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Chapter 4: Problem 5

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 \(\pi \quad f=g \quad \pi^{\prime}\) (b) For suitable choices of coordinates in the two copies of \(\mathbf{P}^{1}\), show that \(y\) 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 \(\pi\). Deduce a relation involving the invariant , 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 \(j.\)

Short Answer

Expert verified
The latter steps involve solving the specified equation system for distinct values of \(j\) and subsequently finding the corresponding \(\lambda\) and \(j\) for the elliptic curves, which satisfy an endomorphism of degree \(2\). As a summary, we deduce the key relationship, create the new morphism as per the said format, adjust coordinates suitably, and finally procure specific values for \(j\) and \(\lambda\).

Step by step solution

01

Create New Morphism

Given that \(X\) is represented as a 2-1 covering of \(\mathbf{P}^{1}\) by a morphism \(\pi: X \rightarrow \mathbf{P}^{1}\), we can create a new morphism, \(\pi^{\prime} : X \rightarrow \mathbf{P}^{1}\) of the form \(\pi \quad f = g \quad \pi^{\prime}\). Here also \(g: \mathbf{P}^{1} \rightarrow \mathbf{P}^{1}\) has degree 2.
02

Setting Coordinates & Deduce Relation

If we appropriately set the coordinates in two copies of \(\mathbf{P}^{1}\), we observe that the morphism \(y\) can be mapped with the form \(x \rightarrow x^{2}\). Further, we conclude that \(g\) is branched over two of the branch points of \(\pi\), and the inverse \(g^{-1}\) of the other two branch points of \(\pi\) consists of the four branch points of \(\pi\). Hence, we deduce a unique relation involving the invariant of \(X\).
03

Finding Values

Having established the relations and deduced the association with the invariant, we can solve the equation system for \(j\). From this, find that there are three specific values of \(j\) that correspond to elliptic curves with an endomorphism of degree \(2\). The corresponding values of \(\lambda\) and \(j\) can be then determined.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Geometry
Algebraic geometry is an area of mathematics that combines algebra, number theory, and geometry. It focuses on studying solutions to algebraic equations and their generalizations, which can describe a variety of geometric structures, known as algebraic varieties. Among these structures are elliptic curves, which are smooth, projective algebraic curves of genus one, with a specified point 'at infinity' making them a group.

In the context of elliptic curves, algebraic geometry allows us to explore their properties through equations and morphisms鈥攕uch as endomorphisms. An endomorphism is a morphism, or mathematical mapping, from the elliptic curve to itself that respects its structure. When considering exercises like the one provided, where the endomorphism is of degree 2, algebraic geometry provides frameworks to visualize and understand these mappings as well as to solve problems related to the structure of elliptic curves.
Covering Space
In mathematics, a covering space is a topological concept that deals with the idea of unfolding a given space into layers that locally look like the original space. For the set of points in one layer, a continuous and surjective function, called the covering map, projects them down to the base space in such a way that around every point in the base space, there exists a neighborhood that is evenly covered by the map.

When working with elliptic curves in algebraic geometry, the notion of a covering space manifests when we map an elliptic curve (denoted by X) onto the complex projective line, denoted by \(\mathbb{P}^1\). In the exercise, \(X\) is represented as a 2-1 covering of \(\mathbb{P}^1\), implying that for each point on \(\mathbb{P}^1\) (excluding the branch points), there are two corresponding points on the elliptic curve \(X\).
Morphism
In the broad umbrella of algebraic geometry, a morphism (or map) translates to a function that respects the structures which are present on the objects being dealt with. Specifically, a morphism between algebraic varieties preserves the equations defining these varieties. For example, an endomorphism is a particular type of morphism where the source and target are the same algebraic structure.

