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 ∈ M where a divisor G ∈ |D| 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 |D| be a base-point free complete linear series on M with r = dim|D| and d = deg|D|.

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 |D| be a base-point free complete linear series on M with r = dim|D| and d = deg|D|.

Deﬁnition A point x ∈ M where a divisor G ∈ |D| 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 |D| = |nK| 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 |D| = |nK| 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 {f1,..., fg } 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 |D|. 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, {f1(z),..., fr (z)}, 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, {f1(z),..., fr (z)}, 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 {f1,..., fg } 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 |D|.

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, {f1(z),..., fr (z)}, 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 {f1,..., fg } 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 |D|. 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

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 ε is an order 7 automorphism. ε and its conjugates each fix 3 points. We will call these 24 points a-points. σ is an order 3 automorphism. σ and its conjugates each fix 2 points. We will call these 56 points b-points. τ is an order 2 automorphism. τ and its conjugates each fix 4 points. We will call these 84 points c-points.

Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Automorphisms

We deﬁne three generators for Aut(X). Let ζ = e2πi/7, 2 5 6 −1 x1 = ζ + ζ and ω1 = (ζ + ζ ) .       ζ 0 0 0 1 0 x1 ω1 1 5 ε = 0 ζ 0 σ = 0 0 1 τ = ω1 1 x1  0 0 1 1 0 0 1 x1 ω1

Farrington Weierstrass Points and Klein’s Quartic 9 / 20 σ is an order 3 automorphism. σ and its conjugates each ﬁx 2 points. We will call these 56 points b-points. τ is an order 2 automorphism. τ and its conjugates each ﬁx 4 points. We will call these 84 points c-points.

Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Automorphisms

ε is an order 7 automorphism. ε and its conjugates each ﬁx 3 points. We will call these 24 points a-points.

Farrington Weierstrass Points and Klein’s Quartic 9 / 20 τ is an order 2 automorphism. τ and its conjugates each ﬁx 4 points. We will call these 84 points c-points.

Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Automorphisms

Farrington Weierstrass Points and Klein’s Quartic 9 / 20 Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Automorphisms

ε is an order 7 automorphism. ε and its conjugates each fix 3 points. We will call these 24 points a-points. σ is an order 3 automorphism. σ and its conjugates each fix 2 points. We will call these 56 points b-points. τ is an order 2 automorphism. τ and its conjugates each fix 4 points. We will call these 84 points c-points.

Farrington Weierstrass Points and Klein’s Quartic 9 / 20 Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Visualizing Klein’s Quartic Curve

Identify edges: 1 ↔ 6 3 ↔ 8 5 ↔ 10 7 ↔ 12 9 ↔ 14 11 ↔ 2 13 ↔ 4

Farrington Weierstrass Points and Klein’s Quartic 10 / 20 The 24 a-points of X are the ordinary Weierstrass points, each with weight 1.

Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Ordinary Weierstrass Points

As a curve of genus 3, X has at most (g − 1)g(g + 1) = 24 ordinary Weierstrass points, counted with weight.

Farrington Weierstrass Points and Klein’s Quartic 11 / 20 Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Ordinary Weierstrass Points

As a curve of genus 3, X has at most (g − 1)g(g + 1) = 24 ordinary Weierstrass points, counted with weight. The 24 a-points of X are the ordinary Weierstrass points, each with weight 1.

Farrington Weierstrass Points and Klein’s Quartic 11 / 20 In the special case where the pluricanonical divisor, mK, is linearly equivalent to a multiple of the point x ∈ M in question, we have more precise results about the weights of the higher-order Weierstrass points.

Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Weights of Ordinary Weierstrass Points

Theorem (Accola) Let |D| be a complete linear series without ﬁxed points of index i(D). If x ∈ M then

w(D, x) ≤ (g − i(D))(g − i(D) + 1)/2.

If we have equality, then M is hyperelliptic.

Farrington Weierstrass Points and Klein’s Quartic 12 / 20 Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Weights of Ordinary Weierstrass Points

Theorem (Accola) Let |D| be a complete linear series without ﬁxed points of index i(D). If x ∈ M then

w(D, x) ≤ (g − i(D))(g − i(D) + 1)/2.

If we have equality, then M is hyperelliptic. In the special case where the pluricanonical divisor, mK, is linearly equivalent to a multiple of the point x ∈ M in question, we have more precise results about the weights of the higher-order Weierstrass points.

Farrington Weierstrass Points and Klein’s Quartic 12 / 20 Corollary Let mK ≡ m(2g − 2)x as above. Then w(nK, x) = w(vK, x) for n, v ≥ 2 and n ≡ v (mod m).

Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Weights of Higher-Order Weierstrass Points

Theorem (Accola) Suppose for x ∈ M we have mK ≡ m(2g − 2)x. Then (1) If m = 1, l = 2, 3,... , we have w(lK, x) = g + w(K, x). (2) If m ≥ 2, and l = 1, 2,... we have w(lmK, x) = w((lm + 1)K, x) = g + w(K, x). (3) If m ≥ 3, l = 1, 2,... and k = 2, 3,..., m − 1 we have w((lm + k)K, x) = w(kK, x). (4) If m ≥ 4, l = 1, 2,... and k = 2, 3,..., m − 1 we have w((lm − k + 1)K, x) = w(kK, x).

Farrington Weierstrass Points and Klein’s Quartic 13 / 20 Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Weights of Higher-Order Weierstrass Points

Corollary Let mK ≡ m(2g − 2)x as above. Then w(nK, x) = w(vK, x) for n, v ≥ 2 and n ≡ v (mod m).

