Visualizing Dessins d Enfants on the Torus

Transcription

1 Department of Mathematics Purdue University Summer Project Presentation Purdue University October 04, 2015

2 Overview of the algorithm Input: An elliptic curve in the form: y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 A Belyĭ Map: β : E(C) P 1 (C) Output: Points and coloring corresponding to a Dessin d Enfant embedded on the torus, T 2 (R).

3 Step 1: Setup Algorithm We find a set of points (x, y) that approximate β 1 on the curve from 0 to 1: β 1( [0, 1] ) E(C). Recall: β 1 (0) = Red Vertices β 1 (1) = Black Vertices β 1 ([0, 1]) = Edges

4 Step 1: Setup Algorithm Given our curve : y 2 + a 1 x y + a 3, y = x 3 + a 2 x 2 + a 4 x + a 6 We simplify it to the following 4(x 3 + a 2 x 2 + a 4 x + a6) + (a 1 x + a 3 ) 3 = 4(x e 1 )(x e 2 )(x e 3 ) Where e 1, e 2, and e 3 are roots of the elliptic curve

5 Step 2: The elliptic logarithm Recall the lattice Λ = { m ω 1 + n ω 2 m, n Z } in terms of the periods ω 1 = e3 e 1 dt (t e 1 ) (t e 2 ) (t e 3 ) and ω 2 = e3 e 2 dt (t e 1 ) (t e 2 ) (t e 3 ) For every point (x, y) calculated in step 1 we compute an elliptic logarithm, z by log E : E(C) C/Λ.: dt z = sgn(y) x (t e 1 ) (t e 2 ) (t e 3 ) = m ω 1 + n ω 2

6 Step 3: Projecting to the torii We exploit the following isomorphisms E(C) (x, y) C/Λ dt sgn(y) x (t e 1 ) (t e 2 ) (t e 3 ) T 2 (R) ( ) u = R + r cos 2πm cos 2πn ( ) v = R + r cos 2πm sin 2πn = m ω 1 + n ω 2 w = r sin 2πn

7 John Cremona and Thotasaphon Thongjunthug s Method One of the major issues was calculating the elliptic logarithm in Sage. We used the Arithmetic Geometric Mean(AGM) to approximate the elliptic integral. Without using an integral within a few steps you can acquire the elliptic integral quickly.

8 Overview of the Algorithm The algorithm is broken up into two steps 1 Finding the periods ω 1 and ω 2 2 Finding the elliptic logarithm To find the periods you set up one "for" loop that converges to ω 1 and ω 2 as you increase the iterations To find the elliptic logarithm you set up a nested "for" loop that calculates a complex number for every point on the elliptic curve.

9 Step 1 - Calculating the Periods To begin calculate four values in terms of the roots of the elliptic curve. A 0 = e 1 e 3 B 0 = e 1 e 2 C 0 = e 2 e 3 D 0 = e 2 e 1 Next we run through the "for" loop from p = 0, 1, 2, 3...N 1 A p+1 = A p + B p 2 B p+1 = A p B p C p+1 = C p + D p 2 D p+1 = C p D p As p iterates to N-1 the following ratio converges to the period ω 1 = π = π ω 2 = π = π A N B N C N D N Remark: There is a consistent way to choose the signs to guarantee convergence

10 Step 2 - Calculating the Elliptic Logarithm Given a point (x, y) on the elliptic curve, compute the following To calculate the elliptic logarithm find the following values I 1 = x e1 J 1 = (2y + a 1x + a 3 ) x e 2 2I 1 (x e 2 ) Then as p = 0, 1, 2, 3, 4...N 1 calculate the following values I p+1 = Ap (I p + 1) B p 1 I p + A p 1 J p+1 = I p+1 J p This returns the elliptic logarithm z = arctan A N J N A N

11 Dessin of Degree 3 Algorithm E : y 2 = x β(x, y) = (y+1) 2

12 Dessin of Degree 4 Algorithm E : y 2 = x 3 + x x β(x, y) = (x 2 +4y+56) (108)

13 Dessin of Degree 5 Algorithm E : y 2 = x 3 + 5x + 10 β(x, y) = (x 5)y+16 32

14 Dessin of Degree 9 Algorithm E : y 2 = x β(x, y) = (216x 3 ) (y+36) 3

15 Acknowledgments Algorithm Dr. Edray Herber Goins Hongshan Li Avi Steiner Dr. Steve Bell Dr. David Goldberg Dr. Lazslo Lempert Dr. Joel and Mrs. Ruth Spira Dr. Uli Walther Dr. Mark Ward Dr. Lowell W. Beineke Dr. Gregery Buzzard / Department of Mathematics College of Science UREP-C / Universidad Nacional de Colombia National Science Foundation

16 Thank You! Questions?