 Solution

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.