Klein's curve

@article{Braden2010KleinsC,
  title={Klein's curve},
  author={Harry W. Braden and Timothy P. Northover},
  journal={Journal of Physics A},
  year={2010},
  volume={43},
  pages={434009}
}
Riemann surfaces with symmetries arise in many studies of integrable systems. We illustrate new techniques in investigating such surfaces by means of an example. By giving a homology basis well adapted to the symmetries of Klein's curve, presented as a plane curve, we derive a new expression for its period matrix. This is explicitly related to the hyperbolic model and results of Rauch and Lewittes. 

Figures and Tables from this paper

Bring's Curve: its Period Matrix and the Vector of Riemann Constants
Bring's curve is the genus 4 Riemann surface with automorphism group of maximal size, S5. Riera and Rodr guez have provided the most detailed study of the curve thus far via a hyperbolic model. We
Computation of the topological type of a real Riemann surface
TLDR
An algorithm for the computation of the topological type of a real compact Riemann surface associated to an algebraic curve is presented, i.e., its genus and the properties of the set of fixed points of the anti-holomorphic involution, and whether this set divides the surface into one or two connected components.
Period matrices of some hyperelliptic Riemann surfaces
In this paper, we calculate period matrices of algebraic curves defined by w = z(z − 1)(z − a21)(z 2 − a22) · · · (z 2 − a g−1) for any g ≥ 2 and a1, a2, . . . , ag−1 ∈ R with 1 < a1 < a2 < · · · <
A canonical form for a symplectic involution
We present a canonical form for a symplectic involution , $$S^2=\mathop {\mathrm{Id}}\nolimits $$S2=Id; the construction is algorithmic. Application is made in the Riemann surface setting.
Construction of period matrices by algebraic techniques
In this paper, we show how to construct in an algebraic way period matrices of compact Riemann surfaces which permit automorphisms.
The period matrix of the hyperelliptic curve $w^2=z^{2g+1}-1$
A geometric algorithm is introduced for finding a symplectic basis of the first integral homology group of a compact Riemann surface, which is a $p$-cyclic covering of ${\mathbb C} P^1$ branched over
Spectral curves are transcendental
  • H. Braden
  • Mathematics
    Letters in Mathematical Physics
  • 2021
Some arithmetic properties of spectral curves are discussed: the spectral curve, for example, of a charge $$n\ge 2$$ n ≥ 2 Euclidean BPS monopole is not defined over $$\overline{\mathbb {Q}}$$ Q ¯ if
Algorithmic approaches to Siegel's fundamental domain
Siegel determined a fundamental domain using the Minkowski reduction of quadratic forms. He gave all the details concerning this domain for genus 1. It is the determination of the Minkowski

References

SHOWING 1-10 OF 24 REFERENCES
MODULI FOR SPECIAL RIEMANN SURFACES OF GENUS 2
Introduction. This paper is an investigation of moduli for the 2-complex parameter family of Riemann surfaces of genus 2 that admit of automorphisms (conformal self-homeomorphisms) of order 2 other
Geodesics, periods, and equations of real hyperelliptic curves
In this paper we start a new approach to the uniformization problem of Riemann surfaces and algebraic curves by means of computational procedures. The following question is studied: Given a compact
A nontrivial algebraic cycle in the Jacobian variety of the Klein quartic
We prove some value of the harmonic volume for the Klein quartic C is nonzero modulo $$\frac{1}{2}{\mathbb{Z}}$$ , using special values of the generalized hypergeometric function 3F2. This result
Klein's surface of genus three and associated theta constants
This surface is famous because the conformal automorphism group Aut(R) has order 168 = 84(3 ―1), the maximum possible. In 1970, Rauch and Lewittes wrote the beautiful paper [4] in which they found
SU(2)-monopoles, curves with symmetries and Ramanujan's heritage
We develop the Ercolani-Sinha construction of SU(2) monopoles for a five-parameter family of centred charge 3 monopoles. In particular we show how to solve the transcendental constraints arising on
On charge-3 cyclic monopoles
We determine the spectral curve of charge 3 BPS su(2) monopoles with C3 cyclic symmetry. The symmetry means that the genus 4 spectral curve covers a (Toda) spectral curve of genus 2. A well adapted
On the Tetrahedrally Symmetric Monopole
We study SU(2) BPS monopoles with spectral curves of the form η3+χ(ζ6+bζ3−1) = 0. Previous work has established a countable family of solutions to Hitchin’s constraint that L2 was trivial on such a
Cyclic Monopoles, Affine Toda and Spectral Curves
We show that any cyclically symmetric monopole is gauge equivalent to Nahm data given by Sutcliffe’s ansatz, and so obtained from the affine Toda equations. Further the direction (the Ercolani-Sinha
Symmetric Monopoles
We discuss SU(2) Bogomolny monopoles of arbitrary charge k invariant under various symmetry groups. The analysis is largely in terms of the spectral curves, the rational maps, and the Nahm equations
...
...