Minimal Genus One Curves

Details Minimal Genus One Curves Minimal Genus One Curves

2010-02-02

arxiv.org/abs/1002.0451v1

Funct. Approx. Comment. Math. Volume 46, Number 1 (2012), 117-131

Mathematics

Number Theory

16 pages, 1 table

Scientific paper

10.7169/facm/2012.46.1.9

In this paper we consider genus one equations of degree n, namely a (generalised) binary quartic when n = 2, a ternary cubic when n = 3, and a pair of quaternary quadrics when n = 4. A new definition for the minimality of genus one equations of degree n is introduced. The advantage of this definition is that it does not depend on invariant theory of genus one curves. We prove that this definition coincides with the classical definition of minimality when n <= 4. As an application, we give a new proof for the existence of global minimal genus one equations over number fields of class number 1.

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