On the indecomposability of the image of the universal pro-${l}$ outer monodromy representation of the moduli stack of once-punctured elliptic curves (Algebraic Number Theory and Related Topics 2015)

Abstract

Minamide proved that the pro‐l Grothendieck–Teichmüller group GT_{l} and the image of the absolute Galois group of a number field in GT_{l} are indecomposable, i.e., do not have a nontrivial direct product decomposition. This Galois image may be identified with the image of the universal pro-{l} outer monodromy representation of the moduli stack of projective lines minus three points over the number field. In the present paper, we prove the indecomposability of the image of the universal pro-{l} outer monodromy representation of the moduli stack of once-punctured elliptic curves over either a number field or an algebraically closed field of characteristic zero.