Arithmetic monodromy actions on the pro-metabelian fundamental group of punctured elliptic curves

Arithmetic monodromy actions on the pro-metabelian fundamental group of punctured elliptic curves

###### Number Theory Seminar

#### William Chen (IAS)

**Wednesday, February 21, 2018 -3:15pm to 4:15pm**

For a finite 2-generated group *G*, one can consider the moduli of elliptic curves equipped with *G*-structures, which is roughly a *G*-Galois cover of the elliptic curve, unramified away from the origin. The resulting moduli spaces are quotients of the upper half plane by possibly noncongruence subgroups of *SL(2,Z)*. When *G* is abelian, it is easy to see that such level structures are equivalent to classical congruence level structures, but in general it is difficult to classify the groups *G* which yield congruence level structures. In this talk I will focus on a recent joint result with Pierre Deligne, where we show that for any metabelian *G*, *G*-structures are congruence in an arithmetic sense. We do this by studying the monodromy action of the fundamental group of the moduli stack of elliptic curves (over *Q*) on the pro-metabelian fundamental group of a punctured elliptic curve.