Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 75 / 5 / Solution

Let ${\mu }_N=[PS{L}_2(\mathbb{Z}):\overline{\Gamma (N)}]$, where the bar denotes the effective projective image. This is the degree of the modular curve projection, rather than the full $S{L}_2$ index when $-I$ is not in $\Gamma (N)$.

Reduction modulo $N$ is onto $S{L}_2(\mathbb{Z}/N\mathbb{Z})$. One proof uses the Chinese remainder theorem and elementary matrices: over $\mathbb{Z}/p^r\mathbb{Z}$, a unimodular first column has a unit entry, so elementary row operations reduce a determinant-one matrix to the identity. Each elementary matrix lifts integrally. Over ${\mathbb{F}}_p$, the first column has $p^2-1$ choices and the second has $p$ choices with prescribed determinant, giving $p(p^2-1)$. Each subsequent prime-power lift has $p^3$ choices, because the determinant-one linearized condition is one trace equation on four entries. HenceFor $N=1,2$, the subgroup contains $-I$; for $N\ge 3$ it does not. ConsequentlyFor $N\ge 2$ there are no effective elliptic stabilizer subgroups in the subgroup. Indeed, write $\gamma =I+NB\in \Gamma (N)$. The determinant condition gives $\mathrm{tr}\gamma =2-N^2\det B$. A noncentral elliptic element in $S{L}_2(\mathbb{Z})$ has trace $-1,0$ or $1$, none congruent to $2$ modulo $N^2$ for $N\ge 2$. Thus the projective action is torsion-free.

All cusp widths are $N$. At infinity, $T^h$ lies in the effective subgroup exactly when $h$ is divisible by $N$: the alternative $T^h\equiv -I(\mathrm{mod}N)$ can only occur for $N=1,2$, and for $N=2$ it imposes the same condition. Every rational modular cusp is a modular translate of infinity, and the subgroup is normal, so the same width holds there. The sum of modular cusp analytic ramification indices equals ${\mu }_N$, giving ${\mu }_N/N$ modular cusps.

The base $X(1)$ is a sphere. For instance $j=E_4^3/\Delta$ is a meromorphic function on it with a single simple pole at the modular cusp and no other poles, using the nonvanishing of $\Delta$ proved above. It therefore defines a degree-one map to the Riemann sphere, making the base genus zero.

The only branch values of $X(N)\to X(1)$ are the two elliptic orbits and the modular cusp. For $N\ge 2$, the numbers of points above them and their analytic ramification indices are respectively ${\mu }_N/2$ with index two, ${\mu }_N/3$ with index three, and ${\mu }_N/N$ with index $N$. Applying the Riemann-Hurwitz formula directly,This is the elliptic and cusp ramification for principal level calculation. Level one is the identity projection. The complete result isFor example the genera for $N=3,4,5,6,7$ are $0,0,0,1,3$. Keeping the exceptional projective index at level two is essential; applying the factor one-half there would give a nonintegral answer. This agrees with the principal congruence modular curve genus formula obtained from these branch counts.