## IV.4: Elliptic Curves


### II.4.1.  #to-work

Let $X$ be an elliptic curve over $k$, with char $k \neq 2$, let $P \in X$ be a point, and let $R$ be the graded ring $R=\bigoplus_{n \geqslant 0} H^0\left(X, \mathcal{O}_X(n P)\right)$. Show that for suitable choice of $t, x, y$
\[
R \cong k[t, x, y] /\left(y^2-x\left(x-t^2\right)\left(x-\lambda t^2\right)\right),
\]
as a graded ring, where $k[t, x, y]$ is graded by setting $\operatorname{deg} t=1, \operatorname{deg} x=2$, $\operatorname{deg} y=3$.

### II.4.2.  #to-work

If $D$ is any divisor of degree $\geqslant 3$ on the elliptic curve $X$, and if we embed $X$ in $\mathbf{P}^n$ by the complete linear system $|D|$, show that the image of $X$ in $\mathbf{P}^n$ is projectively normal.[^rmk.2.4.2]

[^rmk.2.4.2]: 
Note. It is true more generally that if $D$ is a divisor of degree $\geqslant 2 g+1$ on a curve of genus $g$, then the embedding of $X$ by $|D|$ is projectively normal (Mumford $[4$, p. 55$])$

### II.4.3.  #to-work

Let the elliptic curve $X$ be embedded in $\mathbf{P}^2$ so as to have the equation $y^2=$ $x(x-1)(x-\lambda)$. Show that any automorphism of $X$ leaving $P_0=(0,1,0)$ fixed is induced by an automorphism of $\mathbf{P}^2$ coming from the automorphism of the affine $(x, y)$-plane given by
\[
\left\{\begin{array}{l}
x^{\prime}=a x+b \\
y^{\prime}=c y .
\end{array}\right.
\]
In each of the four cases of (4.7), describe these automorphisms of $\mathbf{P}^2$ explicitly, and hence determine the structure of the group $G=\operatorname{Aut}\left(X, P_0\right)$.


### II.4.4.  #to-work

Let $X$ be an elliptic curve in $\mathbf{P}^2$ given by an equation of the form
\[
y^2+a_1 x y+a_3 y=x^3+a_2 x^2+a_4 x+a_6 \text {. }
\]
Show that the $j$-invariant is a rational function of the $a_i$, with coefficients in $\mathbf{Q}$. In particular, if the $a_i$ are all in some field $k_0 \subseteq k$, then $j \in k_0$ also. Furthermore, for every $\alpha \in k_0$, there exists an elliptic curve defined over $k_0$ with $j$-invariant equal to $\alpha$.

### II.4.5.  #to-work

Let $X, P_0$ be an elliptic curve having an endomorphism $f: X \rightarrow X$ of degree 2 .

a. If we represent $X$ as a 2-1 covering of $\mathbf{P}^1$ by a morphism $\pi: X \rightarrow \mathbf{P}^1$ ramified at $P_0$, then as in (4.4), show that there is another morphism $\pi^{\prime}: X \rightarrow \mathbf{P}^1$ and a morphism $g: \mathbf{P}^1 \rightarrow \mathbf{P}^1$, also of degree 2 , such that $\pi \circ f=g \circ \pi^{\prime}$.

b. For suitable choices of coordinates in the two copies of $\mathbf{P}^1$, show that $g$ can be taken to be the morphism $x \rightarrow x^2$.

c. Now show that $g$ is branched over two of the branch points of $\pi$, and that $g^{-1}$ of the other two branch points of $\pi$ consists of the four branch points of $\pi^{\prime}$. Deduce a relation involving the invariant $\lambda$ of $X$.

d. Solving the above, show that there are just three values of $j$ corresponding to elliptic curves with an endomorphism of degree 2 , and find the corresponding values of $\lambda$ and $j$.[^answer.4.4.5]

[^answer.4.4.5]: 
Answers: $j=2^6 \cdot 3^3 ; j=2^6 \cdot 5^3 ; j=-3^3 \cdot 5^3$.

### II.4.6.  #to-work


a. Let $X$ be a curve of genus $g$ embedded birationally in $\mathbf{P}^2$ as a curve of degree $d$ with $r$ nodes. Generalize the method of (Ex. 2.3) to show that $X$ has 
\[
6(g-1)+ 3d
\]
inflection points. A node does not count as an inflection point. Assume char $k=0$.

b. Now let $X$ be a curve of genus $g$ embedded as a curve of degree $d$ in $\mathbf{P}^n, n \geqslant 3$, not contained in any $\mathbf{P}^{n-1}$. For each point $P \in X$, there is a hyperplane $H$ containing $P$, such that $P$ counts at least $n$ times in the intersection $H \cap X$. This is called an **osculating** hyperplane at $P$. It generalizes the notion of tangent line for curves in $\mathbf{P}^2$. 

