arXiv subjects

Michael A. Daas

Publications and source records attributed to Michael A. Daas.

3 recordsLinked to original sources

arXiv2026

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.

math.NT↗

arXiv2025

Congruence conditions for the mod $λ$ values of the Fourier coefficients of classical eigenforms

We classify all instances of the condition $a_{p}(f) \equiv x \bmod λ$ being related to a congruence on the prime $p$, where $a_{p}(f)$ denotes the $p$th Fourier coefficient of a classical normalised cuspidal eigenform $f$ and $λ$ is a prime in the number field generated by the Fourier coefficients of $f$. This classification is done in terms of the (projective) image of the mod $λ$ Galois representation associated with $f$ and extends work by Swinnerton-Dyer. We highlight that for $x = 0$, this condition is more often implied by a congruence on the prime $p$ than the general value of $a_{p}(f) \bmod λ$. Finally, we illustrate various instances of these congruences through examples from the setting of weight 2 newforms attached to rational elliptic curves.

math.NT↗

arXiv2023

CM-values of $p$-adic $Θ$-functions

We prove a $p$-adic version of the work by Gross and Zagier on the differences between singular moduli by proving a set of conjectures by Giampietro and Darmon, who investigated the factorisation of a rational invariant associated to a pair of CM-points on a genus zero Shimura curve, obtained as the ratio of the CM-values of $p$-adic $Θ$-functions. As did Gross and Zagier, we give two proofs; an algebraic proof using CM-theory, and more interestingly, also an analytic proof using $p$-adic infinitesimal deformations of Hilbert Eisenstein series. Since there are no explicit formulae for its cuspidal $p$-adic deformations, we instead compute the Frobenius traces of the appropriate Galois deformation, and show their modularity via an $R = T$ theorem. This approach aims to bridge the gap between classical CM-theory and the more recent $p$-adic advances in the theory of real multiplication.

math.NT↗