Farrington Weierstrass Points and Klein’s Quartic 13 / 20 Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Linear Equivalence to a Multiple of a Point

Theorem (Accola) Pt−1 j Let T be an automorphism of M of order t, j=0 T M = W the orbit space, ϕ : M → W the natural map and x ∈ M a ﬁxed point of T . If W has genus 0, then tK ≡ t(2g − 2)x.

Farrington Weierstrass Points and Klein’s Quartic 14 / 20 Let T have s ≥ 2 ﬁxed points, A1,..., As ∈ M, with images a1,..., as ∈ W under ϕ. Theorem Let T be an involution, then, with notation as above, 1 w(nK , A ) = (−2 + s)s. M l 8

Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Involutions

Suppose however that W is not genus 0.

Farrington Weierstrass Points and Klein’s Quartic 15 / 20 Theorem Let T be an involution, then, with notation as above, 1 w(nK , A ) = (−2 + s)s. M l 8

Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Involutions

Suppose however that W is not genus 0. Let T have s ≥ 2 ﬁxed points, A1,..., As ∈ M, with images a1,..., as ∈ W under ϕ.

Farrington Weierstrass Points and Klein’s Quartic 15 / 20 Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Involutions

Suppose however that W is not genus 0. Let T have s ≥ 2 ﬁxed points, A1,..., As ∈ M, with images a1,..., as ∈ W under ϕ. Theorem Let T be an involution, then, with notation as above, 1 w(nK , A ) = (−2 + s)s. M l 8

Farrington Weierstrass Points and Klein’s Quartic 15 / 20 Corollary Let T be an automorphism of order 3 on a compact Riemann surface M of genus g = 3 with s = 2 ﬁxed points. Then the ﬁxed points of T are not n-Weierstrass points for any n ≥ 2.

Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Automorphisms with 2 ﬁxed points

Theorem Let T be an automorphism of prime order t ≥ 3 on a compact Riemann surface M of genus g ≥ 3 with s = 2 ﬁxed points. If n ≡ 0 or 1 (mod t) then the ﬁxed points of T are not n-Weierstrass points.

Farrington Weierstrass Points and Klein’s Quartic 16 / 20 Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Automorphisms with 2 ﬁxed points

Corollary Let T be an automorphism of order 3 on a compact Riemann surface M of genus g = 3 with s = 2 ﬁxed points. Then the ﬁxed points of T are not n-Weierstrass points for any n ≥ 2.

Farrington Weierstrass Points and Klein’s Quartic 16 / 20 Also, let w0(n) be the combined weight as n-Weierstrass points of the general points of X that are on separate orbits for the automorphism group. Then

2 168w0(n) + 84wc (n) + 56wb(n) + 24wa(n) = (2n − 1) · 12

using the formula (2n − 1)2(g − 1)2g for the total weight of all n-Weierstrass points

Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Total Weights of n-Weierstrass points

Let wa(n) be the weight of the a-points as n-Weierstrass points, wb(n) the weight of the b-points and wc (n) the weight of the c-points.

Farrington Weierstrass Points and Klein’s Quartic 17 / 20 Then

2 168w0(n) + 84wc (n) + 56wb(n) + 24wa(n) = (2n − 1) · 12

using the formula (2n − 1)2(g − 1)2g for the total weight of all n-Weierstrass points

Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Total Weights of n-Weierstrass points

Let wa(n) be the weight of the a-points as n-Weierstrass points, wb(n) the weight of the b-points and wc (n) the weight of the c-points. Also, let w0(n) be the combined weight as n-Weierstrass points of the general points of X that are on separate orbits for the automorphism group.

Farrington Weierstrass Points and Klein’s Quartic 17 / 20 Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Total Weights of n-Weierstrass points

2 168w0(n) + 84wc (n) + 56wb(n) + 24wa(n) = (2n − 1) · 12

using the formula (2n − 1)2(g − 1)2g for the total weight of all n-Weierstrass points

Farrington Weierstrass Points and Klein’s Quartic 17 / 20 Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Main Result

Theorem The ﬁxed points of the automorphisms of Klein’s quartic curve appear as higher-order Weierstrass points as follows: 1 a − points: Let n ≥ 2, then

1 If n ≡ 0 or 1 (mod 7), then wa(n) = 4 2 If n ≡ 2 or 6 (mod 7), then wa(n) = 1. 3 If n ≡ 3 or 5 (mod 7), then wa(n) = 2. 4 If n ≡ 4 (mod 7), then wa(n) = 0.

2 b − points: wb(n) = 0 for all n.

3 c − points: wc (n) = 1 for all n ≥ 2.

Farrington Weierstrass Points and Klein’s Quartic 18 / 20 Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Weights of n-Weierstrass points for Klein’s Quartic

n a-points b-points c-points Other points Total (2n − 1)212 1 1 × 24 0 0 0 24 2 1 × 24 0 1 × 84 0 108 3 2 × 24 0 1 × 84 168 300 4 0 0 1 × 84 504 588 5 2 × 24 0 1 × 84 840 972 6 1 × 24 0 1 × 84 1344 1452 7 4 × 24 0 1 × 84 1848 2028 8 4 × 24 0 1 × 84 2520 2700

Farrington Weierstrass Points and Klein’s Quartic 19 / 20 Background Klein’s Quartic Curve Weights of n- Weierstrass Points Applying the Weight Formulas Thanks!

Special thanks to Emma Previato for her guidance during this work and the invitation to honor Robert Accola today, and to the NSA for supporting this conference.

Farrington Weierstrass Points and Klein’s Quartic 20 / 20