Curves of genus with split Jacobian

Author:
Robert M. Kuhn

Journal:
Trans. Amer. Math. Soc. **307** (1988), 41-49

MSC:
Primary 14H40; Secondary 11G10

DOI:
https://doi.org/10.1090/S0002-9947-1988-0936803-3

MathSciNet review:
936803

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We say that an algebraic curve has *split jacobian* if its jacobian is isogenous to a product of elliptic curves. If is a curve of genus , and a map from to an elliptic curve, then has split jacobian. It is not true that a complement to in the jacobian of is uniquely determined, but, under certain conditions, there is a canonical choice of elliptic curve and algebraic , and we give an algorithm for finding that curve. The construction works in any characteristic other than two. Applications of the algorithm are given to give explicit examples in characteristics 0 and .

**[1]**A. Krazer,*Lehrbuch der Thetafunctionen*Chelsea, New York, 1970.**[2]**L. Moret-Bailley,*Familles de courbes et de variétés abeliennes sur*, Asterisque**86**(1981), 109-124.**[3]**T. Hayashida and M. Nishi,*Existence of curves of genus two on a product of two elliptic curves*, J. Math. Soc. Japan**17**(1965), 1-16. MR**0201434 (34:1318)****[4]**T. Hayashida,*A class number associated with the product of an elliptic curve with itself*, J. Math. Soc. Japan**20**(1968), 26-43. MR**0233804 (38:2125)****[5]**T. Ibukiyama, T. Katsura and F. Oort,*Supersingular curves of genus two and class numbers*, Compositio Math.**57**(1986), 127-152. MR**827350 (87f:14026)****[6]**D. Mumford,*Tata lectures on theta*, Birkhäuser, Boston, Mass., 1984. MR**742776 (86b:14017)****[7]**T. Shioda,*Some remarks on abelian varieties*, J. Fac. Sci. Univ. Tokyo**24**(1977), 11-21. MR**0450289 (56:8585)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
14H40,
11G10

Retrieve articles in all journals with MSC: 14H40, 11G10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1988-0936803-3

Article copyright:
© Copyright 1988
American Mathematical Society