Algebraic Geometry Seminar

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Modular Curves and Finite Groups

Speaker: David Roe, MIT

Location: Warren Weaver Hall 512

Date: Tuesday, February 20, 2024, 3:30 p.m.

Synopsis:

The most famous modular curves are $X_0(N)$ and $X_1(N)$, which parameterize elliptic curves with an N-isogeny or an N-torsion point (respectively).  But there are many more, corresponding to subgroups of $GL(2, \mathbb{Z}/N)$.  Over the last several years I have helped create databases both of modular curves and of finite groups (available online as part of the L-functions and modular forms database).  I will describe some interesting mathematical and algorithmic problems that arose as part of this endeavor, ranging from computing gonalities and models for modular curves, to isomorphism-invariant hashes and minimal degrees for linear and permutation representations of finite groups.