I‑AIMS Institute for Artificial Intelligence
and Mathematical Sciences
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Rational points in the moduli space of genus two

Lubjana Beshaj, Rubén A. Hidalgo, Serge Kruk, Andreas Malmendier, Saúl Quispe, Tanush Shaska

I-AIMS paper 2016-5 · Published in Contemp. Math. 703 (2018), 83–115 · 2018

We build a database of genus 2 curves defined over ℚ which contains all curves with minimal absolute height h ≤ 5, all curves with moduli height h ≤ 20, and all curves with extra automorphisms in standard form y² = f(x²) defined over ℚ with height h ≤ 101. For each isomorphism class in the database, an equation over its minimal field of definition is provided, the automorphism group of the curve, Clebsch and Igusa invariants. The distribution of rational points in the moduli space ℳ₂ for which the field of moduli is a field of definition is discussed and some open problems are presented.

MSC classes: Primary 14H50; Secondary 11G05, 14H10, 14Q05

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