    If $P$ counts at least $n+1$ times in $H \cap X$, we say $H$ is a **hyperosculating hyperplane**, and that $P$ is a **hyperosculation point**. Use Hurwitz's theorem as above, and induction on $n$, to show that $X$ has 
  \[
  n(n+1)(g-1)+(n+1) d
  \]
  hyperosculation points.

c. If $X$ is an elliptic curve, for any $d \geqslant 3$, embed $X$ as a curve of degree $d$ in $\mathbf{P}^{d-1}$, and conclude that $X$ has exactly $d^2$ points of order $d$ in its group law.

### II.4.7. The Dual of a Morphism. #to-work

Let $X$ and $X^{\prime}$ be elliptic curves over $k$, with base points $P_0, P_0^{\prime}$.

a. If $f: X \rightarrow X^{\prime}$ is any morphism, use (4.11) to show that $f^*:$ Pic $X^{\prime} \rightarrow$ Pic $X$ induces a homomorphism $\hat{f}:\left(X^{\prime}, P_0^{\prime}\right) \rightarrow\left(X, P_0\right)$. We call this the **dual** of $f$.

b. If $f: X \rightarrow X^{\prime}$ and $g: X^{\prime} \rightarrow X^{\prime \prime}$ are two morphisms, then $(g \circ f)\hat{} = \hat{f} \circ \hat{g}$.

c. Assume $f\left(P_0\right)=P_0^{\prime}$, and let $n=\operatorname{deg} f$. Show that if $Q \in X$ is any point, and $f(Q)=Q^{\prime}$, then $\hat{f}\left(Q^{\prime}\right)=n_X(Q)$. (Do the separable and purely inseparable cases separately, then combine.) Conclude that $f \circ \hat{f}=n_{X^{\prime}}$ and $\hat{f} \circ f=n_X$.


d. \* If $f, g: X \rightarrow X^{\prime}$ are two morphisms preserving the base points $P_0, P_0^{\prime}$, then $(f+g) \hat{} = \hat{f}+\hat{g}$.[^hint.3.4.8.d]

[^hint.3.4.8.d]: 
Hints: It is enough to show for any $\mathcal{L} \in$ Pic $X^{\prime}$, that $(f+g)^* \mathcal{L} \cong f^* \mathcal{L} \otimes g^* \mathcal{L}$. For any $f$, let $\Gamma_f: X \rightarrow X \times X^{\prime}$ be the graph morphism. Then it is enough to show (for $\mathcal{L}^{\prime}=p_2^* \mathcal{L}$ ) that
\[
\Gamma_{f+g}^*\left(\mathcal{L}^{\prime}\right)=\Gamma_f^* \mathcal{L}^{\prime} \otimes \Gamma_g^* \mathcal{L}^{\prime} .
\]
Let $\sigma: X \rightarrow X \times X^{\prime}$ be the section $x \rightarrow\left(x, P_0^{\prime}\right)$. Define a subgroup of $\operatorname{Pic}\left(X \times X^{\prime}\right)$ as follows:
\[
\tsm{\Pic(X\times X)}
{
\mcl \text{ has degree 0 along each fiber of } p_1, \\
\sigma^* \mcl = 0 \in \Pic(X)
}
.\]
Note that this subgroup is isomorphic to the group $\operatorname{Pic}^{\circ}\left(X^{\prime} / X\right)$ used in the definition of the Jacobian variety. Hence there is a 1-1 correspondence between morphisms $f: X \rightarrow X^{\prime}$ and elements $\mathcal{L}_f \in$ Pic $_\sigma$ (this defines $\mathcal{L}_f$ ). Now compute explicitly to show that $\Gamma_g^*\left(\mathcal{L}_f\right)=\Gamma_f^*\left(\mathcal{L}_g\right)$ for any $f, g$.
\
\
Use the fact that $\mathcal{L}_{f+g}=\mathcal{L}_f \otimes \mathcal{L}_g$, and the fact that for any $\mathcal{L}$ on $X^{\prime}$, $p_2^* \mathcal{L} \in \mathrm{Pic}_\sigma^{\circ}$ to prove the result.


e. Using (d), show that for any $n \in \mathbf{Z}, \hat{n}_X=n_X$. Conclude that deg $n_X=n^2$.

f. Show for any $f$ that $\operatorname{deg} \hat{f}=\operatorname{deg} f$.


