Arithmetic concerning Poncelet’s closure theorem
Jaap Top
JBI-RuG & DIAMANT
November 17th, 2017
(Barranquilla, math colloquium)
Jean-Victor Poncelet (1788–1867)
1812-1814 Poncelet was prisoner of war in Russia
During this time he wrote texts (7 notebooks) on projective geometry,
in 1822 he published a book loosely based on this:
picture from the book:
Take a polygon with sides
`1, `2, . . . , `n and vertices
P1 = `n ∩ `1, P2 = `1 ∩ `2, . . . , Pn = `n−1 ∩ `n.
It is called “Poncelet figure” if conics C, D exist such that all Pj ∈ C and all `j tangent to D.
It is called “trivial Poncelet figure” if moreover either n = 2k − 2 is even and Pk ∈ C ∩ D, or n = 2k − 1 is odd and `k is tangent to both D and C.
Theorem. Given smooth conics C, D, if a Poncelet figure using C, D exists which is not trivial, then for every P ∈ C there is one having P as vertex.
Poncelet’s “proof ” led to discussions, but Jacobi (1828, using elliptic functions) published in Crelle a new and complete proof:
Carl Gustav Jacob Jacobi (1804–1851)
• The result has consequences for “billiard ball trajectories on an elliptical table”
(picture by Alfonso Sorrentino)
• In 1976 Phillip A. Griffiths published a purely algebraic geo- metric proof of Poncelet’s result (“Variations on a theorem of Abel”)
• aim of today’s talk: show how to use the algebraic geometric approach for constructing examples, say over Fp or over Q.
• For this, we first sketch the approach;
• then we “turn it around” to obtain a method for constructing examples,
• and we illustrate this in specific cases and discuss possible applications.
joint work with Johan Los and Tiemar Mepschen (bachelor’s projects) and Nurdag¨ul Anbar Meidl
Conic C ⊂ P2: the zeros P = (a : b : c) of an irreducible homo- geneous quadratic F .
Lines tangent to a second conic D ⊂ P2:
given by ` : dx + ey + f z = 0 with (d : e : f ) ∈ P2.
The tangency condition means that (d : e : f ) satisfies some homogeneous quadratic equation depending on D.
A condition P ∈ ` is expressed as da + eb + f c = 0.
In this way
X := {(P, `) : P ∈ C, P ∈ `, ` tangent to D}
defines an algebraic curve.
We study this in the general case, namely when C and D intersect without multiplicities (hence in 4 points).
The curve X comes with additional structure:
The morphism X → C given by (P, `) 7→ P has degree 2: for almost all P , there are precisely 2 lines `, `0 containing P and tangent to D.
The morphism X → D given by (P, `) 7→ ` ∩ D also has degree 2:
for almost all Q ∈ D, the tangent line ` of D at Q will intersect C in two points P, P0.
The two involutions of X given by, respectively, τ : (P, `) ↔ (P, `0) and σ: (P, `) ↔ (P0, `) each have 4 fixpoints.
Hence X has genus 1, comes with two involutions, and the quo- tient by any of them has genus 0.
Fixing a point on X, it obtains the structure of an elliptic curve E.
Having as quotient a curve of genus 0, the involutions must be given as τ : x 7→ x1 − x resp. σ: x 7→ x2 − x for certain x1, x2 ∈ E.
The composition τ σ therefore equals x 7→ (x1−x2)+x, translation over x1 − x2 ∈ E.
If a Poncelet figure with edges `j and vertices Pj would exist for C, D, then τ σ acts as
(P1, `1) 7→ (P2, `2) 7→ . . . 7→ (Pn, `n) 7→ (P1, `1), so (στ )n = id. This implies the closure theorem!
A first arithmetic result motivated by this:
Theorem. Let X/Q have genus 1 (possibly with X(Q) = ∅ !).
If σ, τ ∈ AutQ(X) are involutions such that τ σ has finite order, and the quotients by σ resp. τ have genus 0,
then this order is one of {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12}.
The case where X contains a rational point is essentially Mazur’s theorem (1977) on rational torsion. The general case follows by applying Mazur’s result to the Jacobian of X.
Example. Take
C: x2 + y2 = 6z2 and D: x2 + xy + y2 = 9z2/4.
Here #(C ∩ D) = 4 and the resulting curve X is defined over Q and has genus 1.
