Beyond the Giampietro--Darmon Conjecture

Giampietro and Darmon conjectured a formula for the norm of various algebraic numbers, obtained as infinite products of $p$-adic cross-ratios of CM points. These quantities arose from the $p$-adic uniformisation of Shimura curves and displayed strong parallels with the Gross--Zagier factorisation for the norms of the differences between two singular moduli. The conjectured formula was conditional on the genus of the Shimura curve being zero, and in earlier work, this formula was proved in most cases. In this work, we extend the validity of the factorisation formula beyond what was conjectured by Giampietro and Darmon to many more cases, by relating this to the genus of an Atkin--Lehner quotient of the Shimura curve being zero instead. To this end, we solve a $p$-inverted version of a counting problem that was previously considered in work of Howard and Yang.