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The Arithmetic of Elliptic Curves

Joseph H. Silverman

Chapter 3

The Geometry of Elliptic Curves - all with Video Answers

Educators

Chapter Questions

Problem 1

Show that the polynomials

$$
x^4-b_4 x^2-2 b_6 x-b_8 \text { and } 4 x^3+b_2 x^2+2 b_4 x+b_6
$$

appearing in the duplication formula (2.3d) are relatively prime if and only if the discriminant $\Delta$ of the corresponding Weierstrass equation is non-zero.

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Problem 2

(a) Find a triplication formula, analogous to the duplication formula given in (2.3). (I.e. Express $x([3] P)$ as a rational function of $x(P)$ and $a_1, \ldots, a_6$.)
(b) Use the result from (a) to show that if $\operatorname{char}(K) \neq 3$, then $E$ has a non-trivial point of order 3 . Conclude that if $\operatorname{gcd}(m, 3)=1$, then $[m] \neq[0]$. (Warning: This exercise probably requires a computer with a symbolic processor.)

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Problem 3

Assume $\operatorname{char}(K) \neq 3$ and $A \in K^*$. Then the curve

$$
E: X^3+Y^3=A Z^3
$$

has genus 1 (exer. 2.7), so together with the point $O=[1,-1,0]$ it becomes an elliptic curve.
(a) Show that three points of $E$ add to $O$ if and only if they are collinear.
(b) If $P=[X, Y, Z] \in E$, show that

$$
-P=[Y, X, Z]
$$

and

$$
[2] P=\left[-Y\left(X^3+A Z^3\right), X\left(Y^3+A Z^3\right), X^3 Z-Y^3 Z\right]
$$

(c) Develop an analogous formula for the sum of two distinct points.
(d) Prove that $E$ has $j$-invariant 0 .

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00:30

Problem 4

Referring to example (2.4), express each of the points $P_2, P_4, P_5, P_6, P_7, P_8$ in the form $[m] P_1+[n] P_3$ with $m, n \in \mathbb{Z}$.

Brandon Fox
Brandon Fox
Numerade Educator

Problem 5

Let $E / K$ be given by a singular Weierstrass equation.
(a) Suppose that $E$ has a node, and let the tangent lines at the node be $y=\alpha_i x+\beta_i, i=1,2$.
(i) If $\alpha_1 \in K$, prove that $\alpha_2 \in K$ and

$$
E_{n s}(K) \cong K^* .
$$

(ii) If $\alpha_1 \notin K$, prove that $L=K\left(\alpha_1, \alpha_2\right)$ is a quadratic extension of $K$. From (i), $E_{n s}(K) \subset E_{n s}(L) \cong L^*$. Show that

$$
E_{n s}(K) \cong\left\{t \in L^*: N_{L / K}(t)=1\right\}
$$

(b) Suppose that $E$ has a cusp. Prove that

$$
E_{n s}(K) \cong K^{+}
$$

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Problem 6

Let $C$ be a smooth curve of genus $g, P_0 \in C$, and $n \geqslant 2 g+1$ an integer. Let $\left\{f_0, f_1, \ldots, f_m\right\}$ be a basis for $\mathscr{L}\left(n\left(P_0\right)\right)$ and

$$
\phi=\left[f_0, \ldots, f_m\right]: C \rightarrow \mathbb{P}^m
$$

the map determined by the $f_i$ 's.
(a) Prove that the image $C^{\prime}=\phi(C)$ is a curve in $\mathrm{P}^m$.
(b) Prove that the map $\phi: C \rightarrow C^{\prime}$ has degree 1.
(c)* Prove that $C^{\prime}$ is smooth, and so that $\phi: C \rightarrow C^{\prime}$ is an isomorphism.

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Problem 7

This exercise gives an elementary (highly computational) proof that the multiplication-by- $m$ map has degree $m^2$. We will assume $\operatorname{char}(K) \neq 2,3$, and take an elliptic curve

$$
E: y^2=x^3+A x+B .
$$

Define division polynomials $\psi_m \in \mathbb{Z}[A, B, x, y]$ inductively as follows:

$$
\begin{aligned}
\psi_1 & =1, \quad \psi_2=2 y \\
\psi_3 & =3 x^4+6 A x^2+12 B x-A^2 \\
\psi_4 & =4 y\left(x^6+5 A x^4+20 B x^3-5 A^2 x^2-4 A B x-8 B^2-A^3\right) \\
\psi_{2 m+1} & =\psi_{m+2} \psi_m^3-\psi_{m-1} \psi_{m+1}^3 \quad(m \geqslant 2) \\
2 y \psi_{2 m} & =\psi_m\left(\psi_{m+2} \psi_{m-1}^2-\psi_{m-2} \psi_{m+1}^2\right) \quad(m \geqslant 2) .
\end{aligned}
$$

