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Myrberg’s Uniformization of Elliptic Curves

Myrberg’s Uniformization of Elliptic Curves. Mika Seppälä Florida State University and University of Helsinki. Elliptic Curves. Smooth projective elliptic curves are compact Riemann surfaces of genus 1, i.e., tori. Every elliptic curve is isomorphic to curve T

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Myrberg’s Uniformization of Elliptic Curves

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  1. Myrberg’s Uniformization of Elliptic Curves Mika Seppälä Florida State University and University of Helsinki

  2. Elliptic Curves Smooth projective elliptic curves are compact Riemann surfaces of genus 1, i.e., tori. Every elliptic curve is isomorphic to curve T defined by an affine equation of the type: Mika Seppälä FSU and University of Helsinki

  3. Mika Seppälä FSU and University of Helsinki

  4. Classical Uniformization Mika Seppälä FSU and University of Helsinki

  5. Lande’s Transformation The equation defines w as a function of z. For a suitable choice of the branch of the square root, this mapping maps the complement of a slot onto the complement of a disk. Mika Seppälä FSU and University of Helsinki

  6. Lande’s Transformation Lande’s transformation maps the complement of a slot onto the complement of a disk. Here m is the mid-point and m+2s and m-2s are the end-points of the slot in question. Different choice of the branch of the square root in the equation defining Lande’s transformation gives a mapping of the complement of the slot onto the disk pictured above. We have where e is an elliptic rotation mapping the outside of the above disk onto its inside. Mika Seppälä FSU and University of Helsinki

  7. Myrberg’s construction Mika Seppälä FSU and University of Helsinki

  8. Mika Seppälä FSU and University of Helsinki

  9. Parameterization of Elliptic Curves Mika Seppälä FSU and University of Helsinki

  10. Iteration Mika Seppälä FSU and University of Helsinki

  11. Myrberg vs. Classical Uniformization Mika Seppälä FSU and University of Helsinki

  12. References • P. J. Myrberg: Uber die Numerische Ausführung der Uniformisierung. Acta Soc. Sci.Fenn., XLVIII(7):1 – 53, 1920 • M. Seppälä:Myrberg’s Numerical Uniformization of Hyperelliptic Curves. Ann. Acad. Sci. Fenn. 2003 • Recent work and implementations:http://www.math.fsu.edu/~seppala/ComputationsOnCurves/index.html • This talk http://www.math.fsu.edu/~seppala/AMSmeetingBaltimore2003/index.htm Mika Seppälä FSU and University of Helsinki

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