In our elliptic curve exercise, morphisms are central to describing the relationships between different spaces. The problem speaks of different morphisms such as \(f\), \(g\), \(\pi\), and \(\pi'\), each playing a role in how the elliptic curve \(X\) is projected onto or interacts with the projective line \(\mathbb{P}^1\). A degree 2 morphism, in this case, can be thought of as a function that 'doubles' the preimage points, which is linked to the 2-1 covering property.
Branch Point
A branch point is a point on a base space where the local structure of a covering map differs from the rest of the space, typically associated with multiple layers of the covering space meeting. In a 2-1 covering, each point on the base space would normally have two pre-images, but at a branch point, there might be only one pre-image due to the layers 'merging' at that point.

In the exercise provided, the elliptic curve \(X\) maps to \(\mathbb{P}^1\) in a 2-1 fashion except at the branch points; this is where the morphism is 'ramified'. When a second morphism, \(g\), maps \(\mathbb{P}^1\) onto another \(\mathbb{P}^1\), the connection between \(g\) and the branch points of \(\pi\) is of particular interest. The exercise asks students to investigate this relationship and determine how the branch points under one map correspond to those under another, deepening their understanding of complex mappings in algebraic geometry.

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Most popular questions from this chapter

In view of \((3.10),\) one might ask conversely, is every plane curve with nodes a projection of a nonsingular curve in \(\mathbf{P}^{3}\) ? Show that the curve \(x y+x^{4}+y^{4}=0\) (assume char \(k \neq 2\) ) gives a counterexample.

Plame Curres. Let \(X\) be a curve of degree \(d\) in \(\mathbf{P}^{2}\). For each point \(P \in X\), let \(T_{P}(X)\) be the tangent line to \(X\) at \(P(\mathrm{I}, \mathrm{Ex} .7 .3) .\) Considering \(T_{P}(X)\) as a point of the dual projective plane \(\left(\mathbf{P}^{2}\right)^{*},\) the \(\operatorname{map} P \rightarrow T_{P}(X)\) gives a morphism of \(X\) to its dual curre \(X^{*}\) in \(\left(\mathbf{P}^{2}\right)^{*}\) (I, Ex. 7.3). Note that even though \(X\) is nonsingular, \(X^{*}\) in general will have singularities. We assume char \(k=0\) below. (a) \(\mathrm{Fix}\) a line \(L \subseteq \mathrm{P}^{2}\) which is not tangent to \(X .\) Define a morphism \(\varphi: X \rightarrow L \mathrm{by}\) \(\varphi(P)=T_{P}(X) \cap L,\) for each point \(P \in X .\) Show that \(\varphi\) is ramified at \(P\) if and only if either \((1) P \in L,\) or (2)\(P\) is an inflection point of \(X,\) which means that the intersection multiplicity (I, Ex. 5.4) of \(T_{P}(X)\) with \(X\) at \(P\) is \(\geqslant 3 .\) Conclude that \(X\) has only finitely many inflection points. (b) A line of \(\mathbf{P}^{2}\) is a multiple tangent of \(X\) if it is tangent to \(X\) at more than one point. It is a bitangent if it is tangent to \(X\) at exactly two points. If \(L\) is a multiple tangent of \(X\), tangent to \(X\) at the points \(P_{1}, \ldots, P_{r},\) and if none of the \(P_{i}\) is an inflection point, show that the corresponding point of the dual curve \(X^{*}\) is an ordinary \(r\) -fold point, which means a point of multiplicity \(r\) with distinct tangent directions \((\mathrm{I}, \mathrm{Ex} .5 .3) .\) Conclude that \(X\) has only finitely many multiple tangents. (c) Let \(O \in \mathbf{P}^{2}\) be a point which is not on \(X\), nor on any inflectional or multiple tangent of \(X\). Let \(L\) be a line not containing \(O .\) Let \(\psi: X \rightarrow L\) be the morphism defined by projection from \(O .\) Show that \(\psi\) is ramified at a point \(P \in X\) if and only if the line \(O P\) is tangent to \(X\) at \(P,\) and in that case the ramification index is 2. Use Hurwitz's theorem and (I, Ex. 7.2) to conclude that there are exactly \(d(d-1)\) tangents of \(X\) passing through \(O .\) Hence the degree of the dual curve (sometimes called the class of \(X\) ) is \(d(d-1)\) (d) Show that for all but a finite number of points of \(X,\) a point \(O\) of \(X\) lies on exactly \((d+1)(d-2)\) tangents of \(X,\) not counting the tangent at \(O\) (e) Show that the degree of the morphism \(\varphi\) of (a) is \(d(d-1)\). Conclude that if \(d \geqslant 2 .\) then \(X\) has \(3 d(d-2)\) inflection points, properly counted. (If \(T_{P}(X)\) has intersection multiplicity \(r\) with \(X\) at \(P,\) then \(P\) should be counted \(r-2\) times as an inflection point. If \(r=3\) we call it an ordinary inflection point.) Show that an ordinary inflection point of \(X\) corresponds to an ordinary cusp of the dual curve \(X^{*}\) (f) Now let \(X\) be a plane curve of degree \(d \geqslant 2,\) and assume that the dual curve \(X^{*}\) has only nodes and ordinary cusps as singularities (which should be true for sufficiently general \(X\) ). Then show that \(X\) has exactly \(\frac{1}{2} d(d-2)(d-3)(d+3)\) bitangents. [Hint: Show that \(X\) is the normalization of \(X^{*}\). Then calculate \(p_{a}\left(X^{*}\right)\) two ways: once as a plane curve of degree \(d(d-1),\) and once using (Ex. \(1.8 \text { ). }]\) (g) For example, a plane cubic curve has exactly 9 inflection points, all ordinary. The line joining any two of them intersects the curve in a third one. (h) A plane quartic curve has exactly 28 bitangents. (This holds even if the curve has a tangent with four-fold contact, in which case the dual curve \(X^{*}\) has a tacnode.

