Counting points over quadratic extensions of $\ensuremath{\mathbb{Z}}_p$

As we noted in Section 3, for an elliptic curve $C$ defined over $\ensuremath{\mathbb{Z}}_p$, we may have to consider points over an extension $F$ of $\ensuremath{\mathbb{Z}}_p$. In this section we want to show how we can find $\char93 C(\ensuremath{\mathbb{Z}}_{p^2})$, knowing $\char93 C(\ensuremath{\mathbb{Z}}_{p})$.

First we need some notation: let $N_m = \char93 C(\ensuremath{\mathbb{Z}}_{p^m})$. The zeta function of $C$ is defined by

$$\zeta_C(s) = \exp(\sum_{m=1}^\infty \frac{N_m}{m} p^{-m s}).$$

The following result is part of a theorem due to A. Weil.

Theorem 6   $\zeta_C(s)$ is a rational function with poles at $s=0$ and $s=1$. The zeroes of $\zeta_C$ are located on the line ${\mathrm{Re}}(s)=\frac{1}{2}$. Furthermore,

$$\zeta_C(s) = \frac{1-T p^{-s} + p^{1-2s}}{(1-p^{-s})(1-p^{1-s})}$$ (2)

for some $T\in\ensuremath{\mathbb{Z}}_p$.

Remark 7   The previous Theorem is part of a more general statement that has been proved by Weil not only for elliptic curves but also for curves of higher genus. This result was, in turn, extended to higher dimensional algebraic varieties by the work of a number of people, including Dwork, Artin, Grothendieck and Deligne.

Using the definition of $\zeta_C$, and taking logarithm on both sides of (2) we obtain

$$\sum_{m=1}^\infty \frac{N_m}{m} p^{-m s} = \log(1-T p^{-s} + p^{1-2s}) - \log(1-p^{-s}) - \log(1-p^{1-s})$$

Since $p^{-s}$ is small for ${\mathrm{Re}}(s)\gg 0$ we can expand the logarithms using the formula $\log(1-h) = -h - \frac{1}{2} h^2 - \cdots$ and order the result by order of vanishing at $s=\infty$ (that is, in powers of $p^{-s}$):

$$\begin{split}N_1 p^{-s} + \frac{N_2}{2} p^{-2s} + \cdots &= -... ... p^{-s} + \frac{1}{2}(2p-T^2+1+p^2) p^{-2s} +\cdots \end{split}$$

and we conclude that

$$\begin{split}N_1 &= -T+1+p\\ N_2 &= N_1(2(p+1)-N1). \end{split}$$

This last expression allows us to compute $N_2$ given $N_1$. For example, the curves $C_1$ and $C_3$ of Example 3 have $N_1(C_1) = 4$ and $N_1(C_3)=8$. Thus, applying the previous formula we get $N_2(C_1) = N_2(C_2) = 32$. Notice that we expected these numbers to be the same because both curves are isomorphic over $\ensuremath{\mathbb{Z}}_{p^2}$.

Remark 8   The process described above can be carried through to any order giving formulas for all the $N_m$ in terms of $N_1$. Hence, we conclude that two varieties having the same $N_1$ have the same zeta function.