(One easily checks that the $\psi_{2 m}$ 's are polynomials.) Further define polynomials $\phi_m$ and $\omega_m$ by

$$
\begin{aligned}
\phi_m & =x \psi_m^2-\psi_{m+1} \psi_{m-1} \\
4 y \omega_m & =\psi_{m+2} \psi_{m-1}^2-\psi_{m-2} \psi_{m+1}^2
\end{aligned}
$$

(a) Prove that $\psi_m, \phi_m, y^{-1} \omega_m$ (for $m$ odd) and $(2 y)^{-1} \psi_m, \phi_m, \omega_m$ (for $m$ even) are polynomials in $\mathbb{Z}\left[A, B, x, y^2\right]$. Hence replacing $y^2$ by $x^3+A x+B$, we will treat them as polynomials in $\mathbb{Z}[A, B, x]$.
(b) As polynomials in $x$, show that

$$
\begin{gathered}
\phi_m(x)=x^{m^2}+\text { lower order terms } \\
\psi_m(x)^2=m^2 x^{m^2-1}+\text { lower order terms }
\end{gathered}
$$

(c) If $\Delta=-16\left(4 A^3+27 B^2\right) \neq 0$, then $\phi_m(x)$ and $\psi_m(x)^2$ are relatively prime polynomials (in $K[x]$.)
(d) Again assume $\Delta \neq 0$, so $E$ is an elliptic curve. Let $P=\left(x_0, y_0\right) \in E$. Then

$$
[m] P=\left(\frac{\phi_m(P)}{\psi_m(P)^2}, \frac{\omega_m(P)}{\psi_m(P)^3}\right)
$$

(e) The map $[m]: E \rightarrow E$ has degree $m^2$.

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Problem 8

(a) Let $E / \mathrm{C}$ be an elliptic curve. We will later show (VI.5.1.1) that there is a lattice $L \subset \mathbb{C}$ and a complex analytic isomorphism of groups $\mathbb{C} / L \cong E(\mathbb{C})$. (N.B. This isomorphism is given by convergent power series, not by rational functions.) Assuming this, prove that

$$
\operatorname{deg}[m]=m^2 \quad \text { and } \quad E[m] \cong \mathbb{Z} / m \mathbb{Z} \times \mathbb{Z} / m \mathbb{Z}
$$

(b) Let $E / K$ be an elliptic curve with $\operatorname{char}(K)=0$. Using (a), prove that $\operatorname{deg}[m]=m^2$. [Hint: If $K$ can be embedded in $\mathbb{C}$, there is no problem. Reduce to this case.]

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Problem 9

Let $E / K$ be an elliptic curve given by a homogeneous Weierstrass equation $F\left(X_0, X_1, X_2\right)=0$. (Le. $x=X_0 / X_2$ and $y=X_1 / X_2$ are Weierstrass coordinate functions.) Let $P \in E$.
(a) Show that $[3] P=O$ if and only if the tangent line to $E$ at $P$ intersects $E$ only at $P$.
(b) Show that $[3] P=O$ if and only if the Hessian matrix

$$
\left(\left(\partial^2 F / \partial X_i \partial X_j\right)(P)\right)_{0 \leqslant l, j \leqslant 2}
$$

has determinant 0 .
(c) If $\operatorname{char}(K) \neq 3$, show that $E[3]$ consists of 9 points.

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Problem 10

Let $E / K$ be an elliptic curve with Weierstrass coordinate functions $x, y$.
(a) Show that the map

$$
\begin{gathered}
\phi: E \rightarrow \mathbb{P}^3 \\
\phi=\left[1, x, y, x^2\right]
\end{gathered}
$$

maps $E$ isomorphically onto the intersection of two quadric surfaces in $\mathrm{P}^3$. In particular, if $H \subset \mathrm{P}^3$ is a hyperplane, then $H \cap \phi(E)$ consists of 4 points (counted with appropriate multiplicity.)
(b) Show that $\phi(O)=[0,0,0,1]$, and the hyperplane $\left\{T_0=0\right\}$ intersects $\phi(E)$ at the single point $\phi(O)$ with multiplicity 4 .
(c) Let $P, Q, R \in E$. Prove $P+Q+R=O$ if and only if $\phi(P), \phi(Q), \phi(R), \phi(O)$ are coplanar.
(d) Let $P \in E$. Prove that $[4] P=O$ if and only if there exists a hyperplane $H \subset \mathbb{P}^3$ such that $H \cap \phi(E)=\{P\}$. Show that if char $K \neq 2$, then there are exactly 16 such points.
(e) Assume $\operatorname{char}(K) \neq 2$. Show that after a linear change of variables (over $\bar{K}$ ), $E$ has a model of the form

$$
\begin{aligned}
T_0^2+T_2^2 & =T_0 T_3 \\
T_1^2+\alpha T_2^2 & =T_2 T_3 .
\end{aligned}
$$

