Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 23 / 3

The map $X(\Gamma )\to X(\Gamma (1))\cong {\mathbb{P}}^1$ has degree $d$, the effective projective index. Ramification occurs only over the two elliptic points and the modular cusp. Over the order-two elliptic point, $r_2$ points have index one and the other $(d-r_2)/2$ points have index two, contributing $(d-r_2)/2$ to the ramification divisor. The analogous order-three contribution is $2(d-r_3)/3$.

The local degree at a modular cusp is its effective cusp width. The cusp widths sum to $d$, so the modular cusp contribution is $d-r_{\infty }$. The Riemann-Hurwitz formula, $2g(X)-2=d(2g(Y)-2)+{\sum }_P(e_P-1)$, now yieldsTherefore the genus formula for a modular curve isUsing the index in ${\mathrm{SL}}_2$ without accounting for its center would give the wrong degree when $-I\notin \Gamma$.

Reduction modulo $p$ maps ${\mathrm{SL}}_2(\mathbb{Z})$ onto ${\mathrm{SL}}_2({\mathbb{F}}_p)$, of order $p(p^2-1)$. The image of ${\Gamma }_1(p)$ is the upper unipotent subgroup of order $p$. Thus its index in ${\mathrm{SL}}_2$ is $p^2-1$, and, since $-I\notin {\Gamma }_1(p)$ for $p\ge 5$, its effective index isEvery element of ${\Gamma }_1(p)$ has trace congruent to two. An effective elliptic element of order two or three has trace zero or $\pm 1$ in a lift, so neither is possible for $p\ge 5$. Hence $r_2=r_3=0$.

Represent a modular cusp by a primitive column $(a,c)$, modulo sign. Its reduction is a nonzero vector in ${\mathbb{F}}_p^2$ modulo sign, and the unipotent subgroup acts by $(a,c)\mapsto (a+bc,c)$. For $c=0$, the nonzero values of $a$ give $(p-1)/2$ orbits. For $c\ne 0$, $a$ varies freely and $c$ modulo sign gives another $(p-1)/2$ orbits. Thus there are $p-1$ modular cusps. The reduction classification is sufficient as well as necessary: completing two primitive columns to determinant-one matrices and adjusting the second columns by a translation makes congruent columns related by $\Gamma (p)$.

More explicitly, if a determinant-one matrix has first column $(a,c)$, conjugating $T^w$ givesIts least allowable cusp width in ${\Gamma }_1(p)$ is one for $c\equiv 0$ and $p$ otherwise. Both types therefore number $(p-1)/2$, with cusp widths one and $p$; their cusp width sum is $d$. There are no sign-twisted modular cusp periods here, because trace two cannot be congruent to minus two for these primes. Substitution givesThese are the prime Gamma 1 cusp counts and widths.

The effective group is torsion free and every modular cusp is regular in the preceding sense, so orders of a meromorphic weight-$k$ form are integers. At interior points use a local automorphy trivialization; at a modular cusp use the Fourier order of the appropriate slash transform in its cusp width coordinate. Let $C$ be the sum of all modular cusp points, each once, and setA meromorphic function $\phi$ belongs to the Riemann-Roch space $\mathcal{L}(D(f))$ exactly when $\mathrm{div}\phi +D(f)\ge 0$. In the interior this requires $f\phi$ to have no pole; at a modular cusp it requires order at least one. Conversely, the quotient of any weight-$k$ cusp form by $f$ is a meromorphic weight-zero function satisfying precisely those inequalities. This proves the cusp-form divisor presentationTo compute the degree without imposing a valence formula as an extra assumption, use the meromorphic tensor differential $f^{12}(d\tau {)}^{6k}$. Its automorphy factors cancel. Its order at an interior point is $12{\mathrm{ord}}_{P}f$; at a modular cusp it is $12{\mathrm{ord}}_{P}f-6k$, since $d\tau$ is a nonzero constant times $d{q}_c/q_c$. A meromorphic section of the $6k$th tensor power of the canonical bundle has total divisor degree $6k(2g-2)$. The regular-cusp valence formula on a torsion-free modular curve is thereforeFor $k\ge 3$, $\mathrm{deg}D(f)-(2g-2)=(k-2)d/12>0$. The Riemann-Roch theorem says $\ell (D)-\ell (K-D)=\mathrm{deg}D+1-g$, and a divisor of negative degree has no nonzero sections. Thus $\ell (K-D)=0$ andUsing $d=(p^2-1)/2$ and $r_{\infty }=p-1$ givesThe canonical-degree and Riemann-Roch facts used here are general results for compact Riemann surfaces, as permitted.