chaoli/docs/ShimuraCurves.html
Shimura curves

In the 60s, Shimura studied certain algebraic curves as analogues of classical modular curves in order to construct class fields of totally real number fields. These curves were later coined "Shimura curves" and vastly generalized by Deligne. We will take a tour of the rich geometry and arithmetic of Shimura curves. Along the way, we may encounter tessellations of disks, quaternion algebras, abelian surfaces, elliptic curves with CM, Hurwitz curves ... and the answer to life, the universe and everything.

[-] Contents
Review of Modular Curves
Shimura curves
Moduli interpretation and class fields
Hurwitz curves

Briefly speaking, Shimura curves are simply one-dimensional Shimura varieties. I have accomplished my trivial notion task because I have told you a trivial notion. But obviously it does not help much if you do not know what the term Shimura varieties means. It only takes 5 chapters in Milne's notes in order to define them ? not too bad ? but initially Shimura invented them really because they are natural analogues of classical modular curves.

https://math.dartmouth.edu/~jvoight/articles/shimura-clay-proceedings-071707.pdf
Shimura curve computations
John Voight 1991 Mathematics Subject Classification.
Abstract. We introduce Shimura curves first as Riemann surfaces and then
as moduli spaces for certain abelian varieties. We give concrete examples of
these curves and do some explicit computations with them.
1. Introduction: modular curves

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