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Part3 (Continued from: https://www.freemathhelp.com/forum/...on-to-unlock-their-secrets-ii-quantas.128430/ )
www.quantamagazine.org
The Manin-Mumford conjecture has to do with how many times one of these complicated curves, nestled inside its Jacobian, intersects the torsion points of the Jacobian. It predicts that these intersections only occur finitely many times. The conjecture reflects the interrelationship between the algebraic nature of a curve (in the way that torsion points are special solutions to the equations defining the curve) and its life as a geometric object (reflecting how the curve is embedded inside its Jacobian, like one shape inside another). Torsion points are crowded in every region of the Jacobian. If you zoom in on any tiny part of it, you will find them. But the Manin-Mumford conjecture predicts that, surprisingly, the nestled curve still manages to miss all but a finite number of them.
In 1983 Michel Raynaud proved the conjecture true. Since then, mathematicians have been trying to upgrade his result. Instead of just knowing that the number of intersections is finite, they’d like to know it’s below some specific value.
“Now that you know that they have only finitely many points in common, then every mathematician you would meet would say, well, how many?” said Krieger.
But the effort to count the intersection points was impeded by the lack of a clear framework in which to think about the complex numbers that define those points. Arithmetic dynamics ended up providing one.
Translating the Problem
In their 2020 paper, DeMarco, Krieger and Ye established that there is an upper bound on the intersection number for a family of curves. A newer paper by another mathematician, Lars Kühne of the University of Copenhagen, presents a proof establishing an upper bound for all curves. That paper was posted in late January and has not been fully vetted.
Raynaud’s previous result proved simply that the number of intersections is finite — but it left room for that finite number to be as large as you could possibly want (in the sense that you can always make a larger finite number). The trio’s new proof establishes what’s called a uniform bound, a cap on how big that finite number of intersections can be. DeMarco, Krieger and Ye didn’t identify that cap exactly, but they proved it exists, and they also identified a long series of steps that future work could take to calculate the number.
The elliptic curves that make up the Jacobians take their solutions from the complex numbers, which gives their graphs a bulkier appearance than the graphs of elliptic curves whose solutions come from the real numbers. Instead of a wiggly line, they look like the surface of a doughnut. The specific family of curves that DeMarco, Krieger and Ye studied has Jacobians that look like two-holed doughnuts. They break apart nicely into two regular doughnuts, each of which is the graph of one of the two constituent elliptic curves.
The new work focuses on the torsion points of those elliptic curves. The three mathematicians knew that the number they were interested in — the number of intersection points between complicated curves and the torsion points of their Jacobians — could be reframed in terms of the number of times that torsion points from one of those elliptic curves overlap torsion points from the other. So, to put a bound on the Manin-Mumford conjecture, all the authors had to do was count the number of intersections between those torsion points.
They knew this could not be accomplished directly. The two elliptic curves and their torsion points could not be immediately compared because they do not necessarily overlap. The torsion points are sprinkled on the surfaces of the elliptic curves, but the two curves might have very different shapes. It’s like comparing points on the surface of a sphere to points on the surface of a cube — the points can have similar relative positions without actually overlapping.
“You can’t really compare the points on those elliptic curves, because they’re in different places; they’re living on different geometric objects,” said Krieger.
But while the torsion points don’t actually necessarily overlap, it’s possible to think of pairs of them as being in the same relative position on each doughnut. And pairs of torsion points that occupy the same relative position on their respective doughnuts can be thought of as intersecting.
In order to determine precisely where these intersections take place, the authors had to lift the torsion points off their respective curves and transpose them over each other — almost the way you’d fit a star chart to the night sky.
Mathematicians knew about these star charts, but they didn’t have a good perspective that allowed them to count the overlapping points. DeMarco, Krieger and Ye managed it using arithmetic dynamics. They translated the two elliptic curves into two different dynamical systems. The two dynamical systems generated points on the same actual space, the complex plane.
“It’s easier to think of one space with two separate dynamical systems, versus two separate spaces with one dynamical system,” said DeMarco.
The finite orbit points of the two dynamical systems corresponded to the torsion points of the underlying elliptic curves. Now, to put a bound on the Manin-Mumford conjecture, the mathematicians just needed to count the number of times these finite orbit points overlapped. They used techniques from dynamical systems to solve the problem.
