Patching and multiplicity 2k for Shimura curves

Fundamental Domains for Shimura Curves

We describe a process for defining and computing a fundamental domain in the upper half plane H of a Shimura curve X 0 (N) associated to an order in a quaternion algebra A/Q. A fundamental domain for X 0 (N) realizes a finite presentation of the quaternion unit group, modulo units of its center. We give explicit examples of domains for the curves X 0 (1), X 0 (1), and X 35 0 (1). The first exam...

Ramanujan Graphs and Shimura Curves

What follows are some long, rambling notes of mine on Ramanujan graphs. For a period of about two months in 2006 I thought very intensely on this subject and had thoughts running in several different directions. In paricular this document contains the most complete exposition so far of my construction of expander graphs using Hecke operators on Shimura curves. I am posting this document now (Ma...

L -invariants and Shimura curves

In earlier work, the second named author described how one may extract Darmonstyle L -invariants from modular forms on Shimura curves that are special at p. In this paper, we show that these L -invariants are preserved by the Jacquet–Langlands correspondence. As a consequence, we prove the second named author’s period conjecture in the case where the base field is Q. As a further application of...

Notes on Shimura curves

Lectures on Shimura Curves 0: Modular Curves

General remarks about these documents: These are lecture notes. Let us reflect separately on the meaning of both words. The first word is meant to indicate some correspondence between what is said in the lectures and what appears in these notes. This correspondence is not precise: on the one hand, in response to a question or a comment, I may say things in class which do not make it into these ...