TY - JOUR

T1 - Multiplicative relations among differences of singular moduli

AU - Aslanyan, Vahagn

AU - Eterović, Sebastian

AU - Fowler, Guy

N1 - 36 pages

PY - 2024/12/20

Y1 - 2024/12/20

N2 - Let \(n \in \mathbb{Z}_{>0}\). We prove that there exist a finite set \(V\) and finitely many algebraic curves \(T_1, \ldots, T_k\) with the following property: if \((x_1, \ldots, x_n, y)\) is an \((n+1)\)-tuple of pairwise distinct singular moduli such that \(\prod_{i=1}^n (x_i - y)^{a_i}=1\) for some \(a_1, \ldots, a_n \in \mathbb{Z} \setminus \{0\}\), then \((x_1, \ldots, x_n, y) \in V \cup T_1 \cup \ldots \cup T_k\). Further, the curves \(T_1, \ldots, T_k\) may be determined explicitly for a given \(n\).

AB - Let \(n \in \mathbb{Z}_{>0}\). We prove that there exist a finite set \(V\) and finitely many algebraic curves \(T_1, \ldots, T_k\) with the following property: if \((x_1, \ldots, x_n, y)\) is an \((n+1)\)-tuple of pairwise distinct singular moduli such that \(\prod_{i=1}^n (x_i - y)^{a_i}=1\) for some \(a_1, \ldots, a_n \in \mathbb{Z} \setminus \{0\}\), then \((x_1, \ldots, x_n, y) \in V \cup T_1 \cup \ldots \cup T_k\). Further, the curves \(T_1, \ldots, T_k\) may be determined explicitly for a given \(n\).

KW - math.NT

U2 - 10.2422/2036-2145.202309_020

DO - 10.2422/2036-2145.202309_020

M3 - Article

JO - Annali della Scuola normale superiore di Pisa - Classe di scienze

JF - Annali della Scuola normale superiore di Pisa - Classe di scienze

SN - 0391-173X

ER -