The Discriminant Valuation

Let $\Gamma$ be a congruence subgroup of $\SL_2(\mathbb{Z})$, e.g., $\Gamma=\Gamma_0(p)$ or $\Gamma_1(p)$. For any integer $k\geq 1$, let $S_k(\Gamma)$ denote the space of holomorphic weight-$k$ cusp forms for $\Gamma$. Let

$$\mathbb{T}= \mathbb{Z}[\ldots,T_n,\ldots] \subset \End(S_k(\Gamma))$$

be the associated Hecke algebra. Then  $\mathbb{T}$ is a commutative ring that is free and of finite rank as a $\mathbb{Z}$-module. Also of interest is the image $\mathbb{T}^{\new}$ of  $\mathbb{T}$ in $\End(S_k(\Gamma)^{\new})$.

Example 2.3   Let $\Gamma=\Gamma_0(243)$, which is illustrated on my T-shirt. Since $243=3^5$, experts will immediately deduce that $\disc(\mathbb{T}) = 0$. A computation shows that

$$\disc(\mathbb{T}^{\new}) = 2^{13} \cdot 3^{40},$$

which reflects the mod-$2$ and mod-$3$ intersections all over my shirt.

Definition 2.4 (Discriminant Valuation)   Let $p$ be a prime and suppose that $\Gamma=\Gamma_0(p)$ or $\Gamma_1(p)$. The discriminant valuation is

$$d_k(\Gamma) = \ord_p($$the discriminant of $ $$$).$$