Bounding the Torsion From Above

In this section we recall the standard upper bound on the order of $\char93 A(\mathbf{Q})_{\tor}$, and illustrate its usefullness.

Let  $f=\sum a_n q^n$ be a weight $2$ newform on $\Gamma_1(N)$ with Nebentypus character $\varepsilon:(\mathbf{Z}/N\mathbf{Z})^*\rightarrow \mathbf{C}^*$ (recall that $f$ is a form on $\Gamma_0(N)$ if and only if $\varepsilon=1$), and let $A=A_f$ be the corresponding optimal quotient of $J_1(N)$, as in [Shi73]. Shimura proved in [Shi94, Ch. 7] that the local Euler factor of $A_f$ at $p$ is

$$L_p(A_f,s) = \prod_{\sigma : K_f \hookrightarrow \overlin... ...hbf{Q}}} \frac{1} {1 - \sigma(a_p) p^{-s} + \sigma(\varepsilon(p)) p^{1-2s}}$$

by showing that the characteristic polynomial $F_p$ of Frobenius on any $\ell$-adic Tate module of $A_{\mathbf{F}_p}$ (for $\ell\nmid pN$) is

$$F_p(X) = \prod_{\sigma : K_f \hookrightarrow \overline{\mathbf{Q}}} X^2 - \sigma(a_p) X + \sigma(\varepsilon(p))p,$$

where $K_f=\mathbf{Q}(\ldots, a_n, \ldots)$. Let $\mathbf{Q}(\varepsilon)$ be the field generated by the values of  $\varepsilon$, and for any $p\nmid N$ let $G_p(X)\in\mathbf{Q}(\varepsilon)[X]$ be the characteristic polynomial of $a_p \in K_f$ over $\mathbf{Q}(\varepsilon)$, which is a polynomial of degree $d'=[K_f: \mathbf{Q}(\varepsilon)]$. Then

$$F_p(X) = \Norm_{\mathbf{Q}(\varepsilon)/\mathbf{Q}}\left( X^{d'} \cdot G_p\left(X + \frac{\varepsilon(p)p}{X}\right)\right),$$

so

$$\char93 A_{\mathbf{F}_p}(\mathbf{F}_p) = \deg(1-\Frob_p) = \vert\det(1-\Frob_p)\vert$$

$$= \vert F_p(1)\vert = \vert\Norm_{\mathbf{Q}(\varepsilon)/\mathbf{Q}}(G_p(1 + \varepsilon(p)p))\vert.$$

If $p\nmid N$ is odd, standard facts about formal groups imply that the reduction map $A(\mathbf{Q})_{\tor}\rightarrow A_{\mathbf{F}_p}(\mathbf{F}_p)$ is injective, so

$$\char93 A(\mathbf{Q})_{\tor} \mid \gcd \left\{ \char93 A_{\mathbf{F}_p}(\mathbf{F}_p) : \text{\rm primes $p\nmid 2N$}\right\}.$$

Likewise, since $A^{\vee}$ is isogenous to $A$, the same bound applies to $A^{\vee}(\mathbf{Q})_{\tor}$, since $A^{\vee}$ and $A$ have the same $L$-series.

The upper bound given above is the same for every abelian variety isogenous to $A$, so it is not surprising that it is not sharp in general. For example, let $E$ (resp., $F$) be the elliptic curve labeled 30A1 (resp. 30A2) in Cremona's tables [Cre97]. Then $E$ and $F$ are isogenous, $E(\mathbf{Q})\approx \mathbf{Z}/6\mathbf{Z}$, and $F(\mathbf{Q})\approx \mathbf{Z}/12\mathbf{Z}$, so

$$12 \mid \gcd \left\{ \char93 E_{\mathbf{F}_p}(\mathbf{F}_p) : \text{\rm primes $p\nmid 2N$}\right\}.$$

(Incidentally, since $\char93 E(\mathbf{F}_5) = 12$, the gcd is $12$.) For answers to some related deep questions about this gcd, see [Kat81].

Example 3.4   Let

$$f = q + (-1+\sqrt{2})q^2 + q^3 + (-2\sqrt{2} + 1)q^4 - 2\sqrt{2}q^5 +\cdots \in S_2(\Gamma_0(39))$$

be the form 39B. Then $G_5(X) = X^2 - 8$, so

$$\char93 A_f(\mathbf{Q})_{\tor} \mid G_5(1+5) = 28.$$

One can check that $A_f$ is isogenous (but not equal) to the Jacobian $J$ of $y^2 + (x^3 + 1)y = -5x^4 - 2x^3 + 16x^2 -12x +2$ and, by [FpS+01], $\char93 J(\mathbf{Q}) = 28$. The authors don't know whether or not $\char93 A_f(\mathbf{Q}) = 28$, but in Example 3.6 below we show that $14\mid \char93 A_f(\mathbf{Q})$.

Example 3.5   Let

$$f= q + (-\zeta_6 - 1)q^2 + (2\zeta_6 - 2)q^3 + \zeta_6q^4 + (-2\zeta_6 + 1)q^5 + \cdots$$

be one of the two Galois-conjugate newforms in $S_2(\Gamma_1(13))$. This form has character $\varepsilon:(\mathbf{Z}/13\mathbf{Z})^* \rightarrow \mathbf{C}^*$ of order $6$, and $A_f = J_1(13)$. We have $G_3(X) = X - 2\zeta_6 + 2$ and $\varepsilon(3) = -\zeta_6$, so

$$\char93 J_1(13)(\mathbf{Q})_{\tor} \mid \char93 J_1(13)(\mathbf{F}_3) = \vert\Norm(G_3(1-3\zeta_6))\vert$$

$$= \vert\Norm(-5\zeta_6 + 3)\vert = 19.$$

In fact Ogg proved that $J_1(13)(\mathbf{Q})_{\tor}\approx \mathbf{Z}/19\mathbf{Z}$ (see [Ogg73] and [MT74]).

Consider the abelian variety quotient tex2html_wrap_inline$A_f=A_20$ of tex2html_wrap_inline$J_0(389)$, as in Section . Each tex2html_wrap_inline$G_p(x)$ is a polynomial of degree tex2html_wrap_inline$20$. We find that tex2html_wrap_inline$G_3(3+1) = 2^2·97 ·2586967$ and tex2html_wrap_inline$G_5(5+1) = 5^2·97 ·183880478851$. Thus tex2html_wrap_inline$97$ is an upper bound on tex2html_wrap_inline$#A(Q)_$. We will see in Example 3.6 that in fact tex2html_wrap_inline$#A(Q)_=97$.