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Higher-Order Weierstrass Points and Klein's Quartic Curve

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Higher-Order Weierstrass Points and Klein's Quartic Curve Background Klein's Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Higher-Order Weierstrass Points and Klein's Quartic Curve Eleanor Farrington Massachusetts Maritime Academy Conference in Memoriam of R.D.M. Accola (1930-2011) March 19, 2016 Farrington Weierstrass Points and Klein's Quartic 1 / 20 Background Klein's Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Outline of the Talk 1 Background 2 Klein's Quartic Curve 3 Weights of n- Weierstrass Points 4 Applying the Weight Formulas Farrington Weierstrass Points and Klein's Quartic 2 / 20 Definition A point x 2 M where a divisor G 2 jDj has order greater than r is called a generalized Weierstrass point for that linear series. Background Klein's Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Generalized Weierstrass Points Let M be a compact Riemann surface of genus g ≥ 2. Let jDj be a base-point free complete linear series on M with r = dimjDj and d = degjDj. Farrington Weierstrass Points and Klein's Quartic 3 / 20 Background Klein's Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Generalized Weierstrass Points Let M be a compact Riemann surface of genus g ≥ 2. Let jDj be a base-point free complete linear series on M with r = dimjDj and d = degjDj. Definition A point x 2 M where a divisor G 2 jDj has order greater than r is called a generalized Weierstrass point for that linear series. Farrington Weierstrass Points and Klein's Quartic 3 / 20 In this notation an ordinary Weierstrass point corresponds to the case n = 1. Background Klein's Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas High-Order Weierstrass Points Definition If K is the canonical divisor, n is an integer ≥ 2, and jDj = jnKj this is called a higher-order Weierstrass point, or n-Weierstrass point. Farrington Weierstrass Points and Klein's Quartic 4 / 20 Background Klein's Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas High-Order Weierstrass Points Definition If K is the canonical divisor, n is an integer ≥ 2, and jDj = jnKj this is called a higher-order Weierstrass point, or n-Weierstrass point. In this notation an ordinary Weierstrass point corresponds to the case n = 1. Farrington Weierstrass Points and Klein's Quartic 4 / 20 If ff1;:::; fg g is a basis for the linear space L(D), then the zeros g(g+1)=2 of W (f1;:::; fg )(dz) are the generalized Weierstrass points for the curve M for jDj. The mutiplicity of the zeros will be called the weight of the Weierstrass points, denoted w(D; x). Background Klein's Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Wronskian Matrices Let M be an algebraic curve of genus g ≥ 2. The Wronskian function is defined for a set of functions in a local coordinate z, ff1(z);:::; fr (z)g, as the determinant f1(z) ··· fr (z) df1 (z) ··· dfr (z) W (f ;:::; f ) = dz dz 1 r . d r−1 d r−1 ( dz ) f1(z) ··· ( dz ) fr (z) Farrington Weierstrass Points and Klein's Quartic 5 / 20 The mutiplicity of the zeros will be called the weight of the Weierstrass points, denoted w(D; x). Background Klein's Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Wronskian Matrices Let M be an algebraic curve of genus g ≥ 2. The Wronskian function is defined for a set of functions in a local coordinate z, ff1(z);:::; fr (z)g, as the determinant f1(z) ··· fr (z) df1 (z) ··· dfr (z) W (f ;:::; f ) = dz dz 1 r . d r−1 d r−1 ( dz ) f1(z) ··· ( dz ) fr (z) If ff1;:::; fg g is a basis for the linear space L(D), then the zeros g(g+1)=2 of W (f1;:::; fg )(dz) are the generalized Weierstrass points for the curve M for jDj. Farrington Weierstrass Points and Klein's Quartic 5 / 20 Background Klein's Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Wronskian Matrices Let M be an algebraic curve of genus g ≥ 2. The Wronskian function is defined for a set of functions in a local coordinate z, ff1(z);:::; fr (z)g, as the determinant f1(z) ··· fr (z) df1 (z) ··· dfr (z) W (f ;:::; f ) = dz dz 1 r . d r−1 d r−1 ( dz ) f1(z) ··· ( dz ) fr (z) If ff1;:::; fg g is a basis for the linear space L(D), then the zeros g(g+1)=2 of W (f1;:::; fg )(dz) are the generalized Weierstrass points for the curve M for jDj. The mutiplicity of the zeros will be called the weight of the Weierstrass points, denoted w(D; x). Farrington Weierstrass Points and Klein's Quartic 5 / 20 Hurwitz used this