Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 29 / 5

Pass to the effective subgroup $\overline{\Gamma }\subset PS{L}_2(\mathbb{Z})$. Its index is $\mu$: adjoining $-I$ halves the original index precisely when $-I\notin \Gamma$. The natural map of compactified modular curveshas degree $\mu$, since a generic orbit has that many lifts. It branches only over the order-two elliptic orbit, the order-three elliptic orbit, and the modular cusp. Away from these, the modular quotient charts are genuine unramified covering charts.

Over $i$, a lift with its own effective stabilizer of order two is unramified; there are ${\nu }_2$ such lifts. Every other lift has ramification index two. As the sum of all ramification indices in that fibre is $\mu$, their number is $(\mu -{\nu }_2)/2$ and their contribution to the ramification divisor is $(\mu -{\nu }_2)/2$. Similarly, over $\rho$ there are ${\nu }_3$ unramified elliptic lifts and $(\mu -{\nu }_3)/3$ lifts of ramification index three, contributing $2(\mu -{\nu }_3)/3$.

At a modular cusp of width $h$, the local parameters are $q_h=e^{2\pi i\tau /h}$ upstairs and $q=e^{2\pi i\tau }$ downstairs. Thus $q=q_h^h$ and the ramification index is $h$. Summing the widths gives $\mu$, so all ${\nu }_{\infty }$ modular cusps contribute $\mu -{\nu }_{\infty }$ to ramification. The Riemann-Hurwitz formula now givesThereforeThe effective index and the projective modular cusp widths ensure that the formula also applies when the original subgroup does not contain $-I$; no spurious factor of two is introduced.

Reduction modulo $N$ has kernel $\Gamma (N)$ and is onto $S{L}_2(\mathbb{Z}/N\mathbb{Z})$. To see surjectivity, use the Chinese remainder theorem to reduce to $\mathbb{Z}/p^a\mathbb{Z}$. In that local ring a unimodular column has a unit entry. Elementary row operations move a unit to the upper position and clear the lower one; diagonal matrices $\mathrm{diag}(u,u^{-1})$ are products of elementary matrices as well. Explicitly, with $E_{12},E_{21}$ denoting elementary shears,and multiplying this by the same matrix with $u=-1$ gives $\mathrm{diag}(u,u^{-1})$. Thus elementary matrices generate the special linear group over each local ring. Chinese remaindering their parameters gives generation over the product ring, and every such parameter lifts to an integer. This proves the reduction is onto.

Over ${\mathbb{F}}_p$, there are $p^2-1$ choices for a nonzero first column and $p$ choices for a second column with determinant one, so $|S{L}_2({\mathbb{F}}_p)|=p(p^2-1)$. Each reduction from modulus $p^{a+1}$ to $p^a$ has kernel represented by $I+p^{a}B$, with $B$ modulo $p$ and $\mathrm{tr}B=0$, of size $p^3$. Hence, for $N>1$,For $N=1$ the index is one. Since $-I\in \Gamma (N)$ exactly when $N=1,2$,

For $N\ge 2$ there are no nontrivial effective elliptic stabilizers. Indeed $A=I+NB\in \Gamma (N)$ with determinant one has $\mathrm{tr}B=-N\det B$, and thereforeA noncentral elliptic element of $S{L}_2(\mathbb{Z})$ has trace $0$ or $\pm 1$, none congruent to $2$ modulo $N^2$ for $N\ge 2$. The central element $-I$ at level two acts trivially and is not an elliptic point of the effective quotient. Thus ${\nu }_2={\nu }_3=0$ for $N\ge 2$, while at level one both numbers are one.

The subgroup $\Gamma (N)$ is normal, so all its modular cusps have the same effective width as infinity. That width is $N$: the shear $T^h$ belongs to $\pm \Gamma (N)$ exactly when $N\mid h$; for $N\ge 3$ its diagonal entries exclude the minus sign, and for $N=1,2$ that sign changes nothing. Since modular cusp widths sum to $\mu$,For $N\ge 3$ the resulting principal congruence modular curve genus isFor clarity, the small levels, including all exceptional terms, areFor $N>6$, every factor in the genus correction is positive, so $g>1$. Therefore the genus-zero levels are exactly $N=1,2,3,4,5$.