Classification of Curres of Genus 2. Fix an algebraically closed field \(k\) of characteristic \(\neq 2\) (a) If \(X\) is a curve of genus 2 over \(k\), the canonical linear system \(|K|\) determines a finite morphism \(f: X \rightarrow \mathbf{P}^{1}\) of degree 2 (Ex. 1.7 ). Show that it is ramified at exactly 6 points, with ramification index 2 at each one. Note that \(f\) is uniquely determined, up to an automorphism of \(\mathbf{P}^{1},\) so \(X\) determines an (unordered) set of 6 points of \(\mathbf{P}^{1},\) up to an automorphism of \(\mathbf{P}^{1}\). (b) Conversely, given six distinct elements \(\alpha_{1}, \ldots, \alpha_{6} \in k,\) let \(K\) be the extension of \(k(x)\) determined by the equation \(z^{2}=\left(x-\alpha_{1}\right) \cdots\left(x-\alpha_{6}\right) .\) Let \(f: X \rightarrow \mathbf{P}^{1}\) be the corresponding morphism of curves. Show that \(g(X)=2,\) the map \(f\) is the same as the one determined by the canonical linear system, and \(f\) is ramified over the six points \(x=\alpha_{i}\) of \(\mathbf{P}^{1}\), and nowhere else. (Cf. (II, Ex. 6.4 ).) (c) Using (I, Ex. 6.6), show that if \(P_{1}, P_{2}, P_{3}\) are three distinct points of \(\mathbf{P}^{1}\), then there exists a unique \(\varphi \in\) Aut \(\mathbf{P}^{1}\) such that \(\varphi\left(P_{1}\right)=0, \varphi\left(P_{2}\right)=1, \varphi\left(P_{3}\right)=x\) Thus in (a), if we order the six points of \(\mathbf{P}^{1}\), and then normalize by sending the first three to \(0,1, x,\) respectively, we may assume that \(X\) is ramified over \(0,1, \nsim, \beta_{1}, \beta_{2}, \beta_{3},\) where \(\beta_{1}, \beta_{2}, \beta_{3}\) are three distinct elements of \(k, \neq 0,1\) (d) Let \(\Sigma_{6}\) be the symmetric group on 6 letters. Define an action of \(\Sigma_{6}\) on sets of three distinct elements \(\beta_{1}, \beta_{2}, \beta_{3}\) of \(k, \neq 0,1,\) as follows: reorder the set \(0,1, x, \beta_{1}, \beta_{2}, \beta_{3}\) according to a given element \(\sigma \in \Sigma_{6},\) then renormalize as in \((\mathrm{c})\) so that the first three become \(0,1, x\) again. Then the last three are the new \(\beta_{1}^{\prime}, \beta_{2}^{\prime}, \beta_{3}^{\prime}\) (e) Summing up, conclude that there is a one-to-one correspondence between the set of isomorphism classes of curves of genus 2 over \(k\), and triples of distinct elements \(\beta_{1}, \beta_{2}, \beta_{3}\) of \(k, \neq 0,1,\) modulo the action of \(\Sigma_{6}\) described in (d). In particular, there are many non-isomorphic curves of genus \(2 .\) We say that curves of genus 2 depend on three parameters, since they correspond to the points of an open subset of \(\mathbf{A}_{k}^{3}\) modulo a finite group

