Elliptic curves are fundamental objects in modern number theory, connecting algebraic geometry and arithmetic. In this talk, I will give an introductory overview of elliptic curves and their arithmetic, comparing the number field and function field settings and highlighting their similarities and differences. I will then present recent results on elliptic curves of analytic rank one and their ranks over certain ring class fields, based on joint work with Seokhyun Choi and Bo-Hae Im.