Noncongruence subgroups of $\mathrm{SL}(2,\mathbb Z)$

Mathematical Conversations
Topic:Noncongruence subgroups of $\mathrm{SL}(2,\mathbb Z)$
Speaker:William Chen
Affiliation:Member, School of Mathematics
Date:Wednesday, November 23
Time/Room:6:00pm - 7:00pm/Dilworth Room

Over the last century, extraordinary amounts of effort have been put into understanding the congruence subgroups of $\mathrm{SL}(2,\mathbb Z)$, whose study has led to the birth of the Langlands program, the modularity theorem, and as a consequence Fermat's Last Theorem. However, $\mathrm{SL}(2,\mathbb Z)$ also has many noncongruence subgroups, which despite having been known to exist since the time of Fricke and Klein (late 1800's), have (in comparison) been largely ignored. For this talk I will describe how we may extend the moduli interpretations of classical congruence modular curves to quotients of the upper half plane by noncongruence subgroups. This will show that whereas congruence modular curves capture the essence of elliptic curves as abelian varieties, noncongruence modular curves see elliptic curves as "anabelian" curves. In a way this asks more questions than it answers, and in my opinion makes noncongruence subgroups fascinating objects of study in their own right.