Since C has no rational points, neither has X.
The involutions σ, τ as constructed before in this case have a product of order 4.
The example corresponds to the Poncelet figure we saw earlier:
Now we turn things around and start from an elliptic curve E over a field k (say, characteristic not 2).
• As involutions on E we take σ: p 7→ −p and τ : p 7→ T − p with composition the translation over T ∈ E(k).
• We want T to have finite order, so if this order is > 3 we can assume T = (0, 0) ∈ E(k) and E is given by
y2 + uxy + vy = x3 + vx2 for some u, v ∈ k.
• T having order n means that u, v satisfy some explicit poly- nomial equation depending on n.
Next, the conics C, D.
• They come from quotients of E by respectively τ and σ.
• The quotient by σ is described by E → P1 given by (x, y) 7→ c := x.
• The quotient by τ is described by E → P1 given by (x, y) 7→ b := (y + v)/x.
• (b, c) satisfy b2c + ubc − vb − c2 − vc − uv = 0, biquadratic in b, c (a bit like Edwards curves. . .)
In the coordinates b, c:
• σ is given as (b, c) 7→ (b0, c);
• τ is given as (b, c) 7→ (b, c0);
• ((could this be useful for fast arithmetic? No?))
• ((with k = Fp, could this Poncelet type description of E have some cryptographic relevance?))
To arrive at a concrete Poncelet situation involving conics C, D:
• Use the b-coordinate to parametrize a conic C, say the stan- dard circle:
b 7→ Pb := b2 − 1
b2 + 1, 2b b2 + 1
!
.
• Now use the c-coordinate to parametrize lines `c given as y = α(c)x + β(c).
• The condition Pb ∈ `c we wish to be expressing the given bidegree (2, 2) equation for E.
This works if one takes
α(c) = c2 + vc + c + uv
v − uc , β(c) = c2 + vc − c + uv uc − v .
The corresponding lines `c are tangent to the conic D given by (v2 + 2v + 1 − 4uv)x2 + (2uv + 2u + 4v)xy + u2y2
+(8uv − 2v2 + 2)x + (2u − 2uv + 4v)y
= 4uv − v2 + 2v − 1.
This finishes an explicit construction, provided we have an elliptic curve E as above on which T = (0, 0) has order n.
For a really rational example, even with many possible starting points P1 and starting lines `1 rational, one should start with an elliptic curve containing infinitely many rational points and moreover a point of exact order n.
By Mazur’s result this requires n ≤ 12 and n 6= 11.
Using Cremona’s tables we found for each of these n such an example with minimal possible conductor.
Example. (n = 9)
E : y2 − 47xy − 624y = x3 − 624x2
has conductor 1482 and T = (0, 0) ∈ E has order 9 and p := (−6, 90) ∈ E has infinite order.
In the way described above, it leads to a Poncelet figure with 9 vertices.
Taking as conics here C : x2 + y2 = 1 and D given by
270817x2 + 56066xy + 2209y2 = 544126x + 61246y − 273313, one obtains from (p + mT )m≥1 on E the Poncelet figure with points
(3960/3961, 89/3961)
↓
(3843/3845, 124/3845)
↓
(2952/2977, −385/2977)
↓
(17472/17497, 935/17497)
↓
(24/25, 7/25)
↓
(1155/1157, 68/1157)
↓
(12/13, 5/13)
↓
(39456/39505, 1967/39505)
↓
If one allows, e.g., (real) quadratic fields then more possibilities for the order n of a torsion point occur.
The maximal possible order in the quadratic case is n = 18.
A paper by Bosman, P. Bruin, Dujella, and Najman (2014) con- tains an elliptic curve over Q(√
26521) having positive rank and a rational point of order 18.
Using this it is clear how to construct corresponding Poncelet figures over the same quadratic field, and with 18 vertices.
A few questions remain:
• Could the construction over a finite field have any crypto- graphic relevance?
• In constructing examples over quadratic fields, we ignored the issue of making the field and/or conductor small. For exam- ple, infinitely many elliptic curves over Q(√
2) exist having a rational point of order 11. However, we have not exhib- ited one with positive rank. (We did find such curves over Q(
√73), etc.)
• Is there an easy way to construct, like we did for n = 4, a genus 1 curve X/Q with X(Q) = ∅ admitting involutions over Q with product of order n, say for n = 9, 10, 12?
