Counting Models of Genus One Curves

Details Counting Models of Genus One Curves Counting Models of Genus One Curves

2010-02-02

arxiv.org/abs/1002.0467v1

Mathematical Proceedings of the Cambridge Philosophical Society, 2011, volume 150, issue 03, pp. 399-417

Mathematics

Number Theory

22 pages

Scientific paper

10.1017/S0305004110000666

Let C be a soluble smooth genus one curve over a Henselian discrete valuation field. There is a unique minimal Weierstrass equation defining C up to isomorphism. In this paper we consider genus one equations of degree n defining C, namely a (generalised) binary quartic when n = 2, a ternary cubic when n = 3, and a pair of quaternary quadrics when n = 4. In general, minimal genus one equations of degree n are not unique up to isomorphism. We explain how the number of minimal genus one equations of degree n varies according to the Kodaira symbol of the Jacobian of C. Then we count these equations up to isomorphism over a number field of class number 1.

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