For what value(s) of $\alpha$ is this model non-singular?
(f) Using the model in (e) and the addition law described by (c), derive formulas for $-P, P_1+P_2$, and $[2] P$ analogous to those given in (2.3).

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Problem 11

Generalize exercise 3.10 as follows. Let $E / K$ be an elliptic curve, and choose a basis $f_1, \ldots, f_m$ for $\mathscr{L}(m(O))$. Then for $m \geqslant 3$, the map

$$
\begin{gathered}
\phi: E \rightarrow \mathbb{P}^{m-1} \\
\phi=\left[f_1, \ldots, f_m\right]
\end{gathered}
$$

maps $E$ isomorphically onto its image (exer. 3.6).
(a) Show that $\phi(E)$ is a curve of degree $m$. (I.e. The intersection of $\phi(E)$ and a hyperplane, counted with multiplicities, consists of $m$ points.) [Hint: Find a hyperplane which intersects $\phi(E)$ at the single point $\phi(O)$, and show that it intersects with multiplicity $m$.]
(b) Let $P_1, \ldots, P_{m-1} \in E$. Prove that $P_1+\cdots+P_{m-1}=O$ if and only if $\phi\left(P_1\right)$, $\ldots, \phi\left(P_{m-1}\right), \phi(O)$ lie in a hyperplane. (Note that if some of the $P_i$ 's coincide, then we require the hyperplane to intersect $\phi(E)$ with correspondingly higher multiplicity.)
(c)* Let $P \in E$. Show that $[m] P=O$ if and only if there is a hyperplane $H \subset \mathbb{P}^{m-1}$ such that $H \cap \phi(E)=\{P\}$. If $\operatorname{char}(K)=0$ or $\operatorname{char}(K)>m$, prove that there are exactly $m^2$ such points. Deduce that $\operatorname{deg}[m]=m^2$.

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Problem 12

Let $m \geqslant 2$ be an integer, prime to $\operatorname{char}(K)$ if $\operatorname{char}(K)>0$. Prove that the natural map

$$
\operatorname{Aut}(E) \rightarrow \operatorname{Aut}(E[m])
$$

is injective except for $m=2$, when the kernel is $\pm 1$. (Do not use (10.1).)

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Problem 13

Generalize (4.12) as follows. Let $C / \bar{K}$ be a smooth curve, and let Isom( $C$ ) denote the group of isomorphisms from $C$ to itself. (E.g. If $C$ is an elliptic curve, then Isom( $C$ ) contains translation maps and $[ \pm 1]$.) Let $\Phi$ be a finite subgroup of Isom $(C)$.
(a) Prove that there exists a unique smooth curve $C^{\prime} / \bar{K}$ and a finite separable morphism $\phi: C \rightarrow C^{\prime}$ such that $\phi^* \bar{K}\left(C^{\prime}\right)=\bar{K}(C)^{\oplus}$. (Here $\bar{K}(C)^{\oplus}$ denotes the subfield of $\bar{K}(C)$ fixed by $\Phi$, where an element $\alpha \in \Phi$ acts on $\bar{K}(C)$ by $\alpha^*: \bar{K}(C) \rightarrow \bar{K}(C)$.)
(b) Let $P \in C$. Prove that

$$
e_P(\phi)=\#\{\alpha \in \Phi: \alpha P=P\} .
$$

(c) Prove that $\phi$ is unramified if and only if every non-trivial element of $\Phi$ has no fixed points.
(d) Express the genus of $C^{\prime}$ in terms of the genus of $C$, $\# \Phi$, and the fixed points of the elements of $\Phi$.
(e)* Suppose that $C$ is defined over $K$, and that $\Phi$ is $G_{\bar{K} / K}$-invariant. (I.e. If $\alpha \in \Phi$, then $\alpha^\sigma \in \Phi$ for all $\sigma \in G_{\overline{\boldsymbol{K}} / \boldsymbol{K}^*}$.) Prove that it is possible to find a $C^{\prime}$ so that $C^{\prime}$ and $\phi$ are defined over $K$. Further, show that $C^{\prime}$ is then unique up to isomorphism over $K$.

