The Fitting ideal of \(J_ 0(q)(\mathbb{F}_{p^ n})\) over the Hecke algebra (Q1322068)

Let \(q\) be a prime. Let \(X_ 0 (q)\) be the classical modular curve and let \(J = J_ 0 (q)\) be its Jacobian variety. Let \(p \neq q\) be another prime, and let \(J(\mathbb{F}_ p)\) denote the group of \(\mathbb{F}_ p\)-rational points on \(J/ \mathbb{F}_ p\). Let \(T\) be the Hecke algebra generated by the Hecke operators. It is known that \(T\) is a free \(\mathbb{Z}\)-module of finite rank. Let \(\varphi\) be the Frobenius endomorphism of \(J/ \mathbb{F}_ p\). It induces a map \(J(\overline \mathbb{F}_ p) @> \varphi >> J (\overline \mathbb{F}_ p)\). Then one has \(\text{det}_{T \otimes \mathbb{Q}_ \ell} (1 - \varphi) = 1 + p - T_ p \in T\), where \(\ell \neq p\) is a prime. Even though the group structure of \(J(\mathbb{F}_ p)\) and \(T/(1 + p - T_ p)T\) are not necessarily identical, the author proves that they have the same Fitting ideals over \(T\), which is a measure of the ``size'' of a module. Moreover, he shows that \(J(\mathbb{F}_{p^ d})\) is a \(T\)-submodule of \(J(\mathbb{F}_{p^ n})\), and one has the inclusion of ideals \(F_ T (J (\mathbb{F}_{p^ n})) \subseteq F_ T (J (\mathbb{F}_{p^ d}))\) when \(d\) divides \(n\). This is a generalization of the fact that given positive integers \(d\) and \(n\), \(\text{card} (J (\mathbb{F}_{p^ d}))\) divides \(\text{card} (J (\mathbb{F}_{p^ n}))\) when \(d\) divides \(n\).