CUBIC EQUATIONS FOR THE HYPERELLIPTIC LOCUS

SAMUEL
 GRUSHEVSKY

doi:ajm/1087840914

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Home > Journals > Asian J. Math. > Volume 8 > Issue 1 > Article

January, 2004 CUBIC EQUATIONS FOR THE HYPERELLIPTIC LOCUS

SAMUEL GRUSHEVSKY

Asian J. Math. 8(1): 161-172 (January, 2004).

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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.