[proofplan] We compare two descriptions of the same correspondence on the analytic moduli space $Y_0(N)(\mathbb{C})$. First we write an explicit set of representatives for the double coset and identify their action on $\tau \in \mathfrak{H}$. Then we classify the $p+1$ cyclic subgroups of $E_\tau[p]$ and show that the quotients by these subgroups are represented by exactly those same transformed points. The condition $p \nmid N$ ensures that the level-$N$ subgroup survives quotienting by every order-$p$ subgroup. [/proofplan]

[step:Decompose the double coset into $p+1$ analytic branches]Let \begin{align*} \alpha_a &:= \begin{pmatrix}1&a\\0&p\end{pmatrix} \quad \text{for } a \in \{0,\dots,p-1\}, & \alpha_\infty &:= \begin{pmatrix}p&0\\0&1\end{pmatrix}. \end{align*} Since $p \nmid N$, the standard double-coset decomposition is \begin{align*} \Gamma_0(N) \begin{pmatrix}1&0\\0&p\end{pmatrix} \Gamma_0(N) = \coprod_{a=0}^{p-1}\Gamma_0(N)\alpha_a \;\coprod\; \Gamma_0(N)\alpha_\infty. \end{align*} Thus the associated analytic correspondence sends the point $[\tau] \in \Gamma_0(N)\backslash \mathfrak{H}$ to the unordered multiset \begin{align*} \left\{ \left[\frac{\tau+a}{p}\right] : a=0,\dots,p-1 \right\} \cup \{[p\tau]\}. \end{align*}[/step]

[guided]We need an explicit double-coset decomposition because a correspondence on the quotient $\Gamma_0(N)\backslash \mathfrak{H}$ is computed by choosing right coset representatives. For the matrix \begin{align*} \begin{pmatrix}1&0\\0&p\end{pmatrix}, \end{align*} one may choose the representatives \begin{align*} \alpha_a = \begin{pmatrix}1&a\\0&p\end{pmatrix} \quad (a=0,\dots,p-1), \qquad \alpha_\infty = \begin{pmatrix}p&0\\0&1\end{pmatrix}. \end{align*} The hypothesis $p \nmid N$ is exactly the condition ensuring that these representatives are valid for $\Gamma_0(N)$ without introducing extra congruence classes in the lower-left entry. The fractional linear action of a matrix \begin{align*} \begin{pmatrix}r&s\\t&u\end{pmatrix} \end{align*} on $\mathfrak{H}$ is \begin{align*} \tau \longmapsto \frac{r\tau+s}{t\tau+u}. \end{align*} Therefore \begin{align*} \alpha_a \tau = \frac{\tau+a}{p}, \qquad \alpha_\infty \tau = p\tau. \end{align*} Hence the double-coset correspondence sends $[\tau]$ to the unordered multiset \begin{align*} \left\{ \left[\frac{\tau+a}{p}\right] : a=0,\dots,p-1 \right\} \cup \{[p\tau]\}. \end{align*}[/guided]

[step:Classify the cyclic order-$p$ subgroups of $E_\tau$]Let \begin{align*} \Lambda_\tau := \mathbb{Z}\tau+\mathbb{Z} \end{align*} and \begin{align*} E_\tau := \mathbb{C}/\Lambda_\tau. \end{align*} The $p$-torsion group is \begin{align*} E_\tau[p] = \frac{\frac{1}{p}\Lambda_\tau}{\Lambda_\tau} = \left\{ \frac{m\tau+n}{p}+\Lambda_\tau : m,n \in \mathbb{Z} \right\}. \end{align*} Its cyclic subgroups of order $p$ are precisely \begin{align*} H_a &:= \left\langle \frac{\tau+a}{p}+\Lambda_\tau \right\rangle \quad \text{for } a=0,\dots,p-1, \\ H_\infty &:= \left\langle \frac{1}{p}+\Lambda_\tau \right\rangle. \end{align*} Indeed, $E_\tau[p]$ is a two-dimensional [vector space](/page/Vector%20Space) over $\mathbb{F}_p$, and its cyclic order-$p$ subgroups are its one-dimensional $\mathbb{F}_p$-subspaces. The displayed list gives the $p$ finite slopes and the vertical slope, hence all $p+1$ such subspaces.[/step]

