Why does the Hecke correspondence preserve principal divisors?

Question

Let $p$ be prime not dividing $N$. Consider the Hecke correspondence $T_p$ inducing a set valued function on $X_0(N)$. I'd like to understand why it acts on $\text{Pic}^0$, and so I'd like to know why $T_p$ preverves principal divisors.

Answer 1

The action of $T_p$ can be described roughly as follows. There are two natural maps $\pi_1, \pi_2$ from $X_0(Np)$ to $X_0(N)$. If $x \in X_0(N)$, then $\pi_1^{-1}(x)$ is a finite set of points on $X_0(Np)$, and $T_p$ sends $x$ to the set $\{ \pi_2(y): y \in \pi_1^{-1}(x) \}$. So $T_p$ is formally $\pi_2 \circ \pi_1^{-1}$.

However, this isn't quite right (because the maps $\pi_1$ and $\pi_2$ can be ramified, so we need to keep track of multiplicities); it's more accurate to work directly with divisors, and say that $T_p$ acts as $(\pi_2)_* \circ (\pi_1)^*$, where the upper and lower stars denote pullback and pushforward on divisors (which are defined in such a way that they keep track of multiplicity automatically).

Now, suppose we have a principal divisor $\Delta = div(f)$ on $X_0(N)$, with $f \in \operatorname{Rat}(X_0(N))$ (my notation for the field of rational functions on $X_0(N)$). Then $\pi_1^{-1}(\Delta)$ -- or, more correctly, the divisor $\pi_1^*(\Delta)$ -- is a principal divisor, because it's the divisor of the rational function $\pi_1^*(f)$ on $X_0(Np)$. On the other hand, there is a norm map $$ (\pi_2)_* : \operatorname{Rat}(X_0(Np))^\times \to \operatorname{Rat}(X_0(N))^\times $$ associated to $\pi_2$ -- it's just the norm map of the finite field extension $\operatorname{Rat}(X_0(Np))$ over $\pi_2^*\left(\operatorname{Rat}(X_0(N))\right)$. It's easy to see that if $\Delta = div(f)$ is a principal divisor on $X_0(Np)$, then $(\pi_2)_* \Delta$ is the divisor of the function $(\pi_2)_*(f)$. So both $(\pi_1)^*$ and $(\pi_2)_*$ preserve principal divisors, and hence their composite does too.

(There's nothing special about $X_0(N)$ and $X_0(Np)$ here -- the argument shows that any algebraic correspondence preserves principal divisors.)