Isogeny Club - Season 4 - Talk #5: A survey on modular curves and their applications

This is the discussion thread for the fifth session of season four of The Isogeny Club , where Guido presents his talk titled: A survey on modular curves and their applications

In this seminar, I want to give a brief introduction to modular forms and modular curves, algebraic curves parametrizing (i.e., whose points “are”) elliptic curves, or isogenies. These objects were studied by great mathematicians like Hecke, Ramanujan, Deligne, and others, and are fundamentally used, for example, in the proof of Fermat’s Last Theorem. Supersingular curves motivate why modular curves have many points over 𝔽p2, which could be useful for codes (see the work of Tsfasman and Vlăduţ) and vice-versa properties of the Jacobians of modular curves give information about supersingular isogeny graphs. This is mostly work by other people.