We define spherical Heron triangles (spherical triangles with “rational” side-lengths and angles) and parametrize them via rational points of certain families of elliptic curves. We show that the congruent number problem has infinitely many solutions for most areas in the spherical setting and we find a spherical Heron triangle with rational medians. We also explore the question of spherical triangles with a single rational median or a single a rational area bisector (median splitting the triangle in half), and discuss various problems involving isosceles spherical triangles.
Nous définissons les triangles de Héron sphériques (triangles sphériques avec des mesures de côtés et des angles « rationnels ») et les paramétrons par des points rationnels en certaines familles de courbes elliptiques. Nous montrons que le problème des nombres congruents a une infinité de solutions pour la plupart des valeurs de l’aire dans le cas sphérique et nous trouvons un triangle de Héron sphérique avec des médianes rationnelles. Nous explorons également la question des triangles sphériques avec une seule médiane rationnelle ou une seule bissectrice d’aire rationnelle (c’est-à-dire, une médiane divisant le triangle en deux), et nous discutons de divers problèmes impliquant des triangles sphériques isocèles.
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Keywords: spherical triangles, elliptic curves, elliptic surfaces
Huang, Tinghao 1 ; Lalín, Matilde 2 ; Mila, Olivier 2
CC-BY-ND 4.0
@article{JTNB_2023__35_1_219_0,
author = {Huang, Tinghao and Lal{\'\i}n, Matilde and Mila, Olivier},
title = {Spherical {Heron} triangles and elliptic curves},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {219--246},
year = {2023},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {35},
number = {1},
doi = {10.5802/jtnb.1243},
language = {en},
url = {https://www.numdam.org/articles/10.5802/jtnb.1243/}
}
TY - JOUR AU - Huang, Tinghao AU - Lalín, Matilde AU - Mila, Olivier TI - Spherical Heron triangles and elliptic curves JO - Journal de théorie des nombres de Bordeaux PY - 2023 SP - 219 EP - 246 VL - 35 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://www.numdam.org/articles/10.5802/jtnb.1243/ DO - 10.5802/jtnb.1243 LA - en ID - JTNB_2023__35_1_219_0 ER -
%0 Journal Article %A Huang, Tinghao %A Lalín, Matilde %A Mila, Olivier %T Spherical Heron triangles and elliptic curves %J Journal de théorie des nombres de Bordeaux %D 2023 %P 219-246 %V 35 %N 1 %I Société Arithmétique de Bordeaux %U https://www.numdam.org/articles/10.5802/jtnb.1243/ %R 10.5802/jtnb.1243 %G en %F JTNB_2023__35_1_219_0
Huang, Tinghao; Lalín, Matilde; Mila, Olivier. Spherical Heron triangles and elliptic curves. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 1, pp. 219-246. doi: 10.5802/jtnb.1243
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