\(f_{*}\) for Divisors. Let \(f: X \rightarrow Y\) be a finite morphism of curves of degree \(n .\) We define a homomorphism \(f_{*}:\) Div \(X \rightarrow\) Div \(Y\) by \(f_{*}\left(\sum n_{i} P_{i}\right)=\sum n_{i} f\left(P_{i}\right)\) for any divisor \(D=\sum n_{i} P_{i}\) on \(X\) (a) For any locally free sheaf \(\delta\) on \(Y\), of rank \(r,\) we define det \(\mathscr{B}=\wedge^{r} \mathscr{E} \in \operatorname{Pic} Y\) (II, Ex. 6.11 ). In particular, for any invertible sheaf \(\mathscr{M}\) on \(X, f_{*} . / /\) is locally free of rank \(n\) on \(Y\), so we can consider det \(f_{*} \mathscr{H} \in\) Pic \(Y\). Show that for any divisor \(D\) on \(X\) $$\operatorname{det}\left(f_{*} \mathscr{L}(D)\right) \cong\left(\operatorname{det} f_{*} c_{X}\right) \otimes \mathscr{L}\left(f_{*} D\right)$$ Note in particular that \(\operatorname{det}\left(f_{*} \mathscr{L}(D)\right) \neq \mathscr{L}\left(f_{*} D\right)\) in general! \([\text { Hint }:\) First consider an effective divisor \(D,\) apply \(f_{*}\) to the exact sequence \(0 \rightarrow \mathscr{P}(-D) \rightarrow\) \(\left.\varphi_{x} \rightarrow \mathscr{C}_{D} \rightarrow 0, \text { and use (II, Ex. } 6.11 \text { ). }\right]\) (b) Conclude that \(f_{*} D\) depends only on the linear equivalence class of \(D,\) so there is an induced homomorphism \(f_{*}: \operatorname{Pic} X \rightarrow\) Pic \(Y\). Show that \(f_{*} \cdot f^{*}: \operatorname{Pic} Y \rightarrow\) Pic \(Y\) is just multiplication by \(n\) (c) Use duality for a finite flat morphism (III, Ex. 6.10 ) and (III, Ex. 7.2) to show that $$\operatorname{det} f_{*} \Omega_{X} \cong \text { (det } f_{*}(x)^{-1} \otimes \Omega_{Y}^{\otimes n}$$ (d) Now assume that \(f\) is separable, so we have the ramification divisor \(R\). We define the branch divisor \(B\) to be the divisor \(f_{*} R\) on \(Y\). Show that $$\left(\operatorname{det} f_{*} C_{x}\right)^{2} \cong \mathscr{L}(-B)$$

A Funny Curre in Characteristic \(p\). Let \(X\) be the plane quartic curve \(x^{3} y+y^{3} z+\) \(z^{3} x=0\) over a field of characteristic \(3 .\) Show that \(X\) is nonsingular, every point of \(X\) is an inflection point, the dual curve \(X^{*}\) is isomorphic to \(X,\) but the natural \(\operatorname{map} X \rightarrow X^{*}\) is purely inseparable.
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