### II.4.8.  The Algebraic Fundamental Group. #to-work

For any curve $X$, the **algebraic fundamental group** $\pi_1(X)$ is defined as $\cocolim \operatorname{Gal}\left(K^{\prime} / K\right)$, where $K$ is the function field of $X$, and $K^{\prime}$ runs over all Galois extensions of $K$ such that the corresponding curve $X^{\prime}$ is étale over $X$ (III, Ex. 10.3). 

Thus, for example, $\pi_1\left(\mathbf{P}^1\right)=1$. (See 2.5.3)

Show that for an elliptic curve $X$, 
\[
\pi_1(X) =
\begin{cases}
\Prod_{\ell \text{ prime}} \ZZ_\ell \times \ZZ_\ell & \characteristic k = 0,
\\ \\
\Prod_{\ell\neq p} \ZZ_\ell \times \ZZ_\ell & \characteristic k = p \text{ and } \Hasse X = 0, 
\\ \\
\ZZ_p \times \Prod_{\ell\neq p} \ZZ_\ell \times \ZZ_\ell & \characteristic k = p \text{ and } \Hasse X \neq 0
\end{cases}
,\]
where $\mathbf{Z}_l=\lim \mathbf{Z} / l^n$ is the $l$-adic integers.[^hint.4.4.8][^rmk.4.4.8]

[^hint.4.4.8]: 
Hints: Any Galois étale cover $X^{\prime}$ of an elliptic curve is again an elliptic curve. If the degree of $X^{\prime}$ over $X$ is relatively prime to $p$, then $X^{\prime}$ can be dominated by the cover $n_X: X \rightarrow X$ for some integer $n$ with $(n, p)=1$. The Galois group of the covering $n_X$ is $\mathbf{Z} / n \times \mathbf{Z} / n$. Étale covers of degree divisible by $p$ can occur only if the Hasse invariant of $X$ is not zero.

[^rmk.4.4.8]: 
Note: More generally, Grothendieck has shown $[\mathrm{SGA} 1, X, 2.6, p. 272]$ that the algebraic fundamental group of any curve of genus $g$ is isomorphic to a quotient of the completion, with respect to subgroups of finite index, of the ordinary topological fundamental group of a compact Riemann surface of genus $g$, i.e., a group with $2 g$ generators $a_1, \ldots, a_g, b_1, \ldots, b_g$ and the relation $\left(a_1 b_1 a_1^{-1} b_1^{-1}\right) \cdots$ $\left(a_g b_g a_g^{-1} b_g^{-1}\right)=1$.


### II.4.9. Isogenies. #to-work

We say two elliptic curves $X, X^{\prime}$ are **isogenous** if there is a finite morphism $f: X \rightarrow X^{\prime}$.

a. Show that isogeny is an equivalence relation.

b. For any elliptic curve $X$, show that the set of elliptic curves $X^{\prime}$ isogenous to $X$, up to isomorphism, is countable.[^hint.3.4.9.b]

[^hint.3.4.9.b]: 
Hint: $X^{\prime}$ is uniquely determined by $X$ and $\operatorname{ker} f$.


### II.4.10.  #to-work

If $X$ is an elliptic curve, show that there is an exact sequence
\[
0 \rightarrow p_1^* \operatorname{Pic} X \oplus p_2^* \operatorname{Pic} X \rightarrow \operatorname{Pic}(X \times X) \rightarrow R \rightarrow 0,
\]
where $R=\operatorname{End}\left(X, P_0\right)$. In particular, we see that $\operatorname{Pic}(X \times X)$ is bigger than the sum of the Picard groups of the factors.[^4.4.10.ref]

[^4.4.10.ref]: 
Cf. (III, Ex. 12.6), (V, Ex. 1.6).

### II.4.11.  #to-work

Let $X$ be an elliptic curve over $\mathbf{C}$, defined by the elliptic functions with periods $1, \tau$. Let $R$ be the ring of endomorphisms of $X$.

a. If $f \in R$ is a nonzero endomorphism corresponding to complex multiplication by $\alpha$, as in (4.18), show that $\operatorname{deg} f=|\alpha|^2$.

b. If $f \in R$ corresponds to $\alpha \in \mathbf{C}$ again, show that the dual $\hat{f}$ of (Ex. 4.7) corresponds to the complex conjugate $\bar{\alpha}$ of $\alpha$.

c. If $\tau \in \mathbf{Q}(\sqrt{-d})$ happens to be integral over $\mathbf{Z}$, show that $R=\mathbf{Z}[\tau]$.

