4 Corrected description of $y^2=x^3+x$ case: Möbius band, not cylinder

I don't know what Mumford had in mind, but here (in some detail) is a down-to-earth way to topologically identify this space with a cylinder.

Let $C$ be our projective cubic curve with affine equation $y^2=x^3-x$. We're considering complex conjugate pairs of points on $C$, that is, pairs ${(x,y), (\bar x,\bar y)}$ of solutions of $y^2=x^3-x$. While those points are not real, the line joining them is real: there are real numbers $a,b,c$, not all zero, such that the line $l_{a,b,c}: aX + bY + c = 0$ passes through $(x,y)$ and $(\bar x, \bar y)$, and the coefficient vector $(a,b,c)$ is determined uniquely up to multiplication by a nonzero scalar. That is, $(a:b:c)$ is a well-defined point in the "dual projective plane" ${\bf P}^*$ of lines on the projective plane with coordinates $(x:y:1)$ where $C$ lives. Now these points $(x,y)$ and $(\bar x, \bar y)$ are on $l_{a,b,c} \cap C$, which contains three points in all, so there is a third point $(x_0,y_0) =: p_0$, necessarily real. Conversely, any line $l$ meets $C$ in at least one real point, and if there is only one such point (and $l$ is not tangent to $C$ at that point) then the other two points of $l \cap C$ constitute a closed-but-not-rational point of $C$.

That is,

the space we're looking for is homeomorphic with the subset, call it $S$, of ${\bf P}^*$ consisting of lines whose real intersection with $C$, with multiplicity, has size $1$

as opposed to size $3$.

One way to describe $S$ is to start from $p_0 = (x_0,y_0)$. It is geometrically clear that this point must be on the infinite component of $C$, call it $C_0$: the other component $C_1$ is a closed curve in the affine plane ${\bf R}^2$, so any line meets it with even total multiplicity. Given $p_0$, the lines through $p_0$ constitute a real projective line, which is topologically a circle and the lines through $p_0$ that meet $C_0$ in two other points $q,q'$ constitute the union of two closed arcs, one for lines where $q,q' \in C_0$ and the other for lines where $q,q' \in C_1$. [The boundary points correspond to the four points $q$ whose tangent passes through $p$, which are the solutions of $2q=-p$ in the group law of $C$.] So the lines through $p_0$ in $S$ constitute two open intervals. Now the subtlety is that when $p_0$ goes around the closed curve $C_0$, these two intervals switch as each of the boundary points makes a complete cycle around $C_0$ or $C_1$, so we must traverse $C_0$ twice to traverse our cylinder once. In effect we're getting a Möbius band cut down the middle, which is indeed a cylinder (with a "full twist", true, but that is an artifact of the embedding in three-dimensional space that we use to visualize $S$).

For a different kind of explicit picture of $S$, note that a real cubic polynomial has one real root (with multiplicity) if and only if its discriminant is negative. So we can describe $S$ by eliminating of the variables from $aX+bY+c=0$, substituting into $Y^2=X^3-X$, computing the discriminant $\Delta$ of the resulting cubic, and plotting the region $\Delta < 0$. For example, in the affine piece $b \neq 0$ of ${\bf P}^*$, we may set $b=1$, compute $Y = -(aX+c)$, find that $$\Delta = -27c^4 - 4(ac)^3 + 30(ac)^2 + 4 a^5 c + 24 ac + a^4 + 4$$ (I didn't promise it would be pretty), and ask www.wolframalpha.com

plot(-27*c^4-4*a^3*c^3+30*a^2*c^2+4*a^5*c+24*a*c+a^4+4 < 0)


to get a picture with two blue components that join up at infinity to form a topological cylinder:

[The two visible cusps come from the inflection points where $p=q=q'$, which are real 3-torsion points on $C$; there's a third such singularity at infinity. This means that of the two boundary components of $S$ (it looks like four but they pair up at infinity) the one containing the cusps is $C_0$, and the other is $C_1$.] Try also

plot(-27*c^4-4*a^3*c^3-30*a^2*c^2+(24*a-4*a^5)*c+a^4-4 < 0)


for the picture arising from the curve $y^2=x^3+x$ with only one real component; again it's this time it is a cylinder, but Möbius band embedded differently in ${\bf P}^*$ so that there boundary and the complement have only one component each:

To connect this with the usual (but less elementary) picture of an elliptic curve over ${\bf C}$ as a complex torus: as Lubin noted, the complex locus of $C$ is isomorphic as a Riemann surface with ${\bf C} / L$ where $L$ is the Gaussian lattice ${\bf Z} + {\bf Z} i$; this is consistent with complex conjugation, and the real locus consists of the cosets mod $L$ of the complex numbers of integral or half-integral imaginary part, constituting the components $C_0$ and $C_1$ respectively. We're looking to identify the conjugate pairs ${(z,\bar z)} \bmod L$ with a cylinder; in terms of the group law the real point $p_0$ associated above to ${(z,\bar z)}$ is $-2 \phantom. {\rm Re}(z)$, which as before can only be on $C_0$ and goes around $C_0$ twice (and in the opposite direction, as it happens) as $z$ goes around the cylinder once.

