Question

I was hoping that someone could help clarify a source of confusion for me, I must be doing and saying something wrong but I just don't know what: Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $$L(s,E)=\sum_{n=1}^{\infty}a_n(E)n^{-s}$$ be the Hasse-Weil $L$-function of $E$. Finally, let $\tilde{E}$ be the reduction of $E$ mod $p$ and assume that $p$ is a prime for which $E$ has good reduction.

Then $$a_p(E)=p+1-|\tilde{E}(\mathbb{F}(p))|$$ and setting $a_1(E)=1$ the $p$ power coefficients are given by $$a_{p^e}(E)=a_p(E)a_{p^{e-1}}(E)-pa_{p^{e-2}}(E).$$

Now looking at Diamond and Shurman, for instance, I find that also we can write $$a_{p^e}(E)=p^e+1-|\tilde{E}(\mathbb{F}(p^e))|$$ but when I use this expression as a "definition" of $a_{p^e}(E)$ and do some explicit calculations I don't get the right recursion, for instance I seem to get in practice $$a_{p^2}(E)=a_p(E)^2 - 2p$$ instead of $$a_{p^2}(E)=a_p(E)^2-p.$$

I must be misunderstanding something, but I can't figure out what. Any help?

Answer 1

There are two different recursions involved here, one for the points of $E$ over ${\mathbb F}_{p^n}$, and the other for the coefficients of the $L$-function.

If we write $a_p = \alpha + \beta,$ where $\alpha\beta = p$ (so $\alpha$ and $\beta$ are the two roots of the char. poly. of Frobenius), then

$$1 + p^n - E({\mathbb F}_{p^n}) = \alpha^n + \beta^n.$$

On the other hand, the Euler factor at $p$ for the $L$-function of $E$ is $$(1 - \alpha p^{-s})^{-1}(1-\beta p^{-s})^{-1}$$ $$= (1 + \alpha p^{-s} + \alpha^2 p^{-2s} + \cdots )(1 + \beta p^{-s} + \beta^2 p^{-2s} + \cdots )$$ $$= 1 + (\alpha + \beta) p^{-s} + (\alpha^2 + \alpha\beta + \beta^2) p^{-2s} + \cdots ,$$ and so we conclude that $a_{p^n}$ (the coefficient of $p^{-ns}$ in the $L$-function) equals $$\alpha^n + \alpha^{n-1} \beta + \cdots + \alpha\beta^{n-1} + \beta^n.$$

These formulas are simply different, as soon as $n > 1.$ The recursion given in the question describes the second, and not the first.

Answer 2

The mistake is acknowledged and corrected in the errata for the third printing:

http://people.reed.edu/~jerry/MF/mferrata3.pdf