Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 23 / 4

For a pair $(L,t)$ choose an oriented basis $({\omega }_1,{\omega }_2)$ with $\mathrm{Im}({\omega }_1/{\omega }_2)>0$ and $t={\omega }_2/N$ modulo $L$. Such a basis exists because an exact-order point gives a primitive vector modulo $N$, which can be completed to a determinant-one basis. Define the marked-lattice model of a modular form byChanging to a basis with the same marked point uses a matrix with $c\equiv 0$, $d\equiv 1$ and therefore $a\equiv 1(\mathrm{mod}N)$, exactly ${\Gamma }_1(N)$. The weight-$k$ transformation of $f$ cancels the factor from ${\omega }_2^{-k}$, proving independence of the basis.

The resulting function has homogeneity $F_f(uL,ut)=u^{-k}{F}_f(L,t)$ for $u\in {\mathbb{C}}^{\times }$. Conversely evaluating at $(\mathbb{Z}\tau +\mathbb{Z},1/N)$ recovers $f$. Holomorphy in $\tau$ and holomorphy in the local parameters of degenerating lattices at every modular cusp characterize the functions arising from modular forms, rather than arbitrary homogeneous lattice functions.

Use the following normalizations for Hecke operators on marked lattices and the diamond operator:Multiplication by a unit $d$ preserves the order of the marked point. There are $p+1$ prime-index overlattices; when $p\nmid N$ they all preserve that order. When $p\mid N$, precisely the overlattice killing the order-$p$ subgroup generated by $(N/p)t$ is excluded, leaving $p$ terms. Thus the definition is meaningful at bad primes too.

Both operations preserve homogeneity, since scalar multiplication bijects the indexing lattices and multiplies every summand by the same $u^{-k}$. Changing the marked basis merely permutes the overlattices, giving the required ${\Gamma }_1(N)$ transformation. Locally each term is a modular form evaluated after a rational fractional-linear substitution, multiplied by its appropriate automorphy factor; this preserves holomorphy on the half-plane.

For modular cusp holomorphy, factor any such rational substitution at a rational modular cusp into an integral modular substitution followed by an upper triangular map $z\mapsto Az+B$ with $A>0$. An existing holomorphic modular cusp expansion stays bounded under this map. The finite sum has the positive period supplied by its new level, so boundedness makes its singularity removable in the new modular cusp parameter. This is cusp holomorphy under rational slash operators. It proves that the operators preserve the functions arising from $M_k({\Gamma }_1(N))$.

For clarity, the factor $1/p$ is paired with the homogeneity convention $F(uL)=u^{-k}F(L)$; it gives the Fourier normalization requested in part (c). The diamond action on modular forms equals $f{|}_k\gamma$ for a lift $\gamma \in {\Gamma }_0(N)$ with lower-right entry congruent to $d$, since the transformed marked point is $d/N$ modulo the original lattice.

For $L_{\tau }=\mathbb{Z}\tau +\mathbb{Z}$, the index-$p$ overlattices areThe first $p$ preserve the marked point $1/N$. At a good prime the last equals $p^{-1}{L}_{p\tau }$, with marked point corresponding after scaling to $p/N$. Consequently the lattice formula becomesThe first term has coefficient $a_{pn}$, because the sum of $p$th roots of unity is zero unless its exponent is divisible by $p$. The second has coefficient $\chi (p)p^{k-1}{a}_{n/p}$ when $p\mid n$ and zero otherwise. ThusThis includes $b_0=(1+\chi (p)p^{k-1})a_0$ at a good prime.

At $p\mid N$, the last overlattice loses the required exact order and is excluded. The operator is then $U_p$, with $b_n=a_{pn}$. The formula remains valid for all primes if the Dirichlet character is extended to integers by zero on nonunits, so $\chi (p)=0$ at bad primes. Without that extension, the printed $\chi (p)$ is defined only when $p\nmid N$. This is the good-prime and bad-prime Hecke coefficient formula.

Take a nonzero eigenform of positive weight, and put $h(\tau )=f(p\tau )$ and $C=\chi (p)p^{k-1}$. The original form belongs to level $Np$ by inclusion of groups. For $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\Gamma }_0(Np)$, the conjugate $\left(\begin{array}{cc}a& pb\\ c/p& d\end{array}\right)$ lies in ${\Gamma }_0(N)$, soIts Dirichlet character is therefore the specified reduced Dirichlet character; modular cusp holomorphy is permitted in the question. This is the Dirichlet character version of an oldform by argument dilation.

A nonzero positive-weight modular form cannot be constant, because the matrix $\left(\begin{array}{cc}1& 0\\ N& 1\end{array}\right)$ would force a nonzero constant to equal $(N\tau +1{)}^k$ times itself. Let $n_0>0$ be its first nonzero positive Fourier index. In a relation $Af+Bh=0$, the $q^{n_0}$ coefficient forces $A=0$, then $B=0$. Hence their span is two-dimensional, even when the original constant term is nonzero.

At the new level $p$ is a bad prime, so its operator is $U_p$. The good-prime eigenrelation at the old level and part (c) giveThus the matrix in the ordered basis $(f,h)$ iswith characteristic polynomial $X^2-\lambda X+C$. For its distinct roots,This is prime stabilization of an oldform.

The nonzero positive-weight qualification is necessary for the two-dimensional assertion. The zero form gives no such span, and if weight zero is allowed, $f=1$ at trivial Dirichlet character has distinct good-prime roots $1$ and $p^{-1}$, while $f(\tau )=f(p\tau )$ spans only one dimension. The asserted result uses the usual nonzero positive-weight eigenform setting.