### II.4.12.  #to-work

Again let $X$ be an elliptic curve over $\mathbf{C}$ determined by the elliptic functions with periods $1, \tau$, and assume that $\tau$ lies in the region $G$ of (4.15B).

a. If $X$ has any automorphisms leaving $P_0$ fixed other than $\pm 1$, show that either $\tau=i$ or $\tau=\omega$, as in (4.20.1) and (4.20.2). This gives another proof of the fact (4.7) that there are only two curves, up to isomorphism, having automorphisms other than $\pm 1$.

b. Now show that there are exactly three values of $\tau$ for which $X$ admits an endomorphism of degree 2. Can you match these with the three values of $j$ determined in (Ex. 4.5)?[^rmk.4.12.b]

[^rmk.4.12.b]: 
Answers: $\tau=i ; \tau=\sqrt{-2} ; \tau=\frac{1}{2}(-1+\sqrt{-7})$.


### II.4.13.  #to-work

If $p=13$, there is just one value of $j$ for which the Hasse invariant of the corresponding curve is 0 . Find it.[^rmk.4.13]

[^rmk.4.13]: 
Answer: $j=5(\bmod 13)$.


### II.4.14.  #to-work

The Fermat curve $X: x^3+y^3=z^3$ gives a nonsingular curve in characteristic $p$ for every $p \neq 3$. Determine the set $\mathfrak{P}=\left\{p \neq 3 \mid X_{(p)}\right.$ has Hasse invariant 0$\}$, and observe (modulo Dirichlet's theorem) that it is a set of primes of density $\frac{1}{2}$.

### II.4.15.  #to-work

Let $X$ be an elliptic curve over a field $k$ of characteristic $p$. Let $F^{\prime}: X_p \rightarrow X$ be the $k$-linear Frobenius morphism (2.4.1). Use (4.10.7) to show that the dual morphism $\hat{F}^{\prime}: X \rightarrow X_p$ is separable if and only if the Hasse invariant of $X$ is 1 . 

Now use (Ex. 4.7) to show that if the Hasse invariant is 1, then the subgroup of points of order $p$ on $X$ is isomorphic to $\mathbf{Z} / p$; if the Hasse invariant is 0 , it is 0 .

### II.4.16.  #to-work

Again let $X$ be an elliptic curve over $k$ of characteristic $p$, and suppose $X$ is defined over the field $\mathbf{F}_q$ of $q=p^r$ elements, i.e., $X \subseteq \mathbf{P}^2$ can be defined by an equation with coefficients in $\mathbf{F}_q$. Assume also that $X$ has a rational point over $\mathbf{F}_q$. Let $F^{\prime}: X_q \rightarrow X$ be the $k$-linear Frobenius with respect to $q$.

a. Show that $X_q \cong X$ as schemes over $k$, and that under this identification, $F^{\prime}: X \rightarrow X$ is the map obtained by the $q$ th-power map on the coordinates of points of $X$, embedded in $\mathbf{P}^2$.

b. Show that $1_X-F^{\prime}$ is a separable morphism and its kernel is just the set $X\left(\mathbf{F}_q\right)$ of points of $X$ with coordinates in $\mathbf{F}_q$.

c. Using (Ex. 4.7), show that $F^{\prime}+\hat{F}^{\prime}=a_X$ for some integer $a$, and that $N=$ $q-a+1$, where $N=\size X\left(\mathbf{F}_q\right)$.

d. Use the fact that $\operatorname{deg}\left(m+n F^{\prime}\right)>0$ for all $m, n \in \mathbf{Z}$ to show that $|a| \leqslant 2 \sqrt{q}$. This is Hasse's proof of the analogue of the Riemann hypothesis for elliptic curves (App. C, Ex. 5.6).

e. Now assume $q=p$, and show that the Hasse invariant of $X$ is 0 if and only if $a \equiv 0(\bmod p)$. Conclude for $p \geqslant 5$ that $X$ has Hasse invariant 0 if and only if $N=p+1$.

### II.4.17.  #to-work

Let $X$ be the curve $y^2+y=x^3-x$ of $(4.23 .8)$.

a. If $Q=(a, b)$ is a point on the curve, compute the coordinates of the point $P+Q$, where $P=(0,0)$, as a function of $a, b$. Use this formula to find the coordinates of $n P, n=1,2, \ldots, 10$.[^check.4.4.17.a]

b. This equation defines a nonsingular curve over $\mathbf{F}_p$ for all $p \neq 37$.

