Determinant Expressions for Hyperelliptic Functions

We develop the theory of generalized Weierstrass σ-and ℘-functions defined on a general trigonal curve of genus three. In particular, we give a list of the associated partial differential equations satisfied by the ℘-functions, a proof that the coefficients of the power series expansion of the σ-function are polynomials of coefficients of the defining equation of the curve, and the derivation of two addition formulae.

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“…Let B(D) be the Brill-Noether matrix for an effective divisor D of C. Then it is well known that (see for example [24] or [26])…”

Section: Abelian Functions For Trigonal Curves 23mentioning

confidence: 99%

Abelian Functions for Trigonal Curves of Genus Three

Eilbeck

Enolskii

Matsutani³

et al. 2010

International Mathematics Research Notices

104

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“…Remark 6.12. -Given the connection between Theorem 6.3 and theory of Kac and van Moerbeke provided by Theorem 6.11, we note in addition: since it is known that for certain multi-indices γ = (γ 1 , ..., γ g ) and for [45,37], by differentiating the equation, the σ function satisfies (k = 1, . .…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

“…Thus it is a natural generalization of the classical Kiepert formula, or the division polynomial. By taking limits of Proposition 4.2 along the Abelian variables, we can give an expression for ψ n in terms of φ i 's in R g [45,Theorem 9.3]. In [38], we proved the following: Theorem 6.1.…”

Section: Division Polynomials ψ 2nmentioning

confidence: 99%

“…Following [45,37], we introduce a multi-index n . For n with 1 n < g, we let n be the set of positive integers i such that n + 1 i g with i ≡ n + 1 mod 2.…”

Section: Sigma Function and Its Derivativesmentioning

confidence: 99%

“…Appendix). The finite-point condition was investigated by Cantor and Ônishi [45,11] using the division polynomial ψ 2N , an element of R g . Similarly, we investigate the periodic solutions of the Toda lattice.…”

Section: Periodicity Of the Toda-lattice Solutionmentioning

confidence: 99%

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Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function

Kodama¹,

Matsutani²,

Превиато³

2013

Ann. inst. Fourier

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M. Toda in 1967 (J. Phys. Soc. Japan, 22 and 23) considered a lattice model with exponential interaction and proved, as suggested by the Fermi-Pasta-Ulam experiments in the 1950s, that it has exact periodic and soliton solutions. The Toda lattice, as it came to be known, was then extensively studied as one of the completely integrable (differential-difference) non-linear equations which admit exact solutions in terms of theta functions of hyperelliptic curves. In this paper, we extend Toda's original approach to give hyperelliptic solutions of the Toda lattice in terms of hyperelliptic Kleinian (sigma) functions for arbitrary genus. The key identities are given by generalized addition formulae for the hyperelliptic sigma functions (J.C. Eilbeck et al., J. reine angew. Math. 619, 2008). We then show that periodic (in the discrete variable, a standard term in the Toda lattice theory) solutions of the Toda lattice correspond to the zeros of Kiepert-Brioschi's division polynomials, and note these are related to solutions of Poncelet's closure problem. One feature of our solution is that the hyperelliptic curve is related in a non-trivial way to the one previously used.

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Faltings’ Delta-Invariant of a Hyperelliptic Riemann Surface

Jong

2005

Number Fields and Function Fields—Two Parallel Worlds

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In this note we give a closed formula for Faltings' delta-invariant of a hyperelliptic Riemann surface. ROBIN DE JONGtwo natural metrics on the determinant of the Hodge bundle. For P on X, not a Weierstrass point, and z a local coordinate about P , we putFurther we let W z (ω)(P ) be the Wronskian at P in z of an orthonormal basis {ω 1 , . . . , ω g } of the differentials H 0 (X, Ω 1 X ) provided with the standard hermitian inner product (ω, η) → i 2 X ω ∧ η. We define an invariant T (X) of X by just two Weierstrass points. In the case g = 2 it can be shown that G ′ (W, W ′ ) 2 = 2 1/4 · ϕ 2 (X) −3/64 ·

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Determinant Expressions for Hyperelliptic Functions

Cited by 35 publications

References 19 publications

Abelian Functions for Trigonal Curves of Genus Three

Abelian Functions for Trigonal Curves of Genus Three

Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function

Faltings’ Delta-Invariant of a Hyperelliptic Riemann Surface

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