## Asian Journal of Mathematics

- Asian J. Math.
- Volume 8, Number 1 (2004), 161-172.

### CUBIC EQUATIONS FOR THE HYPERELLIPTIC LOCUS

#### Abstract

We prove a conjecture from [BK2] that the multi-dimensional vector addition formula for Baker-Akhiezer functions obtained there characterizes Jacobians among principally polarized abelian varieties. We also show that this addition formula is equivalent to Gunning's multisecant formula for the Kummer variety obtained in [Gu2].

We then use this addition formula to obtain cubic relations among theta functions that characterize the locus of hyperelliptic Jacobians among irreducible abelian varieties. In genus 3 our equations are equivalent to the vanishing of one theta-null, and thus are classical (see [M], [P]), but already for genus 4 they appear to be new.

#### Article information

**Source**

Asian J. Math., Volume 8, Number 1 (2004), 161-172.

**Dates**

First available in Project Euclid: 21 June 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.ajm/1087840914

**Mathematical Reviews number (MathSciNet)**

MR2128303

**Zentralblatt MATH identifier**

1100.14523

#### Citation

GRUSHEVSKY, SAMUEL. CUBIC EQUATIONS FOR THE HYPERELLIPTIC LOCUS. Asian J. Math. 8 (2004), no. 1, 161--172. https://projecteuclid.org/euclid.ajm/1087840914