[guided]The quotient isogenies of degree $p$ out of $E_\tau$ are indexed by cyclic subgroups of $E_\tau[p]$ of order $p$. Since \begin{align*} E_\tau = \mathbb{C}/\Lambda_\tau, \qquad \Lambda_\tau = \mathbb{Z}\tau+\mathbb{Z}, \end{align*} a point $z+\Lambda_\tau$ is killed by multiplication by $p$ exactly when $pz \in \Lambda_\tau$. Therefore \begin{align*} E_\tau[p] = \frac{\frac{1}{p}\Lambda_\tau}{\Lambda_\tau} = \left\{ \frac{m\tau+n}{p}+\Lambda_\tau : m,n \in \mathbb{Z} \right\}. \end{align*} The two generators $\tau/p+\Lambda_\tau$ and $1/p+\Lambda_\tau$ identify $E_\tau[p]$ with $\mathbb{F}_p^2$. The one-dimensional subspaces of $\mathbb{F}_p^2$ are the $p$ finite-slope lines and the vertical line. Translating that description back to $E_\tau[p]$, these are \begin{align*} H_a = \left\langle \frac{\tau+a}{p}+\Lambda_\tau \right\rangle \quad \text{for } a=0,\dots,p-1, \end{align*} and \begin{align*} H_\infty = \left\langle \frac{1}{p}+\Lambda_\tau \right\rangle. \end{align*} This accounts for exactly $p+1$ cyclic order-$p$ subgroups.[/guided]

[step:Identify each quotient elliptic curve with the corresponding transformed lattice]For $a \in \{0,\dots,p-1\}$, define the lattice \begin{align*} \Lambda_a := \mathbb{Z}\frac{\tau+a}{p}+\mathbb{Z}. \end{align*} Since $\Lambda_\tau \subset \Lambda_a$ and $\Lambda_a/\Lambda_\tau = H_a$, the quotient map \begin{align*} q_a: E_\tau \to \mathbb{C}/\Lambda_a, \qquad z+\Lambda_\tau \mapsto z+\Lambda_a \end{align*} has kernel $H_a$. Hence \begin{align*} E_\tau/H_a \cong \mathbb{C}/\Lambda_a = E_{(\tau+a)/p}. \end{align*} Similarly, define \begin{align*} \Lambda_\infty := \mathbb{Z}\tau+\mathbb{Z}\frac{1}{p}. \end{align*} Then $\Lambda_\tau \subset \Lambda_\infty$ and $\Lambda_\infty/\Lambda_\tau = H_\infty$, so \begin{align*} E_\tau/H_\infty \cong \mathbb{C}/\Lambda_\infty. \end{align*} Multiplication by $p$ gives an isomorphism \begin{align*} \mathbb{C}/\Lambda_\infty \longrightarrow \mathbb{C}/(p\Lambda_\infty) = \mathbb{C}/(\mathbb{Z}p\tau+\mathbb{Z}) = E_{p\tau}. \end{align*} Therefore the quotient curves obtained from the $p+1$ cyclic subgroups are represented by \begin{align*} \frac{\tau+a}{p} \quad (a=0,\dots,p-1), \qquad p\tau. \end{align*}[/step]

[guided]We now compare the subgroup classification with the analytic representatives from the double coset. For $a=0,\dots,p-1$, set \begin{align*} \Lambda_a := \mathbb{Z}\frac{\tau+a}{p}+\mathbb{Z}. \end{align*} This is a lattice containing $\Lambda_\tau$, because $\tau = p\cdot \frac{\tau+a}{p}-a$ and $1 \in \Lambda_a$. The [quotient group](/page/Quotient%20Group) $\Lambda_a/\Lambda_\tau$ is generated by the class of $(\tau+a)/p$, so \begin{align*} \Lambda_a/\Lambda_\tau = H_a. \end{align*} The natural quotient map \begin{align*} q_a: E_\tau \to \mathbb{C}/\Lambda_a, \qquad z+\Lambda_\tau \mapsto z+\Lambda_a \end{align*} is a holomorphic group homomorphism whose kernel is exactly $\Lambda_a/\Lambda_\tau = H_a$. Hence \begin{align*} E_\tau/H_a \cong \mathbb{C}/\Lambda_a = E_{(\tau+a)/p}. \end{align*} For the remaining subgroup, define \begin{align*} \Lambda_\infty := \mathbb{Z}\tau+\mathbb{Z}\frac{1}{p}. \end{align*} Again $\Lambda_\tau \subset \Lambda_\infty$, and the quotient $\Lambda_\infty/\Lambda_\tau$ is generated by $1/p+\Lambda_\tau$, so it equals $H_\infty$. Thus \begin{align*} E_\tau/H_\infty \cong \mathbb{C}/\Lambda_\infty. \end{align*} This lattice is not written in the standard form $\mathbb{Z}\tau'+\mathbb{Z}$, so we rescale it. Multiplication by $p$ is a complex-linear isomorphism \begin{align*} \mathbb{C}/\Lambda_\infty \longrightarrow \mathbb{C}/(p\Lambda_\infty). \end{align*} Since \begin{align*} p\Lambda_\infty = \mathbb{Z}p\tau+\mathbb{Z}, \end{align*} we get \begin{align*} E_\tau/H_\infty \cong E_{p\tau}. \end{align*} Thus the quotient isogenies give exactly the analytic points \begin{align*} \left[\frac{\tau+a}{p}\right] \quad (a=0,\dots,p-1), \qquad [p\tau]. \end{align*}[/guided]

