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**Jacobian varieties.**
*(English)*
Zbl 0604.14018

Arithmetic geometry, Pap. Conf., Storrs/Conn. 1984, 167-212 (1986).

[For the entire collection see Zbl 0596.00007.]

This article in a natural continuation of the author’s survey on abelian varieties [ibid. 103-150 (1986; see the review 14028)] and represents a detailed treatment of Jacobian varieties. The paper begins with the construction and the basic properties of the Jacobian variety J(C) associated to a smooth projective curve C. Then the author studies the canonical map from the symmetric powers of C to J(C), and proves that J(C) is self-dual (as an abelian variety) and that every abelian variety is a quotient of a Jacobian variety. As an application of some properties of Jacobian varieties one shows (among other things) that the Shafarevich conjecture implies the Mordell conjecture. Finally, one computes the zeta function of a curve defined over a finite field, and one proves the Torelli theorem for curves. The paper ends with some brief, but useful, historical notes concerning the subject considered before.

This article in a natural continuation of the author’s survey on abelian varieties [ibid. 103-150 (1986; see the review 14028)] and represents a detailed treatment of Jacobian varieties. The paper begins with the construction and the basic properties of the Jacobian variety J(C) associated to a smooth projective curve C. Then the author studies the canonical map from the symmetric powers of C to J(C), and proves that J(C) is self-dual (as an abelian variety) and that every abelian variety is a quotient of a Jacobian variety. As an application of some properties of Jacobian varieties one shows (among other things) that the Shafarevich conjecture implies the Mordell conjecture. Finally, one computes the zeta function of a curve defined over a finite field, and one proves the Torelli theorem for curves. The paper ends with some brief, but useful, historical notes concerning the subject considered before.

Reviewer: L.Bădescu

### MSC:

14H40 | Jacobians, Prym varieties |

14K30 | Picard schemes, higher Jacobians |

14H10 | Families, moduli of curves (algebraic) |

14K10 | Algebraic moduli of abelian varieties, classification |