Using Mazur’s theorem on torsions of elliptic curves, an upper bound 24 for the order of the finite Galois group associated with weighted walks in the quarter plane is obtained. The explicit criterion for to have order 4 or 6 is rederived by simple geometric arguments. Using division polynomials, a recursive criterion for to have order or is also obtained. As a corollary, an explicit criterion for to have order 8 is given through a method simpler than the existing one.
En utilisant le théorème de Mazur sur les torsions de courbes elliptiques, on obtient un majorant 24 pour l’ordre du groupe fini de Galois associé aux marches pondérées dans le quart de plan . Le critère explicite pour que soit d’ordre 4 ou 6 est obtenu par un simple argument géométrique. En utilisant des polynômes de division, un critère récursif pour d’ordre ou est également obtenu. Comme corollaire, un critère explicite pour que soit d’ordre 8 est donné et est beaucoup plus simple que la méthode existante.
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Ruichao Jiang 1; Javad Tavakoli 1; Yiqiang Zhao 2
@article{CRMATH_2021__359_5_563_0, author = {Ruichao Jiang and Javad Tavakoli and Yiqiang Zhao}, title = {An upper bound and finiteness criteria for the {Galois} group of weighted walks with rational coefficients in the quarter plane}, journal = {Comptes Rendus. Math\'ematique}, pages = {563--576}, publisher = {Acad\'emie des sciences, Paris}, volume = {359}, number = {5}, year = {2021}, doi = {10.5802/crmath.196}, language = {en}, }
TY - JOUR AU - Ruichao Jiang AU - Javad Tavakoli AU - Yiqiang Zhao TI - An upper bound and finiteness criteria for the Galois group of weighted walks with rational coefficients in the quarter plane JO - Comptes Rendus. Mathématique PY - 2021 SP - 563 EP - 576 VL - 359 IS - 5 PB - Académie des sciences, Paris DO - 10.5802/crmath.196 LA - en ID - CRMATH_2021__359_5_563_0 ER -
%0 Journal Article %A Ruichao Jiang %A Javad Tavakoli %A Yiqiang Zhao %T An upper bound and finiteness criteria for the Galois group of weighted walks with rational coefficients in the quarter plane %J Comptes Rendus. Mathématique %D 2021 %P 563-576 %V 359 %N 5 %I Académie des sciences, Paris %R 10.5802/crmath.196 %G en %F CRMATH_2021__359_5_563_0
Ruichao Jiang; Javad Tavakoli; Yiqiang Zhao. An upper bound and finiteness criteria for the Galois group of weighted walks with rational coefficients in the quarter plane. Comptes Rendus. Mathématique, Volume 359 (2021) no. 5, pp. 563-576. doi : 10.5802/crmath.196. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.196/
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