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# Asian Journal of Mathematics

## Volume 20 (2016)

### Number 2

### Torsion points on Jacobian varieties via Anderson’s $p$-adic soliton theory

Pages: 323 – 352

DOI: http://dx.doi.org/10.4310/AJM.2016.v20.n2.a5

#### Authors

#### Abstract

Anderson introduced a $p$-adic version of soliton theory. He then applied it to the Jacobian variety of a cyclic quotient of a Fermat curve and showed that torsion points of certain prime order lay outside of the theta divisor. In this paper, we evolve his theory further. As an application, we get a stronger result on the intersection of the theta divisor and torsion points on the Jacobian variety for more general curves. New examples are discussed as well. A key new ingredient is a map connecting the $p$-adic loop group and the formal group.

#### Keywords

Sato Grassmannian, $p$-adic tau function, $p$-adic loop group, formal group

#### 2010 Mathematics Subject Classification

Primary 11G20. Secondary 14H40, 14K25, 37J35.

Published 18 March 2016