Pick a random matrix $\gamma $ in $\Gamma =\mathrm{SL}(2,\mathbb{Z})$. Denote by $\mathcal{O}_\mathbb{K}$ the Dedekind ring generated by its eigenvalues, and let $\Delta _\mathbb{K}$, $\Delta _\gamma $ and $\Delta =\mathrm{Tr}(\gamma )^2-4$ be the respective discriminant of the rings $\mathcal{O}_\mathbb{K}$, the multiplier ring $M(2,\mathbb{Z})\cap \mathbb{Q}[\gamma ]$ and $\mathbb{Z}[\gamma ]$. We show that their ratios admit probability limit distributions when ordered by Frobenius norm. In particular, 42% of the elements of $\Gamma $ have a fundamental discriminant, and $\mathbb{Z}[\gamma ]$ is a ring of integers with probability 32%.
Considérons une matrice aléatoire $\gamma $ de $\Gamma =\mathrm{SL}(2,\mathbb{Z})$. Notons par $\mathcal{O}_\mathbb{K}$ l’anneau de Dedekind engendré par ses valeurs propres, et soient $\Delta _\mathbb{K}$, $\Delta _\gamma $ et $\Delta =\mathrm{Tr}(\gamma )^2-4$ les discriminants respectifs des anneaux $\mathcal{O}_\mathbb{K}$, de l’anneau des multiplicateurs $M(2,\mathbb{Z})\cap \mathbb{Q}[\gamma ]$ et $\mathbb{Z}[\gamma ]$. Nous démontrons que les rapports entre ces discriminants convergent en loi vers une loi de probabilité lorsque les matrices de $\Gamma $ sont ordonnés selon leur norme de Frobenius. En particulier, 42% des éléments de $\Gamma $ ont un discriminant qui est fondamental, et $\mathbb{Z}[\gamma ]$ est un anneau d’entier avec probabilité 32%.
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Keywords: discriminant, random matrix
François Maucourant 1
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@article{JTNB_2025__37_3_795_0,
author = {Fran\c{c}ois Maucourant},
title = {Size of discriminants of periodic geodesics on the modular surface},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {795--835},
year = {2025},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {37},
number = {3},
doi = {10.5802/jtnb.1343},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1343/}
}
TY - JOUR AU - François Maucourant TI - Size of discriminants of periodic geodesics on the modular surface JO - Journal de théorie des nombres de Bordeaux PY - 2025 SP - 795 EP - 835 VL - 37 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1343/ DO - 10.5802/jtnb.1343 LA - en ID - JTNB_2025__37_3_795_0 ER -
%0 Journal Article %A François Maucourant %T Size of discriminants of periodic geodesics on the modular surface %J Journal de théorie des nombres de Bordeaux %D 2025 %P 795-835 %V 37 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1343/ %R 10.5802/jtnb.1343 %G en %F JTNB_2025__37_3_795_0
François Maucourant. Size of discriminants of periodic geodesics on the modular surface. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 3, pp. 795-835. doi: 10.5802/jtnb.1343
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