Equations for the Jacobian

of a hyperelliptic curve

Author:
Paul van Wamelen

Journal:
Trans. Amer. Math. Soc. **350** (1998), 3083-3106

MSC (1991):
Primary 14H40; Secondary 14H42

MathSciNet review:
1432144

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Abstract | References | Similar Articles | Additional Information

Abstract: We give an explicit embedding of the Jacobian of a hyperelliptic curve, , into projective space such that the image is isomorphic to the Jacobian over the splitting field of . The embedding is a modification of the usual embedding by theta functions with half integer characteristics.

**[Fly90]**Eugene Victor Flynn,*The Jacobian and formal group of a curve of genus 2 over an arbitrary ground field*, Math. Proc. Cambridge Philos. Soc.**107**(1990), no. 3, 425–441. MR**1041476**, 10.1017/S0305004100068729**[GG93]**Daniel M. Gordon and David Grant,*Computing the Mordell-Weil rank of Jacobians of curves of genus two*, Trans. Amer. Math. Soc.**337**(1993), no. 2, 807–824. MR**1094558**, 10.1090/S0002-9947-1993-1094558-0**[Gra85]**D. R. Grant.*Theta Functions and Division Points on Abelian Varieties of Dimension Two*. PhD thesis, MIT, 1985.**[Gra90]**David Grant,*Formal groups in genus two*, J. Reine Angew. Math.**411**(1990), 96–121. MR**1072975**, 10.1515/crll.1990.411.96**[Har77]**Robin Hartshorne,*Algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR**0463157****[Iit82]**Shigeru Iitaka,*Algebraic geometry*, Graduate Texts in Mathematics, vol. 76, Springer-Verlag, New York-Berlin, 1982. An introduction to birational geometry of algebraic varieties; North-Holland Mathematical Library, 24. MR**637060****[Lan82]**Serge Lang,*Introduction to algebraic and abelian functions*, 2nd ed., Graduate Texts in Mathematics, vol. 89, Springer-Verlag, New York-Berlin, 1982. MR**681120****[LB92]**Herbert Lange and Christina Birkenhake,*Complex abelian varieties*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 1992. MR**1217487****[Mil86a]**Gary Cornell and Joseph H. Silverman (eds.),*Arithmetic geometry*, Springer-Verlag, New York, 1986. Papers from the conference held at the University of Connecticut, Storrs, Connecticut, July 30–August 10, 1984. MR**861969****[Mil86b]**Gary Cornell and Joseph H. Silverman (eds.),*Arithmetic geometry*, Springer-Verlag, New York, 1986. Papers from the conference held at the University of Connecticut, Storrs, Connecticut, July 30–August 10, 1984. MR**861969****[Mum83]**David Mumford,*Tata lectures on theta. I*, Progress in Mathematics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. MR**688651****[Mum84]**David Mumford,*Tata lectures on theta. II*, Progress in Mathematics, vol. 43, Birkhäuser Boston, Inc., Boston, MA, 1984. Jacobian theta functions and differential equations; With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura. MR**742776****[Pil90]**J. Pila,*Frobenius maps of abelian varieties and finding roots of unity in finite fields*, Math. Comp.**55**(1990), no. 192, 745–763. MR**1035941**, 10.1090/S0025-5718-1990-1035941-X

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Additional Information

**Paul van Wamelen**

Affiliation:
Department of Mathematics, University of California, San Diego, San Diego, California 92093

Address at time of publication:
Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803-4918

Email:
wamelen@math.lsu.edu

DOI:
https://doi.org/10.1090/S0002-9947-98-02056-X

Keywords:
Jacobian,
hyperelliptic curve,
theta function,
theta constant,
Thomae's identity

Received by editor(s):
December 5, 1995

Article copyright:
© Copyright 1998
American Mathematical Society