TY - UNPB

T1 - Multiplicative relations among differences of singular moduli

AU - Aslanyan, Vahagn

AU - Eterović, Sebastian

AU - Fowler, Guy

N1 - 36 pages

PY - 2023/8/23

Y1 - 2023/8/23

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 - 10.48550/arXiv.2308.12244

DO - 10.48550/arXiv.2308.12244

M3 - Preprint

BT - Multiplicative relations among differences of singular moduli

ER -