[step:Check that the level-$N$ subgroup is transported correctly]Let \begin{align*} C_\tau := \left\langle \frac{1}{N}+\Lambda_\tau \right\rangle \subset E_\tau \end{align*} be the cyclic subgroup of order $N$. For every cyclic subgroup $H \subset E_\tau[p]$ of order $p$, we have \begin{align*} C_\tau \cap H = \{0\}, \end{align*} because every element of $C_\tau$ has order dividing $N$, every element of $H$ has order dividing $p$, and $\gcd(N,p)=1$. Therefore the image \begin{align*} (C_\tau+H)/H \subset E_\tau/H \end{align*} is cyclic of order $N$. Under the lattice identifications above, this is exactly the level subgroup attached to the transformed point of $Y_0(N)(\mathbb{C})$.[/step]

[guided]The modular curve $Y_0(N)$ does not parametrize only elliptic curves; it parametrizes pairs $(E,C)$ where $C \subset E$ is cyclic of order $N$. Thus, after quotienting by an order-$p$ subgroup $H$, we must verify that the level structure remains a cyclic subgroup of order $N$. Here \begin{align*} C_\tau = \left\langle \frac{1}{N}+\Lambda_\tau \right\rangle \end{align*} has order $N$. If $x \in C_\tau \cap H$, then the order of $x$ divides $N$ because $x \in C_\tau$, and also divides $p$ because $x \in H$. Since $p \nmid N$, we have $\gcd(N,p)=1$, so the only possible order of $x$ is $1$. Hence \begin{align*} C_\tau \cap H = \{0\}. \end{align*} The quotient map $E_\tau \to E_\tau/H$ is therefore injective on $C_\tau$, and its image \begin{align*} (C_\tau+H)/H \end{align*} is again cyclic of order $N$. Under the identifications \begin{align*} E_\tau/H_a \cong E_{(\tau+a)/p}, \qquad E_\tau/H_\infty \cong E_{p\tau}, \end{align*} this transported subgroup is precisely the cyclic subgroup represented by the corresponding point of $Y_0(N)(\mathbb{C})$.[/guided]

[step:Compare the two correspondences on the open modular curve] The moduli definition of $T_p$ sends \begin{align*} (E_\tau,C_\tau) \end{align*} to the unordered multiset \begin{align*} \left\{ (E_\tau/H_a,(C_\tau+H_a)/H_a) : a=0,\dots,p-1 \right\} \cup \left\{ (E_\tau/H_\infty,(C_\tau+H_\infty)/H_\infty) \right\}. \end{align*} By the preceding identifications, this multiset is represented analytically by \begin{align*} \left\{ \left[\frac{\tau+a}{p}\right] : a=0,\dots,p-1 \right\} \cup \{[p\tau]\}, \end{align*} which is exactly the multiset produced by the double-coset correspondence. Hence the Hecke correspondence $T_p$ agrees on $Y_0(N)(\mathbb{C})$ with the correspondence attached to \begin{align*} \Gamma_0(N) \begin{pmatrix}1&0\\0&p\end{pmatrix} \Gamma_0(N). \end{align*} Since $Y_0(N)(\mathbb{C})$ is the open analytic curve inside $X_0(N)(\mathbb{C})$, this proves the asserted agreement on the open curve. [/step]