Modular curves

Modular curves are mathematical objects that encode relationships between special functions called modular forms, which are linked to complex analysis and number theory. They can be thought of as geometric shapes that parameterize families of elliptic curves—smooth, donut-shaped surfaces—together with additional symmetry data. These curves help mathematicians study how elliptic curves behave, especially regarding their solutions and symmetries, and play a crucial role in understanding deep connections like the proof of Fermat’s Last Theorem. Overall, modular curves serve as a bridge connecting geometry, algebra, and number theory.