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Problem 14

Use the non-degeneracy of the Weil pairing to give a quick proof that the map

$$
\operatorname{Hom}\left(E_1, E_2\right) \rightarrow \operatorname{Hom}\left(T_C\left(E_1\right), T_\lambda\left(E_2\right)\right)
$$

is injective. (Note this is not as strong as (7.4).)

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Problem 15

Let $\phi: E_1 \rightarrow E_2$ be an isogeny of degree $m$, with $m$ prime to $\operatorname{char}(K)$ if $\operatorname{char}(K)>0$.
(a) Mimic the construction in section 8 to construct a pairing

$$
e_\phi: \operatorname{ker} \phi \times \operatorname{ker} \hat{\phi} \rightarrow \boldsymbol{\mu}_m .
$$

(b) Prove that $e_\phi$ is bilinear, non-degenerate, and Galois invariant.
(c) Prove that $e_\phi$ is compatible, in the sense that if $\psi: E_2 \rightarrow E_3$ is another isogeny, $P \in \operatorname{ker}(\psi \circ \phi)$, and $Q \in \operatorname{ker}(\hat{\phi})$, then

$$
e_{\psi \circ \phi}(P, Q)=e_\psi(\phi P, Q) .
$$

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Problem 16

Alternative Definition of the Weil Pairing. Let $E$ be an elliptic curve. We define a pairing

$$
\tilde{e}_m: E[m] \times E[m] \rightarrow \mu_m
$$

as follows: Let $P, Q \in E[m]$, and choose divisors $D_P, D_Q$ in $\operatorname{Div}^0(E)$ which add to $P$ and $Q$ respectively. (I.e. $\sigma\left(D_P\right)=P$ and $\sigma\left(D_Q\right)=Q$, where $\sigma$ is as in (3.4a).) We further assume that $D_P$ and $D_Q$ are chosen with disjoint supports. Since $P$ and $Q$ have order $m$, there are functions $f_p, f_Q \in \bar{K}(E)$ such that

$$
\operatorname{div}\left(f_P\right)=m D_P \quad \text { and } \quad \operatorname{div}\left(f_Q\right)=m D_Q .
$$

Then we define

$$
\tilde{e}_m(P, Q)=f_P\left(D_Q\right) / f_Q\left(D_P\right)
$$

(See exer. 2.10 for the definition of the value of a function at a divisor.)
(a) Prove that $\tilde{e}_m(P, Q)$ is well-defined.
(b) Prove that $\tilde{e}_m(P, Q) \in \mu_m$.
(c)* Prove that $\tilde{e}_m=e_m$, where $e_m$ is the Weil pairing defined in section 8. [Hint: Use Weil reciprocity, exer. 2.11.]

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Problem 17

Let $\mathscr{K}$ be a quaternion algebra. Show that $\mathscr{K}$ is ramified at $\infty$. [Hint: $M_2(\mathbb{R})$ contains zero-divisors.]

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Problem 18

Let $E / K$ be an elliptic curve, and assume that $\mathscr{X}=\operatorname{End}(E) \otimes \mathbb{Q}$ is a quaternion algebra.
(a) Prove that if $p \neq \infty$ and $p \neq \operatorname{char}(K)$, then $\mathscr{K}$ splits at $p$. [Hint: Use (7.4).]
(b) Prove that $\operatorname{char}(K)>0$. [Hint: Use exer. 3.17 and (9.5a).]
(c) Prove that $\mathscr{K}$ is the unique quaternion algebra ramified at precisely $\infty$ and $\operatorname{char}(K)$.
(d)* Prove that $\operatorname{End}(E)$ is the maximal order in $\mathscr{K}$. (I.e. The integral closure of $\mathbb{Z}$ in $\mathscr{K}$.)

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Problem 19

Let $\mathscr{H}$ be a quaternion algebra.
(a) Show that $\mathscr{K} \otimes \overline{\mathbb{Q}} \cong M_2(\overline{\mathbb{Q}})$.
(b) Show that $\mathscr{K} \otimes \mathscr{K} \cong M_4(\mathbb{Q})$. (This proves that $\mathscr{K}$ has order 2 in $\operatorname{Br}(\mathbb{Q})$.) [Hint: First show that $\mathscr{K} \otimes \mathscr{K}$ is simple (i.e. has no two-sided ideals.) Then prove that the map

$$
\mathscr{K} \otimes \mathscr{K} \rightarrow \operatorname{End}(\mathscr{K}), \quad a \otimes b \rightarrow(x \rightarrow a \times \hat{b})
$$

is an isomorphism.]

