An Aggregate CM-Pair Estimate and a 5/3 Bound for Valuation Products of abc Triples
Ruiyi Zhang · Preprint, 2026-10-08 · 18 pages
PDFHTMLDOI 10.5281/zenodo.23247744Source and checks
Abstract
Let a + b = c be pairwise coprime positive integers. Then, we introduce R = rad(abc) and P(abc) = ∏_{p | abc} v_p(abc). We present here the proof of
P(abc), d(abc) ≪_ε R^(5/3+ε),
where d is taken to be the divisor function. The first step in the deduction gives an aggregate fixed-radius estimate for CM points on an indefinite Shimura curve. In the case where the CM discriminants lie in [Δ, 2Δ], the number of nearby ordered pairs is
#{(P, Q) ∈ 𝒫_Δ × 𝒫_Δ : d(P, Q) ≤ r_0} ≪ |𝒫_Δ| + Δ(1 + Δ^2/N) N^ε,
which holds uniformly in all polynomial ranges Δ ≤ N^A. This is established by an elementary local orbit count in the Bruhat–Tits tree, using the characteristic-matrix framework of Bruinier, Li and Möller. The spatial Cauchy–Schwarz then yields a bound for the first absolute moment of the CM-packet. The ordinary AM–GM now applies directly to the logarithmic average in Pasten's Arakelov estimate to complete the argument, and neither a fourth moment nor a pointwise subconvexity estimate is required.
Cite
Ruiyi Zhang, An Aggregate CM-Pair Estimate and a 5/3 Bound for Valuation Products of abc Triples, preprint, Zenodo, 2026. https://doi.org/10.5281/zenodo.23247744