4_4x

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 \(t, x, y\) \begin{align*} R \cong k[t, x, y] /\left(y^2-x\left(x-t^2\right)\left(x-\lambda t^2\right)\right), \end{align*} 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. ¹

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 \begin{align*} \left\{\begin{array}{l} x^{\prime}=a x+b \\ y^{\prime}=c y . \end{array}\right. \end{align*} 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 \begin{align*} y^2+a_1 x y+a_3 y=x^3+a_2 x^2+a_4 x+a_6 \text {. } \end{align*} 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 .

• 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 \(\pi \circ f=g \circ \pi^{\prime}\).

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

• 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 \(X\).

• 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 \(j\). ²

II.4.6. #to_work

• 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 \begin{align*} 6(g-1)+ 3d \end{align*} inflection points. A node does not count as an inflection point. Assume char \(k=0\).

• 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 \begin{align*} n(n+1)(g-1)+(n+1) d \end{align*} hyperosculation points.

• 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 \(P_0, P_0^{\prime}\).

• 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 \(\widehat{f}:\left(X^{\prime}, P_0^{\prime}\right) \rightarrow\left(X, P_0\right)\). We call this the dual of \(f\).

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

• 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 \(\widehat{f}\left(Q^{\prime}\right)=n_X(Q)\). (Do the separable and purely inseparable cases separately, then combine.) Conclude that \(f \circ \widehat{f}=n_{X^{\prime}}\) and \(\widehat{f} \circ f=n_X\).

• * If \(f, g: X \rightarrow X^{\prime}\) are two morphisms preserving the base points \(P_0, P_0^{\prime}\), then \((f+g) \widehat{} = \widehat{f}+\widehat{g}\). ³

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

• Show for any \(f\) that \(\operatorname{deg} \widehat{f}=\operatorname{deg} f\).

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 \(X\), \begin{align*} \pi_1(X) = \begin{cases} \displaystyle\prod_{\ell \text{ prime}} {\mathbf{Z}}_\ell \times {\mathbf{Z}}_\ell & \operatorname{ch}k = 0, \\ \\ \displaystyle\prod_{\ell\neq p} {\mathbf{Z}}_\ell \times {\mathbf{Z}}_\ell & \operatorname{ch}k = p \text{ and } {\mathrm{Hasse}}X = 0, \\ \\ {\mathbf{Z}}_p \times \displaystyle\prod_{\ell\neq p} {\mathbf{Z}}_\ell \times {\mathbf{Z}}_\ell & \operatorname{ch}k = p \text{ and } {\mathrm{Hasse}}X \neq 0 \end{cases} ,\end{align*} where \(\mathbf{Z}_l=\lim \mathbf{Z} / l^n\) is the \(l\)-adic integers. ^{4 5}

II.4.9. Isogenies. #to_work

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

• Show that isogeny is an equivalence relation.

• For any elliptic curve \(X\), show that the set of elliptic curves \(X^{\prime}\) isogenous to \(X\), up to isomorphism, is countable. ⁶

II.4.10. #to_work

If \(X\) is an elliptic curve, show that there is an exact sequence \begin{align*} 0 \rightarrow p_1^* \operatorname{Pic} X \oplus p_2^* \operatorname{Pic} X \rightarrow \operatorname{Pic}(X \times X) \rightarrow R \rightarrow 0, \end{align*} 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. ⁷

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 \(X\).

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

• If \(f \in R\) corresponds to \(\alpha \in \mathbf{C}\) again, show that the dual \(\widehat{f}\) of (Ex. 4.7) corresponds to the complex conjugate \(\mkern 1.5mu\overline{\mkern-1.5mu\alpha\mkern-1.5mu}\mkern 1.5mu\) of \(\alpha\).

• 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).

• 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 \(\pm 1\).

• 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)? ⁸

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. ⁹

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 \(\mathfrak{P}=\left\{p \neq 3 \mathrel{\Big|}X_{(p)}\right.\) 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 \(\widehat{F}^{\prime}: X \rightarrow X_p\) 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 \(q\).

• 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 \(\mathbf{P}^2\).

• 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 \(\mathbf{F}_q\).

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

• 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).

• 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 \((4.23 .8)\).

• 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 \(n P, n=1,2, \ldots, 10\). ¹⁰

• This equation defines a nonsingular curve over \(\mathbf{F}_p\) for all \(p \neq 37\).

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 \(\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu\) 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 \(\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu\) 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.

• 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 \(j \in \mathbf{F}_p\).

• Show that \(p_X\) in \(R\) factors into a product \(\pi \widehat{\pi}\) 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 \(\widehat{\pi}\) are associates in \(R\) (i.e., differ by a unit). (Use (2.5).)

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

• 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 \(R / p \cong { \mathbf{F} }_p\).

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. ¹¹