Lectures on Shimura Curves and Related Objects

University of Sheffield, 2015/16


This academic year, I will deliver lectures on Shimura curves and related objects.

The meetings will take place in room J-11 Hicks Building on Wednesdays at 11am (unless stated otherwise) during the Fall.

Extended notes of the past lectures can be found HERE.

Fall Talks:

Title: Quaternion Algebras, I
Week: 6
Contents: Basic properties.

Title: Quaternion Algebras, II
Week: 8
Contents: Quaternion algebras over local fields, ramification of quaternion algebras over number fields, orders.

Title: Hyperbolic Spaces of Dimensions 2 and 3
Week: 9
Contents: We introduce basics of hyperbolic geometry and the groups of isometries in dimensions 2,3.

Title: Arithmetic manifolds associated to orders in quaternion algebras
Week: 10
Contents: We discuss arithmetic lattices arising from groups of norm one elements of orders in quaternion algebras over number fields and the associated manifolds.

Title: Modular curves and Shimura curves, I
Week: 12
Contents: We focus on modular and Shimura curves and their coarse moduli space interpretations.

Spring Talks:

Title: Modular curves and Shimura curves, II
Week: 2
Contents: We focus on modular and Shimura curves and their coarse moduli space interpretations.

Title: Arithmetic hyperbolic 3-manifolds
Week: 3
Contents: We digress to consider the interesting case of Bianchi groups and compact arithmetic hyperbolic 3-manifolds.

Title: Modular forms
Week: 4
Contents: We introduce modular forms.

Title: Cohomology and modular forms
Week: 5
Contents: We study the cohomology of arithmetic groups and relations with modular forms.

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