Lectures on modular curvesHwajong Yoo (SNU) Abstract: In the ﬁrst talk, we will study the theory of modular curves over the ﬁeld of complex numbers. consists of two parts:1. the theory of elliptic curves over the ﬁeld of complex numbers. 2. the theory of compact Riemann surfaces and their Jacobian varieties. We will review relevant results and give heuristic explanation of the theory instead of giving rigorous proofs of well-known theorems. In the second talk, we will focus on the theory of modular curves over the ﬁeld of rational numbers. We will deal with the following. 1. Constructing Galois representations attached to cuspidal eigenforms of weight 2. 2. Constructing rational torsion points on the Jacobian varieties of modular curves. |