[^check.4.4.17.a]: 
Check: $6P = (6,14)$

### II.4.18.  #to-work

Let $X$ be the curve $y^2=x^3-7 x+10$. This curve has at least 26 points with integer coordinates. Find them (use a calculator), and verify that they are all contained in the subgroup (maybe equal to all of $X(\mathbf{Q})$?) generated by $P=(1,2)$ and $Q=(2,2)$.

### II.4.19.  #to-work

Let $X, P_0$ be an elliptic curve defined over $\mathbf{Q}$, represented as a curve in $\mathbf{P}^2$ defined by an equation with integer coefficients. Then $X$ can be considered as the fibre over the generic point of a scheme $\bar{X}$ over Spec $\mathbf{Z}$. Let $T \subseteq \operatorname{Spec} \mathbf{Z}$ be the open subset consisting of all primes $p \neq 2$ such that the fibre $X_{(p)}$ of $\bar{X}$ over $p$ is nonsingular. 

- For any $n$, show that $n_X: X \rightarrow X$ is defined over $T$, and is a flat morphism. 
- Show that the kernel of $n_X$ is also flat over $T$. 
- Conclude that for any $p \in T$, the natural map $X(\mathbf{Q}) \rightarrow X_{(p)}\left(\mathbf{F}_p\right)$ induced on the groups of rational points, maps the $n$-torsion points of $X(\mathbf{Q})$ injectively into the torsion subgroup of $X_{(p)}\left(\mathbf{F}_p\right)$, for any $(n, p)=1$.

By this method one can show easily that the groups $X(\mathbf{Q})$ in (Ex. 4.17) and (Ex. 4.18) are torsion-free.

### II.4.20.  #to-work

Let $X$ be an elliptic curve over a field $k$ of characteristic $p>0$, and let $R=$ $\operatorname{End}\left(X, P_0\right)$ be its ring of endomorphisms.

a. Let $X_p$ be the curve over $k$ defined by changing the $k$-structure of $X$ (2.4.1). Show that $j\left(X_p\right)=j(X)^{1 / p}$. Thus $X \cong X_p$ over $k$ if and only if $j \in \mathbf{F}_p$.

b. Show that $p_X$ in $R$ factors into a product $\pi \hat{\pi}$ of two elements of degree $p$ if and only if $X \cong X_p$. In this case, the Hasse invariant of $X$ is 0 if and only if $\pi$ and $\hat{\pi}$ are associates in $R$ (i.e., differ by a unit). (Use (2.5).)

c. If $\operatorname{Hasse}(X)=0$ show in any case $j \in \mathbf{F}_{p^2}$.

d. For any $f \in R$, there is an induced map $f^*: H^1\left(\mathcal{O}_X\right) \rightarrow H^1\left(\mathcal{O}_X\right)$. This must be multiplication by an element $\lambda_f \in k$. So we obtain a ring homomorphism $\varphi: R \rightarrow k$ by sending $f$ to $\lambda_f$. Show that any $f \in R$ commutes with the (nonlinear) Frobenius morphism $F: X \rightarrow X$, and conclude that if Hasse $(X) \neq 0$, then the image of $\varphi$ is $\mathbf{F}_p$. Therefore, $R$ contains a prime ideal $\mathfrak{p}$ with $R / p \cong \FF_p$.

### II.4.21.  #to-work

Let $O$ be the ring of integers in a quadratic number field $\mathbf{Q}(\sqrt{-d})$. Show that any subring $R \subseteq O, R \neq \mathbf{Z}$, is of the form $R=\mathbf{Z}+f \cdot O$, for a uniquely determined integer $f \geqslant 1$. This integer $f$ is called the **conductor** of the ring $R$.

### II.4.22 \*.  #to-work

If $X \rightarrow \mathbf{A}_{\mathbf{C}}^1$ is a family of elliptic curves having a section, show that the family is trivial.[^rmk.4.4.22.hints]

[^rmk.4.4.22.hints]: 
Hints: Use the section to fix the group structure on the fibres. Show that the points of order 2 on the fibres form an étale cover of $\mathbf{A}_{\mathbf{C}}^1$, which must be trivial, since $\mathbf{A}_{\mathbf{C}}^1$ is simply connected. This implies that $\lambda$ can be defined on the family, so it gives a map $\mathbf{A}_{\mathbf{C}}^1 \rightarrow \mathbf{A}_{\mathbf{C}}^1-\{0,1\}$. Any such map is constant, so $\lambda$ is constant, so the family is trivial.