Mathematicians Set Numbers in Motion to Unlock Their Secrets
A new proof demonstrates the power of arithmetic dynamics, an emerging discipline that combines insights from number theory and dynamical systems.
www.quantamagazine.org
The Manin-Mumford conjecture has to do with how many times one of these complicated curves, nestled inside its Jacobian, intersects the torsion points of the Jacobian. It predicts that these intersections only occur finitely many times. The conjecture reflects the interrelationship between the algebraic nature of a curve (in the way that torsion points are special solutions to the equations defining the curve) and its life as a geometric object (reflecting how the curve is embedded inside its Jacobian, like one shape inside another). Torsion points are crowded in every region of the Jacobian. If you zoom in on any tiny part of it, you will find them. But the Manin-Mumford conjecture predicts that, surprisingly, the nestled curve still manages to miss all but a finite number of them.
In 1983 Michel Raynaud proved the conjecture true. Since then, mathematicians have been trying to upgrade his result. Instead of just knowing that the number of intersections is finite, they’d like to know it’s below some specific value.
“Now that you know that they have only finitely many points in common, then every mathematician you would meet would say, well, how many?” said Krieger.
But the effort to count the intersection points was impeded by the lack of a clear framework in which to think about the complex numbers that define those points. Arithmetic dynamics ended up providing one.
Translating the Problem
In their 2020 paper, DeMarco, Krieger and Ye established that there is an upper bound on the intersection number for a family of curves. A newer paper by another mathematician, Lars Kühne of the University of Copenhagen, presents a proof establishing an upper bound for all curves. That paper was posted in late January and has not been fully vetted.
Raynaud’s previous result proved simply that the number of intersections is finite — but it left room for that finite number to be as large as you could possibly want (in the sense that you can always make a larger finite number). The trio’s new proof establishes what’s called a uniform bound, a cap on how big that finite number of intersections can be. DeMarco, Krieger and Ye didn’t identify that cap exactly, but they proved it exists, and they also identified a long series of steps that future work could take to calculate the number.
The elliptic curves that make up the Jacobians take their solutions from the complex numbers, which gives their graphs a bulkier appearance than the graphs of elliptic curves whose solutions come from the real numbers. Instead of a wiggly line, they look like the surface of a doughnut. The specific family of curves that DeMarco, Krieger and Ye studied has Jacobians that look like two-holed doughnuts. They break apart nicely into two regular doughnuts, each of which is the graph of one of the two constituent elliptic curves.
The new work focuses on the torsion points of those elliptic curves. The three mathematicians knew that the number they were interested in — the number of intersection points between complicated curves and the torsion points of their Jacobians — could be reframed in terms of the number of times that torsion points from one of those elliptic curves overlap torsion points from the other. So, to put a bound on the Manin-Mumford conjecture, all the authors had to do was count the number of intersections between those torsion points.
They knew this could not be accomplished directly. The two elliptic curves and their torsion points could not be immediately compared because they do not necessarily overlap. The torsion points are sprinkled on the surfaces of the elliptic curves, but the two curves might have very different shapes. It’s like comparing points on the surface of a sphere to points on the surface of a cube — the points can have similar relative positions without actually overlapping.
“You can’t really compare the points on those elliptic curves, because they’re in different places; they’re living on different geometric objects,” said Krieger.
But while the torsion points don’t actually necessarily overlap, it’s possible to think of pairs of them as being in the same relative position on each doughnut. And pairs of torsion points that occupy the same relative position on their respective doughnuts can be thought of as intersecting.
In order to determine precisely where these intersections take place, the authors had to lift the torsion points off their respective curves and transpose them over each other — almost the way you’d fit a star chart to the night sky.
Mathematicians knew about these star charts, but they didn’t have a good perspective that allowed them to count the overlapping points. DeMarco, Krieger and Ye managed it using arithmetic dynamics. They translated the two elliptic curves into two different dynamical systems. The two dynamical systems generated points on the same actual space, the complex plane.
“It’s easier to think of one space with two separate dynamical systems, versus two separate spaces with one dynamical system,” said DeMarco.
The finite orbit points of the two dynamical systems corresponded to the torsion points of the underlying elliptic curves. Now, to put a bound on the Manin-Mumford conjecture, the mathematicians just needed to count the number of times these finite orbit points overlapped. They used techniques from dynamical systems to solve the problem.
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