3 Removed stray ')' after "the other is C_1"

I don't know what Mumford had in mind, but here (in some detail) is a down-to-earth way to topologically identify this space with a cylinder.

Let $C$ be our projective cubic curve with affine equation $y^2=x^3-x$. We're considering complex conjugate pairs of points on $C$, that is, pairs ${(x,y), (\bar x,\bar y)}$ of solutions of $y^2=x^3-x$. While those points are not real, the line joining them is real: there are real numbers $a,b,c$, not all zero, such that the line $l_{a,b,c}: aX + bY + c = 0$ passes through $(x,y)$ and $(\bar x, \bar y)$, and the coefficient vector $(a,b,c)$ is determined uniquely up to multiplication by a nonzero scalar. That is, $(a:b:c)$ is a well-defined point in the "dual projective plane" ${\bf P}^*$ of lines on the projective plane with coordinates $(x:y:1)$ where $C$ lives. Now these points $(x,y)$ and $(\bar x, \bar y)$ are on $l_{a,b,c} \cap C$, which contains three points in all, so there is a third point $(x_0,y_0) =: p_0$, necessarily real. Conversely, any line $l$ meets $C$ in at least one real point, and if there is only one such point (and $l$ is not tangent to $C$ at that point) then the other two points of $l \cap C$ constitute a closed-but-not-rational point of $C$.

That is,

the space we're looking for is homeomorphic with the subset, call it $S$, of ${\bf P}^*$ consisting of lines whose real intersection with $C$, with multiplicity, has size $1$

as opposed to size $3$.

One way to describe $S$ is to start from $p_0 = (x_0,y_0)$. It is geometrically clear that this point must be on the infinite component of $C$, call it $C_0$: the other component $C_1$ is a closed curve in the affine plane ${\bf R}^2$, so any line meets it with even total multiplicity. Given $p_0$, the lines through $p_0$ constitute a real projective line, which is topologically a circle and the lines through $p_0$ that meet $C_0$ in two other points $q,q'$ constitute the union of two closed arcs, one for lines where $q,q' \in C_0$ and the other for lines where $q,q' \in C_1$. [The boundary points correspond to the four points $q$ whose tangent passes through $p$, which are the solutions of $2q=-p$ in the group law of $C$.] So the lines through $p_0$ in $S$ constitute two open intervals. Now the subtlety is that when $p_0$ goes around the closed curve $C_0$, these two intervals switch as each of the boundary points makes a complete cycle around $C_0$ or $C_1$, so we must traverse $C_0$ twice to traverse our cylinder once. In effect we're getting a Möbius band cut down the middle, which is indeed a cylinder (with a "full twist", true, but that is an artifact of the embedding in three-dimensional space that we use to visualize $S$).

For a different kind of explicit picture of $S$, note that a real cubic polynomial has one real root (with multiplicity) if and only if its discriminant is negative. So we can describe $S$ by eliminating of the variables from $aX+bY+c=0$, substituting into $Y^2=X^3-X$, computing the discriminant $\Delta$ of the resulting cubic, and plotting the region $\Delta < 0$. For example, in the affine piece $b \neq 0$ of ${\bf P}^*$, we may set $b=1$, compute $Y = -(aX+c)$, find that $$\Delta = -27c^4 - 4(ac)^3 + 30(ac)^2 + 4 a^5 c + 24 ac + a^4 + 4$$ (I didn't promise it would be pretty), and ask www.wolframalpha.com

plot(-27*c^4-4*a^3*c^3+30*a^2*c^2+4*a^5*c+24*a*c+a^4+4 < 0)


to get a picture with two blue components that join up at infinity to form a topological cylinder:

