The Denominator of $L(A,1)/\Omega _A$

In this section, we prove a result about the denominator of the rational number $L(A,1)/\Omega _A$ and compare it to what is predicted by the Birch and Swinnerton-Dyer conjecture.

Proposition 4.6   Let $z$ be the point in $J_0(N)(\mathbf{Q})$ defined by the degree 0 divisor $(0)-(\infty)$ on $X_0(N)$, and let $n = n_f$ be the order of the image of $z$ in $A(\mathbf{Q})$. Then the denominator of $c_\infty\cdot c_A\cdot L(A,1)/\Omega_A$ divides $n$.

Proof. Let $x$ be the image of $z$ in $A(\mathbf{Q})$, and set $I=\Ann(x)\subset\mathbf{T}$. Since $f$ is a newform, the Hecke operators $T_p$, for $p\mid{}N$, act as 0 or $\pm 1$ on $A(\mathbf{Q})$ (see, e.g., [DI95, §6]). If $p\nmid N$, then a standard calculation (see, e.g., [Cre97, §2.8]) shows that $T_p(x) = (p+1)x$. There is an injection $\mathbf{T}/I\hookrightarrow C=\mathbf{Z}{}x$ sending $T_p$ to $T_p(x)$, where $C$ is the order-$n$ cyclic subgroup of $A(\mathbf{Q})$ generated by $x$. With notation as in the proof of Theorem 4.5, we have

$$\frac{L(A,1)}{\Omega_A}\cdot c_\infty\cdot c_A = [\Phi(H)^+:\Phi(\mathbf{T}{}e)]$$

$$= [\Phi(H)^+:\Phi(I e)]\cdot [\Phi(I e):\Phi(\mathbf{T}{}e)]$$

$$= [\Phi(H)^+:I\Phi(e)]\cdot [I\Phi(e):\mathbf{T}{}\Phi(e)]$$

$$= \frac{[\Phi(H)^+:I\Phi(e)]}{[\mathbf{T}{}\Phi(e):I\Phi(e)]} \in \frac{1}{n}\mathbf{Z}.$$

The final inclusion follows from two observations. First, the index $[\Phi(H)^+:I\Phi(e)]$ is an integer because $I$ is exactly the ideal of elements of $\mathbf{T}$ that send $x=\Phi(e)$ into $\Phi(H)$, and $x\in \Phi(H\otimes \mathbf{Q})^+$. Second, there is a surjective map

$$\mathbf{T}/I \rightarrow \frac{\mathbf{T}{}\Phi(e)}{I P(e)}$$

sending $t$ to $t \Phi(e)$, so $[\mathbf{T}{}\Phi(e):IP(e)]$ divides $n=\char93 C=\char93 (\mathbf{T}/I)$. $\qedsymbol$

Conjecture 2.2 predicts that

$$\char93 A(\mathbf{Q})\cdot \char93 A^{\vee}(\mathbf{Q}) \cdot \fr... ...\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)\cdot \prod c_p \in \mathbf{Z},$$

and since $n\mid \char93 A(\mathbf{Q})$, Proposition 4.6 implies that

$$c_\infty \cdot c_A \cdot \char93 A(\mathbf{Q}) \cdot \frac{L(A,1)}{\Omega_A} \in \mathbf{Z}.$$

Since $c_\infty$ is a power of $2$, and $c_A$ is conjecturally one (if $N$ is prime, then it is known to be a power of $2$), Proposition 4.6 provides theoretical evidence for Conjecture 2.2, and also reflects a surprising amount of cancellation between $\prod c_p$ and $\char93 A^{\vee}(\mathbf{Q})$.