fact to prove that M can only have finitely many automorphisms. Lewittes proved that if a nontrivial automorphism fixes at least 5 points, then all the fixed points are Weierstrass points. Mumford has suggested that n-Weierstrass points will play an analogous role for curves of genus g ≥ 2 as n-torsion points play for elliptic curves. Background Klein's Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Weierstrass points and Automorphisms The automorphisms of M fix as a set the Weierstrass points. Farrington Weierstrass Points and Klein's Quartic 6 / 20 Lewittes proved that if a nontrivial automorphism fixes at least 5 points, then all the fixed points are Weierstrass points. Mumford has suggested that n-Weierstrass points will play an analogous role for curves of genus g ≥ 2 as n-torsion points play for elliptic curves. Background Klein's Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Weierstrass points and Automorphisms The automorphisms of M fix as a set the Weierstrass points. Hurwitz used this fact to prove that M can only have finitely many automorphisms. Farrington Weierstrass Points and Klein's Quartic 6 / 20 Mumford has suggested that n-Weierstrass points will play an analogous role for curves of genus g ≥ 2 as n-torsion points play for elliptic curves. Background Klein's Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Weierstrass points and Automorphisms The automorphisms of M fix as a set the Weierstrass points. Hurwitz used this fact to prove that M can only have finitely many automorphisms. Lewittes proved that if a nontrivial automorphism fixes at least 5 points, then all the fixed points are Weierstrass points. Farrington Weierstrass Points and Klein's Quartic 6 / 20 Background Klein's Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Weierstrass points and Automorphisms The automorphisms of M fix as a set the Weierstrass points. Hurwitz used this fact to prove that M can only have finitely many automorphisms. Lewittes proved that if a nontrivial automorphism fixes at least 5 points, then all the fixed points are Weierstrass points. Mumford has suggested that n-Weierstrass points will play an analogous role for curves of genus g ≥ 2 as n-torsion points play for elliptic curves. Farrington Weierstrass Points and Klein's Quartic 6 / 20 Background Klein's Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Key Contributions from Accola \On Generalized Weierstrass Points on Riemann Surfaces" in Modular Forms in Analysis and Number Theory (1983) Topics in the Theory of Riemann Surfaces (1994) Farrington Weierstrass Points and Klein's Quartic 7 / 20 X has 168 automorphisms, the maximum for a curve of genus 3. Let F be the map of X onto its orbits under the 168 automorphisms. The fixed points of the automorphisms correspond to the branch points of F of which there are: 24 points of multiplicity 7, 56 points of multiplicity 3, and 84 points of multiplicity 2. Background Klein's Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Klein's Quartic Curve Klein's quartic, X, (also known as the modular curve X (7)) is the genus 3 curve canonically modelled by the equation XY 3 + YZ 3 + ZX 3 = 0: Farrington Weierstrass Points and Klein's Quartic 8 / 20 Let F be the map of X onto its orbits under the 168 automorphisms. The fixed points of the automorphisms correspond to the branch points of F of which there are: 24 points of multiplicity 7, 56 points of multiplicity 3, and 84 points of multiplicity 2. Background Klein's Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Klein's Quartic Curve Klein's quartic, X, (also known as the modular curve X (7)) is the genus 3 curve canonically modelled by the equation XY 3 + YZ 3 + ZX 3 = 0: X has 168 automorphisms, the maximum for a curve of genus 3. Farrington Weierstrass Points and Klein's Quartic 8 / 20 Background Klein's Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Klein's Quartic Curve Klein's quartic, X, (also known as the modular curve X (7)) is the genus 3 curve canonically modelled by the equation XY 3 + YZ 3 + ZX 3 = 0: X has 168 automorphisms, the maximum for a curve of genus 3. Let F be the map of X onto its orbits under the 168 automorphisms. The fixed points of the automorphisms correspond to the branch points of F of which there are: 24 points of multiplicity 7, 56 points of multiplicity 3, and 84 points of multiplicity 2. Farrington Weierstrass Points and Klein's Quartic 8 / 20 " is an order 7 automorphism.
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