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Problem 20

Let $\mathscr{K}$ be a quadratic imaginary field with ring of integers $\mathcal{O}$. Show that the orders of $\mathscr{K}$ are precisely the rings $\mathbb{Z}+f \mathcal{O}$ for integers $f>0$. The integer $f$ is called the conductor of the order.

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Problem 21

Let $C / \bar{K}$ be a curve of genus 1 . For any point $O \in C$, we can associate to the elliptic curve ( $C, O$ ) its $j$-invariant $j(C, O)$. This exercise sketches a proof that the value $j(C, O)$ is independent of the choice of the basepoint $O$. Thus we can assign a $j$-invariant $j(C)$ to any curve $C$ of genus 1 . (We assume that $\operatorname{char}(K) \neq 2$. The result is still true for $\operatorname{char}(K)=2$, but the method of proof must be modified and the ensuing algebra is more complicated.)
(a) Choose a Legendre equation

$$
y^2=x(x-1)(x-\lambda)
$$

for the elliptic curve $(C, O)$. Show that the map $x: C \rightarrow \mathrm{P}^1$ has degree 2 and is ramified exactly over the points $\{0,1, \lambda, \infty\}$.
(b) Let $O^{\prime} \in C$ be another point, and choose a Legendre equation

$$
w^2=z(z-1)(z-\mu)
$$
for ( $C, O^{\prime}$ ). Let $\tau: C \rightarrow C$ be the translation-by- $O^{\prime}$ map on the elliptic curve $(C, O)$. Show that there are constants $a \in \bar{K}$ and $b \in \bar{K}^*$ such that $\tau^*(z)= a+b x$. [Hint: Look at the divisor of $\tau^*(z)$.]
(c) Let $f: \mathrm{P}^1 \rightarrow \mathrm{P}^1$ be the map $f(t)=a+b t$. Prove the $f$ maps the set $\{0,1, \lambda\}$ bijectively to the set $\{0,1, \mu\}$. [Hint: Compare the ramification of the maps $z \circ \tau$ and $f \circ x$.]
(d) Show that

$$
\mu \in\{\lambda, 1 / \lambda, 1-\lambda, 1 /(1-\lambda), \lambda /(1-\lambda),(\lambda-1) / \lambda\} .
$$

[Hint: Consider the six ways of matching $\{0,1, \lambda\}$ with $\{0,1, \mu\}$.]
(e) Deduce that $j(C, O)=j\left(C, O^{\prime}\right)$. [Hint: Show that the formula for $j\left(E_\lambda\right)$ in (1.7b) does not change if $\lambda$ is replaced by any of the six expressions given in (d).]

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Problem 22

Let $C$ be a curve of genus 1 defined over $K$.
(a) Prove that $j(C) \in K$.
(b) Prove that $C$ is an elliptic curve over $K$ if and only if $C(K) \neq \varnothing$.
(c) Prove that $C$ is always isomorphic (over $\bar{K}$ ) to an elliptic curve defined over $K$.

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Problem 23

Deuring Normal Form. The following normal form for a Weierstrass equation is sometimes useful when dealing with elliptic curves over (algebraically closed) fields of arbitrary characteristic.
(a) Let $E / K$ be an elliptic curve, and assume that either $\operatorname{char}(K) \neq 3$ or $j(E) \neq 0$. Prove that $E$ has a Weierstrass equation over $\bar{K}$ of the form

$$
E: y^2+\alpha x y+y=x^3, \quad \alpha \in \bar{K}
$$

(b) For the Weierstrass equation given in (a), show that $(0,0) \in E[3]$.
(c) For what value(s) of $\alpha$ is the equation singular?
(d) Verify that

$$
j(E)=\alpha^3\left(\alpha^3-24\right)^3 /\left(\alpha^3-27\right) .
$$

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Problem 24

Let $E / K$ be an elliptic curve with complex multiplication over $K$ (i.e. $\operatorname{End}_K(E)$ is strictly larger that $\mathbb{Z}$.) Prove that for all primes $\ell \neq \operatorname{char}(K)$, the action of $G_{\bar{K} / K}$ on the Tate module $T_{\ell}(E)$ is abelian. [Hint: Use the fact that the non-trivial endomorphisms in $\operatorname{End}_{\boldsymbol{K}}(E)$ commute with the action of $G_{\overline{\boldsymbol{K}} / \boldsymbol{K}}$.]

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