# Determinant Expressions for Abelian Functions in Genus Two

@article{nishi2001DeterminantEF, title={Determinant Expressions for Abelian Functions in Genus Two}, author={Yoshihiro {\^O}nishi}, journal={arXiv: Number Theory}, year={2001} }

In this paper we generalize the formula of Frobenius-Stickelberger and the formula of Kiepert type to the genus-two case.

#### 17 Citations

Determinantal Expressions for Hyperelliptic Functions in Genus Three

- Mathematics
- 2001

In this paper we generalize the formula of Frobenius-Stickelberger and the formula of Kiepert to genus-three case. The latter is well-known determinant expression for any division polynomial of any… Expand

Determinant Expressions in Abelian Functions for Purely Trigonal Curves of Degree Four

- Mathematics
- 2005

This paper gives a natural extension of Frobenius-Stickelberger formula and Kiepert formula to Abelian functions for "Purely Trigonal Curves", especially, of degree four. A description on the theory… Expand

ABELIAN FUNCTIONS FOR TRIGONAL CURVES OF DEGREE FOUR AND DETERMINANTAL FORMULAE IN PURELY TRIGONAL CASE

- Mathematics
- 2005

We give the Frobenius–Stickelberger-type and Kiepert-type determinantal formulae for purely trigonal curves of genus three. We explain also general theory of Abelian functions for any trigonal curves… Expand

DETERMINANT EXPRESSIONS FOR HYPERELLIPTIC FUNCTIONS

- Mathematics
- Proceedings of the Edinburgh Mathematical Society
- 2005

Abstract In this paper we give an elegant generalization of the formula of Frobenius–Stickelberger from elliptic curve theory to all hyperelliptic curves. A formula of Kiepert type is also obtained… Expand

Determinant Expressions for Hyperelliptic Functions (with an Appendix by Shigeki Matsutani)

- Mathematics
- 2001

In this paper we give quite pretty generalization of the formula of Frobenius-Stickelberger to all hyperelliptic curves. The formula of Kiepert type is also obtained by limiting process from this… Expand

On a Generalized Frobenius–Stickelberger Addition Formula

- Mathematics
- 2003

In this Letter we obtain a generalization of the Frobenius–Stickelberger addition formula for the (hyperelliptic) σ-function of a genus 2 curve in the case of three vector-valued variables. The… Expand

Division polynomials and canonical local heights on hyperelliptic Jacobians

- Mathematics
- 2011

We generalize the division polynomials of elliptic curves to hyperelliptic Jacobians over the complex numbers. We construct them by using the hyperelliptic sigma function. Using the division… Expand

Recursion relation of hyperelliptic PSI-functions of genus two

- Mathematics, Physics
- 2001

A recursion relation of hyperelliptic ψ functions of genus two, which was derived by D. G. Cantor (J. reine angew. Math. 447 (1994) 91–145), is studied. As Cantor's approach is algebraic, another… Expand

Genus Two Quasi--Siegel Modular Forms and Gromov--Witten Theory of Toric Calabi-Yau Threefolds

- Mathematics, Physics
- 2019

We first develop theories of differential rings of quasi-Siegel modular and quasi-Siegel Jacobi forms for genus two. Then we apply them to the Eynard-Orantin topological recursion of certain local… Expand

The hyperelliptic ζ–function and the integrable massive Thirring model

- Mathematics
- Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
- 2003

We provide a treatment of algebro–geometric solutions of the classical massive Thirring system in the framework of the Weierstrass–Klein theory of hyperelliptic functions. We show that the equations… Expand

#### References

SHOWING 1-10 OF 36 REFERENCES

On Gunning’s Prime Form in Genus 2

- Mathematics
- Canadian Mathematical Bulletin
- 2002

Abstract Using a classical generalization of Jacobi’s derivative formula, we give an explicit expression for Gunning’s prime form in genus 2 in terms of theta functions and their derivatives.

Determinant Expressions for Hyperelliptic Functions (with an Appendix by Shigeki Matsutani)

- Mathematics
- 2001

In this paper we give quite pretty generalization of the formula of Frobenius-Stickelberger to all hyperelliptic curves. The formula of Kiepert type is also obtained by limiting process from this… Expand

Theta Functions on Riemann Surfaces

- Mathematics
- 1973

Riemann's theta function.- The prime-form.- Degenerate Riemann surfaces.- Cyclic unramified coverings.- Ramified double coverings.- Bordered Riemann surfaces.

A generalization of Jacobi's derivative formula to dimension two.

- Mathematics
- 1988

Various 19th-century authors provided generalizations of (0. 2) to g-dimensional theta functions, expressing the Jacobian of g distinct odd theta functions at zero explicitly s a rational function in… Expand

A Generalization of a Formula of Eisenstein

- Mathematics
- 1991

where E[p\ denotes the non-zero p-torsion of E. (Equation (0.1) was probably known to Eisenstein: he published a similar formula. See [1] for a proof of (0.1) and related history.) The automorphism… Expand

Complex Multiplication Formulae for Hyperelliptic Curves of Genus Three

- Mathematics
- 1998

p.387, `.5, “[1]” should be “[2]”. p.389, `.−12, “ d dt`+1 ” should be “ d ` dt` ”. p.389, `.−9, “(i, n1 + · · · + nj−1 + `)-entry” should be “(n1 + · · · + nj−1 + `, i)-entry with 1 ≤ ` 5 nj”.… Expand

Zur Theorie der elliptischen Functionen.

- Mathematics
- 1877

Uie merkwürdige Formel, welche Herr Hermite in einer kürzlich erschienenen Notiz über elliptische Functionen mitgetheilt hat (dieses Journal Bd. 82, S. 343), veranlasst uns, auf den Zusammenhang… Expand

A Course of Modern Analysis

- Physics, Computer Science
- Nature
- 1916

The volume now gives a somewhat exhaustive account of the various ramifications of the subject, which are set out in an attractive manner and should become indispensable, not only as a textbook for advanced students, but as a work of reference to those whose aim is to extend the knowledge of analysis. Expand

Wirkliche Ausführung der ganzzahligen Multiplication der elliptischen Functionen.

- Mathematics
- 1873

JL/as Problem der rationalen Multiplication der elliptischen Functionen ist der Theorie nach vollständig gelöst, und zwar sind für seine Lösung besonders zwei Methoden bekannt, von denen die eine auf… Expand