Moduli for rational genus 2 curves with real multiplication for discriminant 5
Alex Cowan1; Kimball Martin2
1 Department of Pure Mathematics University of Waterloo 200 University Ave W Waterloo, ON N2L 3G1, Canada
2 Department of Mathematics University of Oklahoma Norman, OK 73019, USA
Journal de théorie des nombres de Bordeaux, Volume 36 (2024) no. 2, pp. 493-525.

Principally polarized abelian surfaces with prescribed real multiplication (RM) are parametrized by certain Hilbert modular surfaces. Thus rational genus 2 curves with RM correspond to rational points on Hilbert modular surfaces via their Jacobians, but the converse is not true. We give a simple generic description of which rational moduli points correspond to rational curves, as well as give associated Weierstrass models, in the case of RM by the ring of integers of ℚ(5). To prove this, we provide some techniques for reducing quadratic forms over polynomial rings.

Les surfaces abéliennes principalement polarisées à multiplications réelles (RM) par un anneau donné sont parametrisées par les points d’une surface modulaire de Hilbert. Si une courbe de genre 2 à RM est définie sur ℚ, alors elle correspond, via sa jacobienne, à un point rationnel sur la surface modulaire de Hilbert appropriée. Cependant, l’implication réciproque est fausse en générale. Dans le cas de RM par l’anneau des entiers de ℚ(5), nous donnons une description générique simple des points rationnels de la variété des modules correspondant à des courbes rationnelles, ainsi que des équations de Weierstrass associées. Pour ce faire, nous fournissons quelques techniques pour réduire des formes quadratiques définies sur des anneaux de polynômes.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1286
Classification: 11F41, 11G10, 11G30, 14H10, 14K10, 11E12
Keywords: Courbes de genre 2, multiplications réelles, surfaces modulaires de Hilbert, coniques de Mestre

Alex Cowan 1; Kimball Martin 2

1 Department of Pure Mathematics University of Waterloo 200 University Ave W Waterloo, ON N2L 3G1, Canada
2 Department of Mathematics University of Oklahoma Norman, OK 73019, USA
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Alex Cowan; Kimball Martin. Moduli for rational genus 2 curves with real multiplication for discriminant 5. Journal de théorie des nombres de Bordeaux, Volume 36 (2024) no. 2, pp. 493-525. doi : 10.5802/jtnb.1286. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1286/

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