Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.02306

Beyond the Giampietro--Darmon Conjecture

Michael A. Daas
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michael A. Daas. 2026-07-02. Beyond the Giampietro--Darmon Conjecture. https://arxiv.org/abs/2607.02306

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case

Given a cusp form $f$ which is supersingular at a fixed prime $p$ away from the level, and a Coleman family $F$ through one of its $p$-stabilisations, we construct a $2$-variable meromorphic $p$-adic $L$-function for the symmetric square of $F$. We prove that this new $p$-adic $L$-function interpolates values of complex imprimitive symmetric square $L$-functions, for the various specialisations of the family $F$. We use this $p$-adic $L$-function to prove a $p$-adic factorisation formula, expressing the geometric $p$-adic $L$-function attached to the Rankin--Selberg convolution of $f$ with itself as a the product of the $p$-adic symmetric square $L$-function of $f$ and a Kubota-Leopoldt $L$-function. This extends a result of Dasgupta in the ordinary case.

math.NT↗
arXiv2026

Exceptional poles of archimedean Rankin-Selberg L-functions for irreducible generic representations of GL(n,R)

For irreducible generic representations $π_1$ and $π_2$ of $\operatorname{GL}_n(\mathbb R)$, we prove that the notions of exceptional pole of type $1$ and type $2$ coincide at every level. When both representations are in general position, we use this identification to express the Rankin--Selberg $L$-function $L(s,π_1\timesπ_2)$ in terms of the exceptional $L$-factors attached to the irreducible constituents of their derivatives.

math.NT↗