Let $N$ be a positive integer, and let $f$ be a newform for $S_2(\Gamma_0(N))$. Then by Shimura's construction, the variety $J_0(N)$ has a quotient $A_f$ which is an abelian variety attaced to $f$. $$\pi:J_0(N) \to A_f$$ Dualizing this map we get a map $A_f^\vee \to J_0(N)^\vee$, and since the latter is isomorphic to $J_0(N)$ (it is a Jacobian), we get a map $$A_f^\vee \to A_f$$ This map is an isogeny of degree $d^2$ for some positive integer $d$, and the modular degree is defined precisely as $d$. The curve $X_0(N)$ has a natural inclusion into its Jacobian, so it make sence to look at the map $$ X_0(N) \hookrightarrow J_0(N) \to A_f$$ Question: is any relation between the degree of the map $X_0(N) \to A_f$ (which I mean the degree of $X_0(N)$ onto its image) and $d$? I would guess that to prove an inequality should not be that hard, but is it an equality? (in the case $A_f$ has dimension $1$ it is)
2 Answers
Let $\tau : X_0(N) \to A_f$ be the name of your map. And let $C := \tau(X_0(N))$ then if $A_f \cong J(C)$ and furthermore the inclusion $C \subseteq A_f$ can be identified with the Abel-Jacobi map, then the relation you are asking for does hold. This just stems from a more general fact of maps between curves, namely if $g: X \to Y$ is a map of curves then the composition of pullback and pushforward $J(Y) \to J(X) \to J(Y)$ is just multiplication by $\deg g$. So there is a relation this case, namely $$\text{deg }( A_f^\vee \to A_f) = (\text{deg }X_0(N) \to C)^{2 genus(C)}.$$ So the same formula holds except that you have to include the genus of the image curve.
However in general I do not expect such a relation to hold. To give an explicit counter example, if $N=311$ then according to the LMFDB there are only two (galois orbits of) newforms. Namely $f_a := 311.2.a$ and $f_b := 311.2.b$. Now consider the map $\pi_b: X_0(N) \to A_{f_b}$. Since $A_{f_b}$ is a simple abelian variety of dimension $22$ any curve inside $A_{f_b}$ has to have genus $\geq 22$. In particular $C := \pi_b(X_0(N))$ has to have genus $\geq 22$. However $genus(X_0(311))=26$ so the riemann hurwitz formula forces deg$(\pi_b)=1$. However $$\text{deg }( A_{f_b}^\vee \to A_{f_b}) = \text{deg }( A_{f_a}^\vee \to A_{f_a}) = 2^{2genus(X_0(N)^+)}=2^{8}$$ So in this case there is no relation of the form you want. See section 3.4 of these lecture notes for the first equality.
Note that the above is not an isolated or pathological example. There are many more examples in the LFMDB where $N$ is prime and there are only 2 galois orbits of newforms in $S_2(\Gamma_0(N))$ and in all these cases a similar argument works to show there is no interesting relation between the degrees for the modular form whose Fricke sign is $-1$. Aside from the LMFDB examples this situation is also something that is conjectured to happen quite often (see https://mathoverflow.net/a/396522/23501).
The degree of the map is the modular degree. This is Lemma 5.2 of Component Groups of Purely Toric Quotients by Stein and Conrad. The idea is to examine the size of the kernel of the map.
