The Stacks project

64.24 The Legendre family

Let $k$ be a finite field of odd characteristic, $X = \mathop{\mathrm{Spec}}(k[\lambda , \frac{1}{\lambda (\lambda - 1)}])$, and consider the family of elliptic curves $f : E \to X$ on $\mathbf{P}^2_ X$ whose affine equation is $y^2 = x(x - 1)(x - \lambda )$. We set $\mathcal{F} = Rf_*^1\mathbf{Q}_\ell = \left\{ R^1f_*\mathbf{Z}/\ell ^ n\mathbf{Z}\right\} _{n\geq 1} \otimes \mathbf{Q}_\ell $. In this situation, the following is true

• for each $n \geq 1$, the sheaf $R^1f_*(\mathbf{Z}/\ell ^ n\mathbf{Z})$ is finite locally constant – in fact, it is free of rank 2 over $\mathbf{Z}/\ell ^ n\mathbf{Z}$,

• the system $\{ R^1f_*\mathbf{Z}/\ell ^ n\mathbf{Z}\} _{n\geq 1}$ is a lisse $\ell $-adic sheaf, and

• for all $x\in |X|$, $\det (1 - \pi _ x\ T^{\deg x} |_{\mathcal{F}_{\bar x}}) = (1 - \alpha _ x T^{\deg x})(1 - \beta _ x T^{\deg x})$ where $\alpha _ x, \beta _ x$ are the eigenvalues of the geometric frobenius of $E_ x$ acting on $H^1(E_{\bar x}, \mathbf{Q}_\ell )$.

Note that $E_ x$ is only defined over $\kappa (x)$ and not over $k$. The proof of these facts uses the proper base change theorem and the local acyclicity of smooth morphisms. For details, see [SGA4.5]. It follows that

Applying Theorem 64.20.2 we get

and we see in particular that this is a rational function. Furthermore, it is relatively easy to show that $H_ c^0(X_{\bar k}, \mathcal{F}) = H_ c^2(X_{\bar k}, \mathcal{F}) = 0$, so we merely have

To compute this determinant explicitly, consider the Leray spectral sequence for the proper morphism $f : E \to X$ over $\mathbf{Q}_\ell $, namely

which degenerates. We have $f_*\mathbf{Q}_\ell = \mathbf{Q}_\ell $ and $R^1f_*\mathbf{Q}_\ell = \mathcal{F}$. The sheaf $R^2f_*\mathbf{Q}_\ell = \mathbf{Q}_\ell (-1)$ is the Tate twist of $\mathbf{Q}_\ell $, i.e., it is the sheaf $\mathbf{Q}_\ell $ where the Galois action is given by multiplication by $\# \kappa (x)$ on the stalk at $\bar x$. It follows that, for all $n\geq 1$,

where the first equality follows from Theorem 64.20.5, the second one from the Leray spectral sequence and the third one by writing down the higher direct images of $\mathbf{Q}_\ell $ under $f$. Alternatively, we could write

and use the trace formula for each curve. We can also find the number of $k_ n$-rational points simply by counting. The zero section contributes $q^ n -2$ points (we omit the points where $\lambda = 0, 1$) hence

Now we have

where $\varepsilon _ n = 1$ if $-1$ is a square in $k_ n$, 0 otherwise, i.e.,

Thus $ \# E(k_ n) = q^{2n} - q^ n - 2+ 2\varepsilon _ n$. Comparing with the previous formula, we find

which implies, by elementary algebra of complex numbers, that if $-1$ is a square in $k_ n^*$, then $\dim H_ c^1(X_{\bar k}, \mathcal{F}) = 2$ and the eigenvalues are $1$ and $1$. Therefore, in that case we have