Quotients of arbitrary dimension

For $N\geq 4$ , let $\Gamma$ a subgroup of  $\Gamma_1(N)$ that contains  $\Gamma_0(N)$ , let $X$ be the modular curve over ${\bf {Q}}$ associated to $\Gamma$ , and let $J$ be the Jacobian of $X$ . Let $I$ be a saturated ideal of the corresponding Hecke algebra ${\bf {T}}$ , so ${\bf {T}}/ I$ is torsion free. Then $A = A_I = J/IJ$ is an optimal quotient of $J$ .

For a newform $f = \sum a_n(f) q^n \in S_2(\Gamma)$ , consider the ring homomorphism ${\bf {T}}\to {\bf {Z}}[\ldots, a_n(f), \ldots]$ that sends $T_n$ to $a_n(f)$ . The kernel $I_f\subset {\bf {T}}$ of this homomorphism is a saturated prime ideal of ${\bf {T}}$ . The newform quotient $A_f$ associated to $f$ is the quotient $J/I_f J$ . Shimura introduced $A_f$ in [Shi73] where he proved that $A_f$ is an abelian variety over ${\bf {Q}}$ of dimension equal to the degree of the field ${\bf {Q}}(\ldots,a_n(f),\ldots)$ . He also observed that there is a natural map ${\bf {T}}\to {\rm End}(A_f)$ with kernel $I_f$ .

For the rest of this section, fix a quotient $A$ associated to a saturated ideal $I$ of ${\bf {T}}$ ; note that $A$ may or may not be attached to a newform. The modular degree tex2html_wrap_inline$m_A$ of an optimal quotient tex2html_wrap_inline$A$ of tex2html_wrap_inline$J$ is the (positive) square root of the degree of the induced composite tex2html_wrap_inline$A^&or#vee; &rarr#to;J^&or#vee;&cong#cong;J &rarr#to;A$.