[The two visible cusps come from the inflection points where $p=q=q'$, which are real 3-torsion points on $C$; there's a third such singularity at infinity. This means that of the two boundary components of $S$ (it looks like four but they pair up at infinity) the one containing the cusps is $C_0$, and the other is $C_1$).] C_1$.] Try also plot(-27*c^4-4*a^3*c^3-30*a^2*c^2+(24*a-4*a^5)*c+a^4-4 < 0)  for the picture arising from the curve$y^2=x^3+x$with only one real component; again it's a cylinder, but embedded differently in${\bf P}^*$so that there boundary and the complement have only one component each: To connect this with the usual (but less elementary) picture of an elliptic curve over${\bf C}$as a complex torus: as Lubin noted, the complex locus of$C$is isomorphic as a Riemann surface with${\bf C} / L$where$L$is the Gaussian lattice${\bf Z} + {\bf Z} i$; this is consistent with complex conjugation, and the real locus consists of the cosets mod$L$of the complex numbers of integral or half-integral imaginary part, constituting the components$C_0$and$C_1$respectively. We're looking to identify the conjugate pairs${(z,\bar z)} \bmod L$with a cylinder; in terms of the group law the real point$p_0$associated above to${(z,\bar z)}$is$-2 \phantom. {\rm Re}(z)$, which as before can only be on$C_0$and goes around$C_0$twice (and in the opposite direction, as it happens) as$z$goes around the cylinder once. 2 Identified how the boundary components of$S$correspond with the components of$C$I don't know what Mumford had in mind, but here (in some detail) is a down-to-earth way to topologically identify this space with a cylinder. Let$C$be our projective cubic curve with affine equation$y^2=x^3-x$. We're considering complex conjugate pairs of points on$C$, that is, pairs${(x,y), (\bar x,\bar y)}$of solutions of$y^2=x^3-x$. While those points are not real, the line joining them is real: there are real numbers$a,b,c$, not all zero, such that the line$l_{a,b,c}: aX + bY + c = 0$passes through$(x,y)$and$(\bar x, \bar y)$, and the coefficient vector$(a,b,c)$is determined uniquely up to multiplication by a nonzero scalar. That is,$(a:b:c)$is a well-defined point in the "dual projective plane"${\bf P}^*$of lines on the projective plane with coordinates$(x:y:1)$where$C$lives. Now these points$(x,y)$and$(\bar x, \bar y)$are on$l_{a,b,c} \cap C$, which contains three points in all, so there is a third point$(x_0,y_0) =: p_0$, necessarily real. Conversely, any line$l$meets$C$in at least one real point, and if there is only one such point (and$l$is not tangent to$C$at that point) then the other two points of$l \cap C$constitute a closed-but-not-rational point of$C$. That is, the space we're looking for is homeomorphic with the subset, call it$S$, of${\bf P}^*$consisting of lines whose real intersection with$C$, with multiplicity, has size$1$as opposed to size$3$. One way to describe$S$is to start from$p_0 = (x_0,y_0)$. It is geometrically clear that this point must be on the infinite component of$C$, call it$C_0$: the other component$C_1$is a closed curve in the affine plane${\bf R}^2$, so any line meets it with even total multiplicity. Given$p_0$, the lines through$p_0$constitute a real projective line, which is topologically a circle and the lines through$p_0$that meet$C_0$in two other points$q,q'$constitute the union of two closed arcs, one for lines where$q,q' \in C_0$and the other for lines where$q,q' \in C_1$. [The boundary points correspond to the four points$q$whose tangent passes through$p$, which are the solutions of$2q=-p$in the group law of$C$.] So the lines through$p_0$in$S$constitute two open intervals. Now the subtlety is that when$p_0$goes around the closed curve$C_0$, these two intervals switch as each of the boundary points makes a complete cycle around$C_0$or$C_1$, so we must traverse$C_0$twice to traverse our cylinder once. In effect we're getting a Möbius band cut down the middle, which is indeed a cylinder (with a "full twist", true, but that is an artifact of the embedding in three-dimensional space that we use to visualize$S$). For a different kind of explicit picture of$S$, note that a real cubic polynomial has one real root (with multiplicity) if and only if its discriminant is negative. So we can describe$S$by eliminating of the variables from$aX+bY+c=0$, substituting into$Y^2=X^3-X$, computing the discriminant$\Delta$of the resulting cubic, and plotting the region$\Delta < 0$. For example, in the affine piece$b \neq 0$of${\bf P}^*$, we may set$b=1$, compute$Y = -(aX+c)$, find that $$\Delta = -27c^4 - 4(ac)^3 + 30(ac)^2 + 4 a^5 c + 24 ac + a^4 + 4$$ (I didn't promise it would be pretty), and ask www.wolframalpha.com plot(-27*c^4-4*a^3*c^3+30*a^2*c^2+4*a^5*c+24*a*c+a^4+4 < 0)  to get a picture with two blue components that join up at infinity to form a topological cylinder: [The two visible cusps come from the inflection points where$p=q=q'$, which are real 3-torsion points on$C$; there's a third such singularity at infinity.] infinity. This means that of the two boundary components of$S$(it looks like four but they pair up at infinity) the one containing the cusps is$C_0$, and the other is$C_1$).] Try also plot(-27*c^4-4*a^3*c^3-30*a^2*c^2+(24*a-4*a^5)*c+a^4-4 < 0)  for the picture arising from the curve$y^2=x^3+x$with only one real component; again it's a cylinder, but embedded differently in${\bf P}^*$so that there boundary and the complement has have only one component each: To connect this with the usual (but less elementary) picture of an elliptic curve over${\bf C}$as a complex torus: as Lubin noted, the complex locus of$C$is isomorphic as a Riemann surface with${\bf C} / L$where$L$is the Gaussian lattice${\bf Z} + {\bf Z} i$; this is consistent with complex conjugation, and the real locus consists of the cosets mod$L$of the complex numbers of integral or half-integral imaginary part, constituting the components$C_0$and$C_1$respectively. We're looking to identify the conjugate pairs${(z,\bar z)} \bmod L$with a cylinder; in terms of the group law the real point$p_0$associated above to${(z,\bar z)}$is$-2 \phantom. {\rm Re}(z)$, which as before can only be on$C_0$and goes around$C_0$twice (and in the opposite direction, as it happens) as$z\$ goes around the cylinder once.

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