A New Automorphism Of X0(108)

TL;DR

This paper identifies a mistake in previous work on automorphisms of modular curves, specifically correcting the case N=108 by constructing a new automorphism of order 2.

Contribution

It corrects the analysis of automorphisms of X0(108) and explicitly constructs a new automorphism of order 2, expanding understanding of automorphism groups of modular curves.

Findings

B0(108) has index 2 in A0(108)

Constructed an explicit automorphism of order 2

Corrects previous analysis by Kenku and Momose

Abstract

Let X0(N) denote the modular curve classifying elliptic curves with a cyclic N-isogeny, A0(N) its group of algebraic autmorphisms and B0(N) the subgroup of automorphisms coming from matrices acting on the upper halfplane. In a well-known paper, Kenku and Momose showed that A0(N) and B0(N) are equal (all automorphisms come from matrix action) when X0(N) has genus at least 2, except for N = 37 and 63. However, there is a mistake in their analysis of the N = 108 case. In the style of Kenku and Momose, we show that B0(108) is of index 2 in A0(108) and construct an explicit new automorphism of order 2 on a canonical model of X0(108).