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, Tome 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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1286
Classification : 11F41, 11G10, 11G30, 14H10, 14K10, 11E12
Mots-clés : 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
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{JTNB_2024__36_2_493_0,
     author = {Alex Cowan and Kimball Martin},
     title = {Moduli for rational genus 2 curves with real multiplication for discriminant 5},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {493--525},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {36},
     number = {2},
     year = {2024},
     doi = {10.5802/jtnb.1286},
     mrnumber = {4830940},
     zbl = {07948975},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1286/}
}
TY  - JOUR
AU  - Alex Cowan
AU  - Kimball Martin
TI  - Moduli for rational genus 2 curves with real multiplication for discriminant 5
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2024
SP  - 493
EP  - 525
VL  - 36
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1286/
DO  - 10.5802/jtnb.1286
LA  - en
ID  - JTNB_2024__36_2_493_0
ER  - 
%0 Journal Article
%A Alex Cowan
%A Kimball Martin
%T Moduli for rational genus 2 curves with real multiplication for discriminant 5
%J Journal de théorie des nombres de Bordeaux
%D 2024
%P 493-525
%V 36
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1286/
%R 10.5802/jtnb.1286
%G en
%F JTNB_2024__36_2_493_0
Alex Cowan; Kimball Martin. Moduli for rational genus 2 curves with real multiplication for discriminant 5. Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 2, pp. 493-525. doi : 10.5802/jtnb.1286. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1286/

[1] Wieb Bosma; John Cannon; Catherine Playoust The Magma algebra system. I. The user language, J. Symb. Comput., Volume 24 (1997) no. 3-4, pp. 235-265 Computational algebra and number theory (London, 1993) | DOI | MR | Zbl

[2] Armand Brumer The rank of J0(N), Columbia University Number Theory Seminar (New York, 1992) (Astérisque), Volume 228, Société Mathématique de France, 1995, pp. 41-68 | Zbl

[3] Gabriel Cardona; Jordi Quer Field of moduli and field of definition for curves of genus 2, Computational aspects of algebraic curves (Lecture Notes Series on Computing), Volume 13, World Scientific, 2005, pp. 71-83 | DOI | MR | Zbl

[4] Edgar Costa; Nicolas Mascot; Jeroen Sijsling; John Voight Rigorous computation of the endomorphism ring of a Jacobian, Math. Comput., Volume 88 (2019) no. 317, pp. 1303-1339 | DOI | MR | Zbl

[5] Noam Elkies Shimura curve computations via K3 surfaces of Néron-Severi rank at least 19, Algorithmic number theory (Lecture Notes in Computer Science), Volume 5011, Springer, 2008, pp. 196-211 | DOI | MR | Zbl

[6] Noam Elkies; Abhinav Kumar K3 surfaces and equations for Hilbert modular surfaces, Algebra Number Theory, Volume 8 (2014) no. 10, pp. 2297-2411 | DOI | MR | Zbl

[7] Gerard van der Geer Hilbert modular surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 16, Springer, 1988, x+291 pages | MR

[8] Ki-ichiro Hashimoto On Brumer’s family of RM-curves of genus two, Tôhoku Math. J., Volume 52 (2000) no. 4, pp. 475-488 | MR | Zbl

[9] Ki-ichiro Hashimoto; Yukiko Sakai General form of Humbert’s modular equation for curves with real multiplication of Δ=5, Proc. Japan Acad., Ser. A, Volume 85 (2009) no. 10, pp. 171-176 | MR | Zbl

[10] Chandrashekhar Khare; Jean-Pierre Wintenberger Serre’s modularity conjecture. I, Invent. Math., Volume 178 (2009) no. 3, pp. 485-504 | DOI | MR | Zbl

[11] Abhinav Kumar; Ronen E. Mukamel Real multiplication through explicit correspondences, LMS J. Comput. Math., Volume 19A (2016), pp. 29-42 | DOI | MR | Zbl

[12] Jean-François Mestre Construction de courbes de genre 2 à partir de leurs modules, Effective methods in algebraic geometry (Castiglioncello, 1990) (Progress in Mathematics), Volume 94, Birkhäuser, 1991, pp. 313-334 | DOI | MR | Zbl

[13] Jean-François Mestre Familles de courbes hyperelliptiques à multiplications réelles, Arithmetic algebraic geometry (Texel, 1989) (Progress in Mathematics), Volume 89, Birkhäuser, 1991, pp. 193-208 | DOI | MR

[14] Kenneth A. Ribet Abelian varieties over Q and modular forms, Modular curves and abelian varieties (Progress in Mathematics), Volume 224, Birkhäuser, 2004, pp. 241-261 | DOI | MR | Zbl

[15] The Sage Developers Sagemath, the Sage Mathematics Software System (Version 9.4) (2021) (https://www.sagemath.org)

[16] Yukiko Sakai Poncelet’s theorem and curves of genus two with real multiplication of Δ=5, J. Ramanujan Math. Soc., Volume 24 (2009) no. 2, pp. 143-170 | MR | Zbl

[17] John Wilson Explicit moduli for curves of genus 2 with real multiplication by Q(5), Acta Arith., Volume 93 (2000) no. 2, pp. 121-138 | DOI | MR | Zbl

Cité par Sources :