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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)

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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.

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